ploeh blog 2019-11-11T09:16:21+00:00 Mark Seemann danish software design https://blog.ploeh.dk Diamond rock https://blog.ploeh.dk/2019/11/11/diamond-rock 2019-11-11T09:15:00+00:00 Mark Seemann <div id="post"> <p> <em>A diamond kata implementation written in Rockstar</em> </p> <p> I've <a href="/2015/01/10/diamond-kata-with-fscheck">previously written about the diamond kata</a>, which has become one of my favourite programming exercises. I wanted to take the <a href="https://codewithrockstar.com">Rockstar</a> language out for a spin, and it seemed a good candidate problem. </p> <h3 id="ab491bf89ae945f48821fe7b324febeb"> Rockstar <a href="#ab491bf89ae945f48821fe7b324febeb" title="permalink">#</a> </h3> <p> If you're not aware of the Rockstar programming language, it started with a tweet from <a href="http://paulstovell.com">Paul Stovell</a>: <blockquote> <p> "To really confuse recruiters, someone should make a programming language called Rockstar." </p> <footer><cite><a href="https://twitter.com/paulstovell/status/1013960369465782273">Paul Stovell</a></cite></footer> </blockquote> This inspired <a href="http://www.dylanbeattie.net">Dylan Beattie</a> to create the Rockstar programming language. The language's <a href="https://codewithrockstar.com">landing page</a> already sports an <a href="/2015/08/03/idiomatic-or-idiosyncratic">idiomatic</a> implementation of the <a href="http://codingdojo.org/kata/FizzBuzz">FizzBuzz kata</a>, so I had to pick another exercise. The <a href="http://claysnow.co.uk/recycling-tests-in-tdd">diamond kata</a> was a natural choice for me. </p> <h3 id="c17e0b1d17f04d19b9e965ad531f10d9"> Lyrics <a href="#c17e0b1d17f04d19b9e965ad531f10d9" title="permalink">#</a> </h3> <p> I'll start with the final result. What follows here are the final lyrics to <em>Diamond rock</em>. As you'll see, it starts with a short prologue after which it settles into a repetitive pattern. I imagine that this might make it useful for audience participation, in the anthem rock style of e.g. <a href="https://en.wikipedia.org/wiki/We_Will_Rock_You">We Will Rock You</a>. </p> <p> After 25 repetitions the lyrics change. I haven't written music to any of them, but I imagine that this essentially transitions into another song, just like <em>We Will Rock You</em> traditionally moves into <a href="https://en.wikipedia.org/wiki/We_Are_the_Champions">We Are the Champions</a>. The style of the remaining lyrics does, however, suggest something musically more complex than a rock anthem. </p> <p> <pre>Your memory says&nbsp;&nbsp; The way says A Despair takes your hope The key is nothing Your courage is a tightrope Let it be without it If your hope is your courage Let the key be the way Build your courage up If your hope is your courage The key says B Build your courage up If your hope is your courage The key says C Build your courage up If your hope is your courage The key says D Build your courage up If your hope is your courage The key says E Build your courage up If your hope is your courage The key says F Build your courage up If your hope is your courage The key says G Build your courage up If your hope is your courage The key says H Build your courage up If your hope is your courage The key says I Build your courage up If your hope is your courage The key says J Build your courage up If your hope is your courage The key says K Build your courage up If your hope is your courage The key says L Build your courage up If your hope is your courage The key says M Build your courage up If your hope is your courage The key says N Build your courage up If your hope is your courage The key says O Build your courage up If your hope is your courage The key says P Build your courage up If your hope is your courage The key says Q Build your courage up If your hope is your courage The key says R Build your courage up If your hope is your courage The key says S Build your courage up If your hope is your courage The key says T Build your courage up If your hope is your courage The key says U Build your courage up If your hope is your courage The key says V Build your courage up If your hope is your courage The key says W Build your courage up If your hope is your courage The key says X Build your courage up If your hope is your courage The key says Y Build your courage up If your hope is your courage The key says Z Give back the key Dream takes a tightrope Your hope's start'n'stop While Despair taking your hope ain't a tightrope Build your hope up Give back your hope Raindrop is a wintercrop Light is a misanthrope Let it be without raindrop Darkness is a kaleidoscope Let it be without raindrop Diamond takes your hour, love, and your day If love is the way Let your sorrow be your hour without light Put your sorrow over darkness into flight Put your memory of flight into motion Put it with love, and motion into the ocean Whisper the ocean If love ain't the way Let ray be your day of darkness without light Let your courage be your hour without darkness, and ray Put your courage over darkness into the night Put your memory of ray into action Let satisfaction be your memory of the night Let alright be satisfaction with love, action, love, and satisfaction Shout alright Listen to the wind Let flight be the wind Let your heart be Dream taking flight Let your breath be your heart of darkness with light Your hope's stop'n'start While your hope is lower than your heart Let away be Despair taking your hope Diamond taking your breath, away, and your hope Build your hope up Renown is southbound While your hope is as high as renown Let away be Despair taking your hope Diamond taking your breath, away, and your hope Knock your hope down</pre> </p> <p> Not only do these look like lyrics, but it's also an executable program! </p> <h3 id="994b9a9d46dc467383c528a17dc99f65"> Execution <a href="#994b9a9d46dc467383c528a17dc99f65" title="permalink">#</a> </h3> <p> If you want to run the program, you can copy the text and paste it into <a href="https://codewithrockstar.com/online">the web-based Rockstar interpreter</a>; that's what I did. </p> <p> When you <em>Rock!</em> the lyrics, the interpreter will prompt you for an input. There's no instructions or input validation, but <em>the only valid input is the letters</em> <code>A</code> to <code>Z</code>, and <em>only in upper case</em>. If you type anything else, I don't know what'll happen, but most likely it'll just enter an infinite loop, and you'll have to reboot your computer. </p> <p> If you input, say, <code>E</code>, the output will be the expected diamond figure: </p> <p> <pre> A B B C C D D E E D D C C B B A </pre> </p> <p> When you paste the code, be sure to include everything. There's significant whitespace in those lyrics; I'll explain later. </p> <h3 id="12f2c388f5c8499594a3bf90f37e07f4"> Readable code <a href="#12f2c388f5c8499594a3bf90f37e07f4" title="permalink">#</a> </h3> <p> As the Rockstar documentation strongly implies, <em>singability</em> is more important than readability. You can, however, write more readable Rockstar code, and that's what I started with: </p> <p> <pre>Space says&nbsp;&nbsp; LetterA says A GetLetter takes index If index is 0 Give back LetterA If index is 1 retVal says B Give back retVal If index is 2 retVal says C Give back retVal GetIndex takes letter Index is 0 While GetLetter taking Index ain't letter Build Index up Give back Index PrintLine takes width, l, lidx If l is LetterA Let completeSpaceCount be width minus 1 Let paddingCount be completeSpaceCount over 2 Let padding be Space times paddingCount Let line be padding plus l plus padding Say line Else Let internalSpaceSize be lidx times 2 minus 1 Let filler be Space times internalSpaceSize Let totalOuterPaddingSize be width minus 2, internalSpaceSize Let paddingSize be totalOuterPaddingSize over 2 Let padding be Space times paddingSize Let line be padding plus l, filler, l, padding Say line Listen to input Let idx be GetIndex taking input Let width be idx times 2 plus 1 Let counter be 0 While counter is lower than idx Let l be GetLetter taking counter PrintLine taking width, l, counter Build counter up While counter is as high as 0 Let l be GetLetter taking counter PrintLine taking width, l, counter Knock counter down</pre> </p> <p> This prototype only handled the input letters <code>A</code>, <code>B</code>, and <code>C</code>, but it was enough to verify that the algorithm worked. I've done the diamond kata several times before, so I only had to find the most imperative implementation on my hard drive. It wasn't too hard to translate to Rockstar. </p> <p> Although Rockstar supports mainstream quoted strings like <code>"A"</code>, <code>"B"</code>, and so on, you can see that I went straight for <em>poetic string literals</em>. Before I started persisting Rockstar code to a file, I experimented with the language using the online interpreter. I wanted the program to look as much like rock lyrics as I could, so I didn't want to have too many statements like <code>Shout "FizzBuzz!"</code> in my code. </p> <h3 id="1ae1c1c329894b438bc934097090901b"> Obscuring space <a href="#1ae1c1c329894b438bc934097090901b" title="permalink">#</a> </h3> <p> My first concern was whether I could obscure the space character. Using a poetic string literal, I could: </p> <p> <pre>Space says&nbsp;&nbsp;</pre> </p> <p> The rules of poetic string literals is that everything between <code>says&nbsp;</code> and the newline character becomes the value of the string variable. So there's an extra space after <code>says&nbsp;</code>! </p> <p> After I renamed all the variables and functions, that line became: </p> <p> <pre>Your memory says&nbsp;&nbsp;</pre> </p> <p> Perhaps it isn't an unprintable character, but it <em>is</em> unsingable. </p> <h3 id="d1ae4568cc104312a3c20fe8019ccb18"> No else <a href="#d1ae4568cc104312a3c20fe8019ccb18" title="permalink">#</a> </h3> <p> The keyword <code>Else</code> looks conspicuously like a programming construct, so I wanted to get rid of that as well. That was easy, because I could just invert the initial <code>if</code> condition: </p> <p> <pre>If l ain't LetterA</pre> </p> <p> This effectively switches between the two alternative code blocks. </p> <h3 id="cbaa9c218905453bb3df2155b9e9b822"> Obscuring letter indices <a href="#cbaa9c218905453bb3df2155b9e9b822" title="permalink">#</a> </h3> <p> I also wanted to obscure the incrementing index values <code>1</code>, <code>2</code>, <code>3</code>, etcetera. Since the indices are monotonically increasing, I realised that I could use a counter and increment it: </p> <p> <pre>number is 0 If index is number Let retVal be LetterA Build number up If index is number retVal says B Build number up If index is number retVal says C</pre> </p> <p> The function initialises <code>number</code> to <code>0</code> and assigns a value to <code>retVal</code> if the input <code>index</code> is also <code>0</code>. </p> <p> If not, it increments the <code>number</code> (so that it's now <code>1</code>) and again compares it to <code>index</code>. This sufficiently obscures the indices, but if there's a way to hide the letters of the alphabet, I'm not aware of it. </p> <p> After I renamed the variables, the code became: </p> <p> <pre>Your courage is a tightrope Let it be without it If your hope is your courage Let the key be the way Build your courage up If your hope is your courage The key says B Build your courage up If your hope is your courage The key says C</pre> </p> <p> There's one more line of code in the final lyrics, compared to the above snippet. The line <code>Let it be without it</code> has no corresponding line of code in the readable version. What's going on? </p> <h3 id="aef4cf50f4484327b37cd61995c74d99"> Obscuring numbers <a href="#aef4cf50f4484327b37cd61995c74d99" title="permalink">#</a> </h3> <p> Like poetic string literals, Rockstar also supports <em>poetic number literals</em>. Due to its modulo-ten-based system, however, I found it difficult to come up with a good ten-letter word that fit the song's lyrical theme. I <em>could</em> have done something like this to produce the number <code>0</code>: </p> <p> <pre>Your courage is barbershop</pre> </p> <p> or some other ten-letter word. My problem was that regardless of what I chose, it didn't sound good. Some article like <code>a</code> or <code>the</code> would sound better, but that would change the value of the poetic number literal. <code>a tightrope</code> is the number <em>19</em>, because <code>a</code> has one letter, and <code>tightrope</code> has nine. </p> <p> There's a simple way to produce <em>0</em> from any number: just subtract the number from itself. That's what <code>Let it be without it</code> does. I could also have written it as <code>Let your courage be without your courage</code>, but I chose to take advantage of Rockstar's <em>pronoun</em> feature instead. I'd been looking for an opportunity to include the phrase <a href="https://en.wikipedia.org/wiki/Let_It_Be_(Beatles_song)">Let It Be</a> ever since I learned about the <code>Let x be y</code> syntax. </p> <p> The following code snippet initialises the variable <code>Your courage</code> to <code>19</code>, but on the next line subtracts 19 from 19 and updates the variable so that its value is now <code>0</code>. </p> <p> <pre>Your courage is a tightrope Let it be without it</pre> </p> <p> I had the same problem with initialising the numbers <em>1</em> and <em>2</em>, so further down I resorted to similar tricks: </p> <p> <pre>Raindrop is a wintercrop Light is a misanthrope Let it be without raindrop Darkness is a kaleidoscope Let it be without raindrop </pre> </p> <p> Here I had the additional constraint that I wanted the words to rhyme. The rhymes are a continuation of the previous lines' <code>up</code> and <code>hope</code>, so I struggled to come up with a ten-letter word that rhymes with <code>up</code>; <code>wintercrop</code> was the best I could do. <code>a wintercrop</code> is <em>10</em>, and the strategy is to define <code>Light</code> and <code>Darkness</code> as <em>11</em> and <em>12</em>, and then subtract <em>10</em> from both. At the first occurrence of <code>Let it be without raindrop</code>, <code>it</code> refers to <code>Light</code>, whereas the second time <code>it</code> refers to <code>Darkness</code>. </p> <h3 id="46b60c0d25e54f9599e5bd40b7e88d80"> Lyrical theme <a href="#46b60c0d25e54f9599e5bd40b7e88d80" title="permalink">#</a> </h3> <p> Once I had figured out how to obscure strings and numbers, it was time to rename all the readable variables and function names into idiomatic Rockstar. </p> <p> At first, I thought that I'd pattern my lyrics after <a href="https://en.wikipedia.org/wiki/Shine_On_You_Crazy_Diamond">Shine On You Crazy Diamond</a>, but I soon ran into problems with the keyword <code>taking</code>. I found it difficult to find words that would naturally succeed <code>taking</code>. Some options I thought of were: <ul> <li>taking the bus</li> <li>taking a chance</li> <li>taking hold</li> <li>taking flight</li> <li>taking time</li> </ul> Some of these didn't work for various reasons. In Rockstar <code>times</code> is a keyword, and apparently <code>time</code> is reserved as well. At least, the online interpreter choked on it. </p> <p> <code>Taking a chance</code> sounded <a href="https://en.wikipedia.org/wiki/Take_a_Chance_on_Me">too much like ABBA</a>. <code>Taking hold</code> created the derived problem that I had to initialise and use a variable called <code>hold</code>, and I couldn't make that work. </p> <p> <code>Taking flight</code>, on the other hand, turned out to provide a fertile opening. </p> <p> I soon realised, though, that my choice of words pulled the lyrical theme away from idiomatic Rockstar vocabulary. While I do get the <a href="https://en.wikipedia.org/wiki/Livin%27_on_a_Prayer">Tommy and Gina</a> references, I didn't feel at home in that poetic universe. </p> <p> On the other hand, I thought that the words started to sound like <a href="https://en.wikipedia.org/wiki/Yes_(band)">Yes</a>. I've listened to a lot of Yes. The lyrics are the same kind of lame and vapid as what was taking form in my editor. I decided to go in that direction. </p> <p> Granted, this is no longer idiomatic Rockstar, since it's more <a href="https://en.wikipedia.org/wiki/Progressive_rock">prog rock</a> than <a href="https://en.wikipedia.org/wiki/Glam_metal">hair metal</a>. I invoke creative license. </p> <p> Soon I also conceived of the extra ambition that I wanted the verses to rhyme. Here, it proved fortunate that the form <code>let x be y</code> is interchangeable with the form <code>put y into x</code>. Some words, like <em>darkness</em>, are difficult to rhyme with, so it helps that you can hide them within a <code>put y into x</code> form. </p> <p> Over the hours(!) I worked on this, a theme started to emerge. I'm particularly fond of the repeated motifs like: </p> <p> <pre>Your hope's start'n'stop</pre> </p> <p> which rhymes with <code>up</code>, but then later it appears again as </p> <p> <pre>Your hope's stop'n'start</pre> </p> <p> which rhymes with <code>heart</code>. Both words, by the way, represent the number <em>0</em>, since there's ten letters when you ignore the single quotes. </p> <h3 id="234366a7cb1a4998813719845f3c08d7"> Conclusion <a href="#234366a7cb1a4998813719845f3c08d7" title="permalink">#</a> </h3> <p> I spent more time on this that I suppose I ought to, but once I got started, it was hard to stop. I found the translation from readable code into 'idiomatic' Rockstar at least as difficult as writing working software. There's a lesson there, I believe. </p> <p> Rockstar is still a budding language, so I did miss a few features, chief among which would be <em>arrays</em>, but I'm not sure how one would make arrays sufficiently rock'n'roll. </p> <p> A unit testing framework would also be nice. </p> <p> If you liked this article, please <a href="https://www.linkedin.com/in/ploeh/">endorse my <em>Rockstar</em> skills on LinkedIn</a> so that we can <em>"confuse recruiters."</em> </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. The 80/24 rule https://blog.ploeh.dk/2019/11/04/the-80-24-rule 2019-11-04T06:51:00+00:00 Mark Seemann <div id="post"> <p> <em>Write small blocks of code. How small? Here's how small.</em> </p> <p> One of the most common questions I get is this: </p> <p> <em>If you could give just one advice to programmers, what would it be?</em> </p> <p> That's easy: </p> <p> <em>Write small blocks of code.</em> </p> <p> Small methods. Small functions. Small procedures. </p> <p> How small? </p> <h3 id="849d07675e3a4c5681641eb0c67dfafb"> Few lines of code <a href="#849d07675e3a4c5681641eb0c67dfafb" title="permalink">#</a> </h3> <p> You can't give a universally good answer to that question. Among other things, it depends on the programming language in question. Some languages are much denser than others. The densest language I've ever encountered is <a href="https://en.wikipedia.org/wiki/APL_(programming_language)">APL</a>. </p> <p> Most mainstream languages, however, seem to be verbose to approximately the same order of magnitude. My experience is mostly with C#, so I'll use that (and similar languages like Java) as a starting point. </p> <p> When I write C# code, I become uncomfortable when my method size approaches fifteen or twenty lines of code. C# is, however, a fairly wordy language, so it sometimes happens that I have to allow a method to grow larger. My limit is probably somewhere around 25 lines of code. </p> <p> That's an arbitrary number, but if I have to quote a number, it would be around that size. Since it's arbitrary anyway, let's make it <em>24</em>, for reasons that I'll explain later. </p> <p> The maximum line count of a C# (or Java, or JavaScript, etc.) method, then, should be 24. </p> <p> To repeat the point from before, this depends on the language. I'd consider a 24-line <a href="https://www.haskell.org">Haskell</a> or <a href="https://fsharp.org">F#</a> function to be so huge that if I received it as a pull request, I'd reject it <a href="/2015/01/15/10-tips-for-better-pull-requests">on the grounds of size</a> alone. </p> <h3 id="8506ebcba585459b9739d84a7bcad758"> Narrow line width <a href="#8506ebcba585459b9739d84a7bcad758" title="permalink">#</a> </h3> <p> Most languages allow for flexibility in layout. For example, C-based languages use the <code>;</code> character as a delimiter. This enables you to write more than one statement per line: </p> <p> <pre><span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">foo</span>&nbsp;=&nbsp;32;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">bar</span>&nbsp;=&nbsp;<span style="font-weight:bold;color:#1f377f;">foo</span>&nbsp;+&nbsp;10;&nbsp;<span style="color:#2b91af;">Console</span>.<span style="color:#74531f;">WriteLine</span>(<span style="font-weight:bold;color:#1f377f;">bar</span>);</pre> </p> <p> You could attempt to avoid the 24-line-height rule by writing wide lines. That would, however, be to defeat the purpose. </p> <p> The purpose of writing small methods is to nudge yourself towards writing readable code; code that fits in your brain. The smaller, the better. </p> <p> For completeness sake, let's institute a maximum line width as well. If there's any accepted industry standard for maximum line width, it's 80 characters. I've used that maximum for years, and it's a good maximum. </p> <p> Like all other programmers, other people's code annoys me. The most common annoyance is that people write too wide code. </p> <p> This is probably because most programmers have drunk the Cool Aid that bigger screens make you more productive. When you code on a big screen, you don't notice how wide your lines become. </p> <p> There's many scenarios where wide code is problematic: <ul> <li>When you're comparing changes to a file side-by-side. This often happens when you review pull requests. Now you have only half of your normal screen width.</li> <li>When you're looking at code on a smaller device.</li> <li>When you're getting old, or are otherwise visually impaired. After I turned 40, I discovered that I found it increasingly difficult to see small things. I still use a 10-point font for programming, but I foresee that this will not last much longer.</li> <li>When you're <a href="https://en.wikipedia.org/wiki/Mob_programming">mob programming</a> you're limited to the size of the shared screen.</li> <li>When you're sharing your screen via the web, for remote pair programming or similar.</li> <li>When you're presenting code at meetups, user groups, conferences, etc.</li> </ul> What most programmers need, I think, is just a <a href="https://en.wikipedia.org/wiki/Nudge_theory">nudge</a>. In Visual Studio, for example, you can install the <a href="https://marketplace.visualstudio.com/items?itemName=PaulHarrington.EditorGuidelines">Editor Guidelines</a> extension, which will display one or more vertical guidelines. You can configure it as you'd like, but I've mine set to 80 characters, and bright red: </p> <p> <img src="/content/binary/vertical-guideline-at-80-characters.png" alt="Screen shot of editor with code, showing red vertical line at 80 characters."> </p> <p> Notice the red dotted vertical line that cuts through <code>universe</code>. It tells me where the 80 character limit is. </p> <h3 id="d94a52940215425e9cb19492d9f51a41"> Terminal box <a href="#d94a52940215425e9cb19492d9f51a41" title="permalink">#</a> </h3> <p> The 80-character limit has a long and venerable history, but what about the 24-line limit? While both are, ultimately, arbitrary, both fit the size of the popular <a href="https://en.wikipedia.org/wiki/VT100">VT100</a> terminal, which had a display resolution of 80x24 characters. </p> <p> A box of 80x24 characters thus reproduces the size of an old terminal. Does this mean that I suggest that you should write programs on terminals? No, people always misunderstand this. That should be the maximum size of a method. On larger screens, you'd be able to see multiple small methods at once. For example, you could view a unit test and its target in a split screen configuration. </p> <p> The exact sizes are arbitrary, but I think that there's something fundamentally right about such continuity with the past. </p> <p> I've been using the 80-character mark as a soft limit for years. I tend to stay within it, but I occasionally decide to make my code a little wider. I haven't paid quite as much attention to the number of lines of my methods, but only for the reason that I know that I tend to write methods shorter than that. Both limits have served me well for years. </p> <h3 id="e5301bcefd8f444487906af03df293b0"> Example <a href="#e5301bcefd8f444487906af03df293b0" title="permalink">#</a> </h3> <p> Consider this example: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">ActionResult</span>&nbsp;<span style="font-weight:bold;color:#74531f;">Post</span>(<span style="color:#2b91af;">ReservationDto</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">dto</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">validationMsg</span>&nbsp;=&nbsp;<span style="color:#2b91af;">Validator</span>.<span style="color:#74531f;">Validate</span>(<span style="font-weight:bold;color:#1f377f;">dto</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#8f08c4;">if</span>&nbsp;(<span style="font-weight:bold;color:#1f377f;">validationMsg</span>&nbsp;!=&nbsp;<span style="color:#a31515;">&quot;&quot;</span>) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#8f08c4;">return</span>&nbsp;<span style="font-weight:bold;color:#74531f;">BadRequest</span>(<span style="font-weight:bold;color:#1f377f;">validationMsg</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:#2b91af;">Mapper</span>.<span style="color:#74531f;">Map</span>(<span style="font-weight:bold;color:#1f377f;">dto</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">reservations</span>&nbsp;=&nbsp;Repository.<span style="font-weight:bold;color:#74531f;">ReadReservations</span>(<span style="font-weight:bold;color:#1f377f;">reservation</span>.Date); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">accepted</span>&nbsp;=&nbsp;maîtreD.<span style="font-weight:bold;color:#74531f;">CanAccept</span>(<span style="font-weight:bold;color:#1f377f;">reservations</span>,&nbsp;<span style="font-weight:bold;color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#8f08c4;">if</span>&nbsp;(!<span style="font-weight:bold;color:#1f377f;">accepted</span>) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#8f08c4;">return</span>&nbsp;<span style="font-weight:bold;color:#74531f;">StatusCode</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">StatusCodes</span>.Status500InternalServerError, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#a31515;">&quot;Couldn&#39;t&nbsp;accept.&quot;</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">id</span>&nbsp;=&nbsp;Repository.<span style="font-weight:bold;color:#74531f;">Create</span>(<span style="font-weight:bold;color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#8f08c4;">return</span>&nbsp;<span style="font-weight:bold;color:#74531f;">Ok</span>(<span style="font-weight:bold;color:#1f377f;">id</span>); }</pre> </p> <p> This method is 18 lines long, which includes the method declaration, curly brackets and blank lines. It easily stays within the 80-character limit. Note that I've deliberately formatted the code so that it behaves. You can see it in this fragment: </p> <p> <pre><span style="font-weight:bold;color:#8f08c4;">return</span>&nbsp;<span style="font-weight:bold;color:#74531f;">StatusCode</span>( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">StatusCodes</span>.Status500InternalServerError, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#a31515;">&quot;Couldn&#39;t&nbsp;accept.&quot;</span>);</pre> </p> <p> Most people write it like this: </p> <p> <pre><span style="font-weight:bold;color:#8f08c4;">return</span>&nbsp;<span style="font-weight:bold;color:#74531f;">StatusCode</span>(<span style="color:#2b91af;">StatusCodes</span>.Status500InternalServerError,&nbsp;<span style="color:#a31515;">&quot;Couldn&#39;t&nbsp;accept.&quot;</span>);</pre> </p> <p> That doesn't look bad, but I've seen much worse examples. </p> <p> Another key to writing small methods is to call other methods. The above <code>Post</code> method doesn't look like much, but significant functionality could be hiding behind <code>Validator.Validate</code>, <code>Repository.ReadReservations</code>, or <code>maîtreD.CanAccept</code>. I hope that you agree that each of these objects and methods are named well enough to give you an idea about their purpose. </p> <h3 id="ca4e5a8403244f93a95c2736cdaf7eee"> Code that fits in your brain <a href="#ca4e5a8403244f93a95c2736cdaf7eee" title="permalink">#</a> </h3> <p> As I describe in my <a href="https://cleancoders.com/episode/humane-code-real-episode-1/show">Humane Code</a> video, the human brain can only keep track of <a href="https://en.wikipedia.org/wiki/The_Magical_Number_Seven,_Plus_or_Minus_Two">about seven things</a>. I think that this rule of thumb applies to the way we read and interpret code. If you need to understand and keep track of more than seven separate things at the same time, the code becomes harder to understand. </p> <p> This could explain why small methods are good. They're only good, however, if they're self-contained. When you look at a method like the above <code>Post</code> method, you'll be most effective if you don't need to have a deep understanding of how each of the dependencies work. If this is true, the method only juggles about five dependencies: <code>Validator</code>, <code>Mapper</code>, <code>Repository</code>, <code>maîtreD</code>, and its own base class (which provides the methods <code>BadRequest</code>, <code>StatusCode</code>, and <code>Ok</code>). Five dependencies is fewer than seven. </p> <p> Another way to evaluate the cognitive load of a method is to measure its <a href="https://en.wikipedia.org/wiki/Cyclomatic_complexity">cyclomatic complexity</a>. The <code>Post</code> method's cyclomatic complexity is <em>3</em>, so that should be easily within the brain's capacity. </p> <p> These are all heuristics, so read this for inspiration, not as law. They've served me well for years, though. </p> <h3 id="1c2120e143784dde8e520026b67651d4"> Conclusion <a href="#1c2120e143784dde8e520026b67651d4" title="permalink">#</a> </h3> <p> You've probably heard about the <em>80/20 rule</em>, also known as the <a href="https://en.wikipedia.org/wiki/Pareto_principle">Pareto principle</a>. Perhaps the title lead you to believe that this article was a misunderstanding. I admit that I went for an arresting title; perhaps a more proper name is the <em>80x24 rule</em>. </p> <p> The exact numbers can vary, but I've found a maximum method size of 80x24 characters to work well for C#. </p> </div> <div id="comments"> <hr> <h2 id="comments-header"> Comments </h2> <div class="comment" id="cbdf43b5551242efa330a163f01119ca"> <div class="comment-author">Jiehong</div> <div class="comment-content"> <p> As a matter of fact, <em>terminals</em> had 80 characters lines, because <a href="https://en.wikipedia.org/wiki/Punched_card#/media/File:FortranCardPROJ039.agr.jpg">IBM punch cards</a>, representing only 1 line, had 80 symbols (even though only the first 72 were used at first). However, I don't know why terminals settled for 24 lines! In Java, which is similar to C# in term of verbosity, Clean Code tend to push towards 20-lines long functions or less. One of the danger to make functions even smaller is that many more functions can create many indirections, and that becomes harder to keep track within our brains. </p> </div> <div class="comment-date">2019-11-04 11:13 UTC</div> </div> <div class="comment" id="7fc45f8fd537ba9907ad73daa2c85b52"> <div class="comment-author">Terrell</div> <div class="comment-content"> <p> Some additional terminal sizing history in Mike Hoye's recent similarly-named post: <a href="http://exple.tive.org/blarg/2019/10/23/80x25/">http://exple.tive.org/blarg/2019/10/23/80x25/</a> Spoiler - Banknotes! </p> </div> <div class="comment-date">2019-11-04 13:25 UTC</div> </div> </div> <hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. A basic Haskell solution to the robot journeys coding exercise https://blog.ploeh.dk/2019/10/28/a-basic-haskell-solution-to-the-robot-journeys-coding-exercise 2019-10-28T04:34:00+00:00 Mark Seemann <div id="post"> <p> <em>This article shows an idiomatic, yet beginner-friendly Haskell solution to a coding exercise.</em> </p> <p> <a href="https://twitter.com/mikehadlow/status/1186332184086495233">Mike Hadlow tweeted</a> a coding exercise that involves parsing and evaluating instruction sets. <a href="https://www.haskell.org">Haskell</a> excels at such problems, so I decided to give it a go. Since this was only an exercise for the fun of it, I didn't want to set up a complete Haskell project. Rather, I wanted to write one or two <code>.hs</code> files that I could interact with via <em>GHCi</em>. This means no lenses, monad transformers, or other fancy libraries. </p> <p> Hopefully, this makes the code friendly to Haskell beginners. It shows what I consider <a href="/2015/08/03/idiomatic-or-idiosyncratic">idiomatic</a>, but basic Haskell, solving a problem of moderate difficulty. </p> <h3 id="79b308125b2d4bcc9df79f7c6569a6e9"> The problem <a href="#79b308125b2d4bcc9df79f7c6569a6e9" title="permalink">#</a> </h3> <p> <a href="https://github.com/mikehadlow/Journeys">Mike Hadlow has a detailed description of the exercise</a>, but in short, you're given a file with a set of instructions that look like this: </p> <p> <pre>1 1 E RFRFRFRF 1 1 E</pre> </p> <p> The first and last lines describe the position and orientation of a robot. The first line, for example, describes a robot at position (1, 1) facing east. A robot can face in one of the four normal directions of the map: north, east, south, and west. </p> <p> The first line gives the robot's start position, and the last line the <em>expected</em> end position. </p> <p> The middle line is a set of instructions to the robot. It can turn left or right, or move forward. </p> <p> The exercise is to evaluate whether journeys are valid; that is, whether the robot's end position matches the expected end position if it follows the commands. </p> <h3 id="cf43d781d49545c38e30f487867e904f"> Imports <a href="#cf43d781d49545c38e30f487867e904f" title="permalink">#</a> </h3> <p> I managed to solve the exercise with a single <code>Main.hs</code> file. Here's the module declaration and the required imports: </p> <p> <pre><span style="color:blue;">module</span>&nbsp;Main&nbsp;<span style="color:blue;">where</span> <span style="color:blue;">import</span>&nbsp;Data.Foldable <span style="color:blue;">import</span>&nbsp;Data.Ord <span style="color:blue;">import</span>&nbsp;Text.Read&nbsp;(<span style="color:#2b91af;">readPrec</span>) <span style="color:blue;">import</span>&nbsp;Text.ParserCombinators.ReadP <span style="color:blue;">import</span>&nbsp;Text.ParserCombinators.ReadPrec&nbsp;(<span style="color:#2b91af;">readPrec_to_P</span>,&nbsp;<span style="color:#2b91af;">minPrec</span>)</pre> </p> <p> These imports are only required to support parsing of input. Once parsed, you can evaluate each journey using nothing but the functions available in the standard <code>Prelude</code>. </p> <h3 id="8329bba327f54dbfb8474f9102ba8662"> Types <a href="#8329bba327f54dbfb8474f9102ba8662" title="permalink">#</a> </h3> <p> Haskell is a statically typed language, so it often pays to define some types. Granted, the exercise hardly warrants all of these types, but as an example of idiomatic Haskell, I think that this is still good practice. After all, Haskell types are easy to declare. Often, they are one-liners: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;Direction&nbsp;=&nbsp;North&nbsp;|&nbsp;East&nbsp;|&nbsp;South&nbsp;|&nbsp;West&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> The <code>Direction</code> type enumerates the four corners of the world. </p> <p> <pre><span style="color:blue;">data</span>&nbsp;Robot&nbsp;=&nbsp;Robot&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;robotPosition&nbsp;::&nbsp;(Integer,&nbsp;Integer) &nbsp;&nbsp;,&nbsp;robotDirection&nbsp;::&nbsp;Direction&nbsp;} &nbsp;&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> The <code>Robot</code> record type represents the state of a robot: its position and the direction it faces. </p> <p> You'll also need to enumerate the commands that you can give a robot: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;Command&nbsp;=&nbsp;TurnLeft&nbsp;|&nbsp;TurnRight&nbsp;|&nbsp;MoveForward&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> Finally, you can also define a type for a <code>Journey</code>: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;Journey&nbsp;=&nbsp;Journey&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;journeyStart&nbsp;::&nbsp;Robot &nbsp;&nbsp;,&nbsp;journeyCommands&nbsp;::&nbsp;[Command] &nbsp;&nbsp;,&nbsp;journeyEnd&nbsp;::&nbsp;Robot&nbsp;} &nbsp;&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> These are all the types required for solving the exercise. </p> <h3 id="5ada3f24aa4e4858892855ae076f457b"> Parsing <a href="#5ada3f24aa4e4858892855ae076f457b" title="permalink">#</a> </h3> <p> The format of the input file is simple enough that it could be done in an ad-hoc fashion using <code>lines</code>, <code>word</code>, <code>read</code>, and a few other low-level functions. While the format barely warrants the use of parser combinators, I'll still use some to showcase the power of that approach. </p> <p> Since one of my goals is to implement the functionality using a single <code>.hs</code> file, I can't pull in external parser combinator libraries. Instead, I'll use the built-in <code>ReadP</code> module, which I've often found sufficient to parse files like the present exercise input file. </p> <p> First, you're going to have to be able to parse numbers, which can be done using the <code>Read</code> type class. You'll need, however, to be able to compose <code>Integer</code> parsers with other <code>ReadP</code> parsers. </p> <p> <pre><span style="color:#2b91af;">parseRead</span>&nbsp;::&nbsp;<span style="color:blue;">Read</span>&nbsp;a&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;<span style="color:blue;">ReadP</span>&nbsp;a parseRead&nbsp;=&nbsp;readPrec_to_P&nbsp;readPrec&nbsp;minPrec</pre> </p> <p> This turns every <code>Read</code> instance value into a <code>ReadP</code> value. (I admit that I wasn't sure which precedence number to use, but <code>minPrec</code> seems to work.) </p> <p> Next, you need a parser for <code>Direction</code> values: </p> <p> <pre><span style="color:#2b91af;">parseDirection</span>&nbsp;::&nbsp;<span style="color:blue;">ReadP</span>&nbsp;<span style="color:blue;">Direction</span> parseDirection&nbsp;= &nbsp;&nbsp;choice&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;char&nbsp;<span style="color:#a31515;">&#39;N&#39;</span>&nbsp;&gt;&gt;&nbsp;<span style="color:blue;">return</span>&nbsp;North, &nbsp;&nbsp;&nbsp;&nbsp;char&nbsp;<span style="color:#a31515;">&#39;E&#39;</span>&nbsp;&gt;&gt;&nbsp;<span style="color:blue;">return</span>&nbsp;East, &nbsp;&nbsp;&nbsp;&nbsp;char&nbsp;<span style="color:#a31515;">&#39;S&#39;</span>&nbsp;&gt;&gt;&nbsp;<span style="color:blue;">return</span>&nbsp;South, &nbsp;&nbsp;&nbsp;&nbsp;char&nbsp;<span style="color:#a31515;">&#39;W&#39;</span>&nbsp;&gt;&gt;&nbsp;<span style="color:blue;">return</span>&nbsp;West&nbsp;]</pre> </p> <p> Notice how declarative this looks. The <code>choice</code> function combines a list of other parsers. When an individual parser in that list encounters the <code>'N'</code> character, it'll parse it as <code>North</code>, <code>'E'</code> as <code>East</code>, and so on. </p> <p> You can now parse an entire <code>Robot</code> using the <code>Applicative</code> <code>&lt;*&gt;</code> and <code>&lt;*</code> operators. </p> <p> <pre><span style="color:#2b91af;">parseRobot</span>&nbsp;::&nbsp;<span style="color:blue;">ReadP</span>&nbsp;<span style="color:blue;">Robot</span> parseRobot&nbsp;= &nbsp;&nbsp;(\x&nbsp;y&nbsp;d&nbsp;-&gt;&nbsp;Robot&nbsp;(x,&nbsp;y)&nbsp;d)&nbsp;&lt;$&gt; &nbsp;&nbsp;(parseRead&nbsp;&lt;*&nbsp;char&nbsp;<span style="color:#a31515;">&#39;&nbsp;&#39;</span>)&nbsp;&lt;*&gt; &nbsp;&nbsp;(parseRead&nbsp;&lt;*&nbsp;char&nbsp;<span style="color:#a31515;">&#39;&nbsp;&#39;</span>)&nbsp;&lt;*&gt; &nbsp;&nbsp;&nbsp;parseDirection</pre> </p> <p> The <code>&lt;*&gt;</code> operator combines two parsers by using the output of both of them, whereas the <code>&lt;*</code> combines two parsers by running both of them, but discarding the output of the right-hand parser. A good mnemonic is that the operator points to the parser that produces an output. Here', the <code>parseRobot</code> function uses the <code>&lt;*</code> operator to require that each number is followed by a space. The space, however, is just a delimiter, so you throw it away. </p> <p> <code>parseRead</code> parses any <code>Read</code> instance. Here, the <code>parseRobot</code> function uses it to parse each <code>Integer</code> in a robot's position. It also uses <code>parseDirection</code> to parse the robot's direction. </p> <p> Similar to how you can parse directions, you can also parse the commands: </p> <p> <pre><span style="color:#2b91af;">parseCommand</span>&nbsp;::&nbsp;<span style="color:blue;">ReadP</span>&nbsp;<span style="color:blue;">Command</span> parseCommand&nbsp;= &nbsp;&nbsp;choice&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;char&nbsp;<span style="color:#a31515;">&#39;L&#39;</span>&nbsp;&gt;&gt;&nbsp;<span style="color:blue;">return</span>&nbsp;TurnLeft, &nbsp;&nbsp;&nbsp;&nbsp;char&nbsp;<span style="color:#a31515;">&#39;R&#39;</span>&nbsp;&gt;&gt;&nbsp;<span style="color:blue;">return</span>&nbsp;TurnRight, &nbsp;&nbsp;&nbsp;&nbsp;char&nbsp;<span style="color:#a31515;">&#39;F&#39;</span>&nbsp;&gt;&gt;&nbsp;<span style="color:blue;">return</span>&nbsp;MoveForward]</pre> </p> <p> Likewise, similar to how you parse a single robot, you can now parse a journey: </p> <p> <pre><span style="color:#2b91af;">parseJourney</span>&nbsp;::&nbsp;<span style="color:blue;">ReadP</span>&nbsp;<span style="color:blue;">Journey</span> parseJourney&nbsp;= &nbsp;&nbsp;Journey&nbsp;&lt;$&gt; &nbsp;&nbsp;(parseRobot&nbsp;&lt;*&nbsp;string&nbsp;<span style="color:#a31515;">&quot;\n&quot;</span>)&nbsp;&lt;*&gt; &nbsp;&nbsp;(many&nbsp;parseCommand&nbsp;&lt;*&nbsp;string&nbsp;<span style="color:#a31515;">&quot;\n&quot;</span>)&nbsp;&lt;*&gt; &nbsp;&nbsp;&nbsp;parseRobot</pre> </p> <p> The only new element compared to <code>parseRobot</code> is the use of the <code>many</code> parser combinator, which looks for zero, one, or many <code>Command</code> values. </p> <p> This gives you a way to parse a complete journey, but the input file contains many of those, separated by newlines and other whitespace: </p> <p> <pre><span style="color:#2b91af;">parseJourneys</span>&nbsp;::&nbsp;<span style="color:blue;">ReadP</span>&nbsp;[<span style="color:blue;">Journey</span>] parseJourneys&nbsp;=&nbsp;parseJourney&nbsp;`sepBy`&nbsp;skipSpaces</pre> </p> <p> Finally, you can parse a multi-line string into a list of journeys: </p> <p> <pre><span style="color:#2b91af;">parseInput</span>&nbsp;::&nbsp;<span style="color:#2b91af;">String</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;[<span style="color:blue;">Journey</span>] parseInput&nbsp;=&nbsp;<span style="color:blue;">fst</span>&nbsp;.&nbsp;minimumBy&nbsp;(comparing&nbsp;<span style="color:blue;">snd</span>)&nbsp;.&nbsp;readP_to_S&nbsp;parseJourneys</pre> </p> <p> When you run <code>readP_to_S</code>, it'll produce a list of alternatives, as there's more than one way to interpret the file according to <code>parseJourneys</code>. Each alternative is presented as a tuple of the parse result and the remaining (or unconsumed) string. I'm after the alternative that consumes as much of the input file as possible (which turns out to be all of it), so I use <code>minimumBy</code> to find the tuple that has the smallest second element. Then I return the first element of that tuple. </p> <p> Play around with <code>readP_to_S parseJourneys</code> in GHCi if you want all the details. </p> <h3 id="34135daae9634db2885e28382760d1fd"> Evaluation <a href="#34135daae9634db2885e28382760d1fd" title="permalink">#</a> </h3> <p> Haskell beginners may still find operators like <code>&lt;*&gt;</code> cryptic, but they're essential to parser combinators. Evaluation of the journeys is, in comparison, simple. </p> <p> You can start by defining a function to turn right: </p> <p> <pre><span style="color:#2b91af;">turnRight</span>&nbsp;::&nbsp;<span style="color:blue;">Robot</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Robot</span> turnRight&nbsp;r@(Robot&nbsp;_&nbsp;North)&nbsp;=&nbsp;r&nbsp;{&nbsp;robotDirection&nbsp;=&nbsp;East&nbsp;} turnRight&nbsp;r@(Robot&nbsp;_&nbsp;&nbsp;East)&nbsp;=&nbsp;r&nbsp;{&nbsp;robotDirection&nbsp;=&nbsp;South&nbsp;} turnRight&nbsp;r@(Robot&nbsp;_&nbsp;South)&nbsp;=&nbsp;r&nbsp;{&nbsp;robotDirection&nbsp;=&nbsp;West&nbsp;} turnRight&nbsp;r@(Robot&nbsp;_&nbsp;&nbsp;West)&nbsp;=&nbsp;r&nbsp;{&nbsp;robotDirection&nbsp;=&nbsp;North&nbsp;}</pre> </p> <p> There's more than one way to write a function that rotates one direction to the right, but I chose one that I found most readable. It trades clarity for verbosity by relying on simple pattern matching. I hope that it's easy to understand for Haskell beginners, and perhaps even for people who haven't seen Haskell code before. </p> <p> The function to turn left uses the same structure: </p> <p> <pre><span style="color:#2b91af;">turnLeft</span>&nbsp;::&nbsp;<span style="color:blue;">Robot</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Robot</span> turnLeft&nbsp;r@(Robot&nbsp;_&nbsp;North)&nbsp;=&nbsp;r&nbsp;{&nbsp;robotDirection&nbsp;=&nbsp;West&nbsp;} turnLeft&nbsp;r@(Robot&nbsp;_&nbsp;&nbsp;West)&nbsp;=&nbsp;r&nbsp;{&nbsp;robotDirection&nbsp;=&nbsp;South&nbsp;} turnLeft&nbsp;r@(Robot&nbsp;_&nbsp;South)&nbsp;=&nbsp;r&nbsp;{&nbsp;robotDirection&nbsp;=&nbsp;East&nbsp;} turnLeft&nbsp;r@(Robot&nbsp;_&nbsp;&nbsp;East)&nbsp;=&nbsp;r&nbsp;{&nbsp;robotDirection&nbsp;=&nbsp;North&nbsp;}</pre> </p> <p> The last command you need to implement is moving forward: </p> <p> <pre><span style="color:#2b91af;">moveForward</span>&nbsp;::&nbsp;<span style="color:blue;">Robot</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Robot</span> moveForward&nbsp;(Robot&nbsp;(x,&nbsp;y)&nbsp;North)&nbsp;=&nbsp;Robot&nbsp;(x,&nbsp;y&nbsp;+&nbsp;1)&nbsp;North moveForward&nbsp;(Robot&nbsp;(x,&nbsp;y)&nbsp;&nbsp;East)&nbsp;=&nbsp;Robot&nbsp;(x&nbsp;+&nbsp;1,&nbsp;y)&nbsp;East moveForward&nbsp;(Robot&nbsp;(x,&nbsp;y)&nbsp;South)&nbsp;=&nbsp;Robot&nbsp;(x,&nbsp;y&nbsp;-&nbsp;1)&nbsp;South moveForward&nbsp;(Robot&nbsp;(x,&nbsp;y)&nbsp;&nbsp;West)&nbsp;=&nbsp;Robot&nbsp;(x&nbsp;-&nbsp;1,&nbsp;y)&nbsp;West</pre> </p> <p> The <code>moveForward</code> function also pattern-matches on the direction the robot is facing, this time to increment or decrement the <code>x</code> or <code>y</code> coordinate as appropriate. </p> <p> You can now evaluate all three commands: </p> <p> <pre><span style="color:#2b91af;">evalCommand</span>&nbsp;::&nbsp;<span style="color:blue;">Command</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Robot</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Robot</span> evalCommand&nbsp;&nbsp;&nbsp;TurnRight&nbsp;=&nbsp;turnRight evalCommand&nbsp;&nbsp;&nbsp;&nbsp;TurnLeft&nbsp;=&nbsp;turnLeft evalCommand&nbsp;MoveForward&nbsp;=&nbsp;moveForward</pre> </p> <p> The <code>evalCommand</code> pattern-matches on all three <code>Command</code> cases and returns the appropriate function for each. </p> <p> You can now evaluate whether a <code>Journey</code> is valid: </p> <p> <pre><span style="color:#2b91af;">isJourneyValid</span>&nbsp;::&nbsp;<span style="color:blue;">Journey</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">Bool</span> isJourneyValid&nbsp;(Journey&nbsp;s&nbsp;cs&nbsp;e)&nbsp;=&nbsp;<span style="color:blue;">foldl</span>&nbsp;(<span style="color:blue;">flip</span>&nbsp;evalCommand)&nbsp;s&nbsp;cs&nbsp;==&nbsp;e</pre> </p> <p> The <code>isJourneyValid</code> function pattern-matches the constituent values out of <code>Journey</code>. I named the <code>journeyStart</code> value <code>s</code> (for <em>start</em>), the <code>journeyCommands</code> value <code>cs</code> (for <em>commands</em>), and the <code>journeyEnd</code> value <code>e</code> (for <em>end</em>). </p> <p> The <code>evalCommand</code> function evaluates a single <code>Command</code>, but a <code>Journey</code> contains many commands. You'll need to evaluate the first command to find the position from which you evaluate the second command, and so on. Imperative programmers would use a <em>for loop</em> for something like that, but in functional programming, a <em>fold</em>, in this case from the left, is how it's done. </p> <p> <code>foldl</code> requires you to supply an initial state <code>s</code> as well as the list of commands <code>cs</code>. The entire <code>foldl</code> expression produces a final <code>Robot</code> state that you can compare against the expected end state <code>e</code>. </p> <h3 id="6267cc3a03594d0fb21cd7cc61430eb0"> Execution <a href="#6267cc3a03594d0fb21cd7cc61430eb0" title="permalink">#</a> </h3> <p> Load the input file, parse it, and evaluate each journey in the <code>main</code> function: </p> <p> <pre><span style="color:#2b91af;">main</span>&nbsp;::&nbsp;<span style="color:#2b91af;">IO</span>&nbsp;() main&nbsp;=&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;input&nbsp;&lt;-&nbsp;parseInput&nbsp;&lt;$&gt;&nbsp;<span style="color:blue;">readFile</span>&nbsp;<span style="color:#a31515;">&quot;input.txt&quot;</span> &nbsp;&nbsp;<span style="color:blue;">mapM_</span>&nbsp;<span style="color:blue;">print</span>&nbsp;$&nbsp;isJourneyValid&nbsp;&lt;$&gt;&nbsp;input</pre> </p> <p> I just load the <code>Main.hs</code> file in GHCi and run the <code>main</code> function: </p> <p> <pre>Prelude&gt; :load Main.hs [1 of 1] Compiling Main ( Main.hs, interpreted ) Ok, one module loaded. *Main&gt; main True True True</pre> </p> <p> I used the same input file as Mike Hadlow, and it turns out that all journeys are valid. That's not what I'd expected from an exercise like this, so I cloned and ran Mike's solution as well, and it seems that it arrives at the same result. </p> <h3 id="683ea9809a774338b16aed1ad41e1984"> Conclusion <a href="#683ea9809a774338b16aed1ad41e1984" title="permalink">#</a> </h3> <p> Haskell is a great language for small coding exercises that require parsing and interpretation. In this article, I demonstrated one solution to the <em>robot journeys</em> coding exercise. My goal was to show some beginner-friendly, but still idiomatic Haskell code. </p> <p> Granted, the use of parser combinators is on the verge of being overkill, but I wanted to show an example; Haskell examples are scarce, so I hope it's helpful. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. A red-green-refactor checklist https://blog.ploeh.dk/2019/10/21/a-red-green-refactor-checklist 2019-10-21T06:49:00+00:00 Mark Seemann <div id="post"> <p> <em>A simple read-do checklist for test-driven development.</em> </p> <p> I recently read <a href="https://amzn.to/35Wk5yD">The Checklist Manifesto</a>, a book about the power of checklists. That may sound off-putting and tedious, but I actually <a href="https://www.goodreads.com/review/show/2949987528">found it inspiring</a>. It explains how checklists empower skilled professionals to focus on difficult problems, while preventing avoidable mistakes. </p> <p> Since I read the book with the intent to see if there were ideas that we could apply in software development, I thought about checklists one might create for software development. Possibly the simplest checklist is one that describes the <em>red-green-refactor</em> cycle of test-driven development. </p> <h3 id="530e04dfea574602952023fa218d8a7c"> Types of checklists <a href="#530e04dfea574602952023fa218d8a7c" title="permalink">#</a> </h3> <p> As the book describes, there's basically two types of checklists: <ul> <li><strong>Do-confirm.</strong> With such a checklist, you perform a set of tasks, and then subsequently, at a sufficient <em>pause point</em> go through the checklist to verify that you remembered to perform all the tasks on the list.</li> <li><strong>Read-do.</strong> With this type of checklist, you read each item for instructions and then perform the task. Only when you've performed the task do you move on to the next item on the list.</li> </ul> I find it most intuitive to describe the red-green-refactor cycle as a <em>read-do</em> list. I did, however, find it expedient to include a <em>do-confirm</em> sub-list for one of the overall steps. </p> <p> This list is, I think, mostly useful if you're still learning test-driven development. It can be easily internalised. As such, I offer this for inspiration, and as a learning aid. </p> <h3 id="05b38ebc9c0c419b9a146be976578bd2"> Red-green-refactor checklist <a href="#05b38ebc9c0c419b9a146be976578bd2" title="permalink">#</a> </h3> <p> Read each of the steps in the list and perform the task. <ol> <li>Write a failing test. <ul> <li>Did you run the test?</li> <li>Did it fail?</li> <li>Did it fail because of an assertion?</li> <li>Did it fail because of the <em>last</em> assertion?</li> </ul> </li> <li>Make all tests pass by doing the simplest thing that could possibly work.</li> <li>Consider the resulting code. Can it be improved? If so, do it, but make sure that all tests still pass.</li> <li>Repeat</li> </ol> Perhaps the most value this checklist provides isn't so much the overall <em>read-do</em> list, but rather the subordinate <em>do-confirm</em> list associated with the first step. </p> <p> I regularly see people write failing tests as an initial step. The reason the test fails, however, is because the implementation throws an exception. </p> <h3 id="24a066fa0b9b401687d47b92473d63d0"> Improperly failing tests <a href="#24a066fa0b9b401687d47b92473d63d0" title="permalink">#</a> </h3> <p> Consider, as an example, the first test you might write when doing the <a href="https://en.wikipedia.org/wiki/Fizz_buzz">FizzBuzz</a> kata. </p> <p> <pre>[<span style="color:#2b91af;">Fact</span>] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">One</span>() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">string</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#2b91af;">FizzBuzz</span>.<span style="color:#74531f;">Convert</span>(1); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">Equal</span>(<span style="color:#a31515;">&quot;1&quot;</span>,&nbsp;<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> I wrote this test first (i.e. before the 'production' code) and used Visual Studio's refactoring tools to generate the implied type and method. </p> <p> When I run the test, it fails. </p> <p> Further investigation, however, reveals that the test fails when <code>Convert</code> is called: </p> <p> <pre>Ploeh.Katas.FizzBuzzKata.FizzBuzzTests.One Source: FizzBuzzTests.cs line: 11 Duration: 8 ms Message: System.NotImplementedException : The method or operation is not implemented. Stack Trace: at FizzBuzz.Convert(Int32 i) in FizzBuzz.cs line: 9 at FizzBuzzTests.One() in FizzBuzzTests.cs line: 13 </pre> </p> <p> This is hardly surprising, since this is the current 'implementation': </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">string</span>&nbsp;<span style="color:#74531f;">Convert</span>(<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">i</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">throw</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">NotImplementedException</span>(); }</pre> </p> <p> This is what the subordinate <em>do-confirm</em> checklist is for. Did the test fail because of an assertion? In this case, the answer is no. </p> <p> This means that you're not yet done with the <em>read</em> phase. </p> <h3 id="97295029364d4cd7ace15be9c9f8dc64"> Properly failing tests <a href="#97295029364d4cd7ace15be9c9f8dc64" title="permalink">#</a> </h3> <p> You can address the issue by changing the <code>Convert</code> method: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">string</span>&nbsp;<span style="color:#74531f;">Convert</span>(<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">i</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:#a31515;">&quot;&quot;</span>; }</pre> </p> <p> This causes the test to fail because of an assertion: </p> <p> <pre> Ploeh.Katas.FizzBuzzKata.FizzBuzzTests.One Source: FizzBuzzTests.cs line: 11 Duration: 13 ms Message: Assert.Equal() Failure ↓ (pos 0) Expected: 1 Actual: ↑ (pos 0) Stack Trace: at FizzBuzzTests.One() in FizzBuzzTests.cs line: 14 </pre> </p> <p> Not only does the test fail because of an assertion - it fails because of the last assertion (since there's only one assertion). This completes the <em>do-confirm</em> checklist, and you're now ready to make the simplest change that could possibly work: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">string</span>&nbsp;<span style="color:#74531f;">Convert</span>(<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">i</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>; }</pre> </p> <p> This passes the test suite. </p> <h3 id="bdb785a78ebf475c95c64197274268c7"> Conclusion <a href="#bdb785a78ebf475c95c64197274268c7" title="permalink">#</a> </h3> <p> It's important to see tests fail. Particularly, it's important to see tests fail for the reason you expect them to fail. You'd be surprised how often you inadvertently write an <a href="/2019/10/14/tautological-assertion">assertion that can never fail</a>. </p> <p> Once you've seen the test fail for the proper reason, make it pass. </p> <p> Finally, refactor the code if necessary. </p> </div> <div id="comments"> <hr> <h2 id="comments-header"> Comments </h2> <div class="comment" id="99be0da15a164d5782afdef808300828"> <div class="comment-author">Tyson Williams</div> <div class="comment-content"> <p> I remember the first time that I realized that I did the red step wrong because my test didn't fail for the intended reason (i.e. it didn't fail because of an assertion). Before that, I didn't realize that I needed to This is a nice programming checklist. Thanks for sharing it :) </p> <blockquote> 3. Consider the resulting code. Can it be improved? If so, do it, but make sure that all tests still pass. </blockquote> <blockquote> Finally, refactor the code if necessary. </blockquote> <p> If I can be a <a href="https://blog.ploeh.dk/2019/10/07/devils-advocate/">Devil's advocate</a> for a moment, then I would say that code can always be improved and few things are necessary. In all honesty though, I think the refactoring step is the most interesting. All three steps include aspects of science and art, but I think the refactor step includes the most of both. On the one hand, it is extremely creative and full of judgement calls about what code should be refactored and what properties the resulting code should have. On the other hand, much of the work of how to (properly) refactor is laid out in books like <a href="https://www.amazon.com/Refactoring-Improving-Existing-Addison-Wesley-Signature/dp/0134757599">Martin Fowler's Refacoring</a> and is akin to algebraic manipulations of an algebraic formula. </p> <p> In other words, I feel like there is room to expand on this checklist in the refactor step. Do you have any thoughts about you might expand it? </p> </div> <div class="comment-date">2019-10-25 00:33 UTC</div> </div> <div class="comment" id="28976782c7984115a65d539eff3d0414"> <div class="comment-author"><a href="/">Mark Seemann</a></div> <div class="comment-content"> <p> Tyson, thank you for writing. I agree that the <em>refactoring</em> step is both important and compelling. I can't, however, imagine how a checklist would be useful. </p> <p> The point of <em>The Checklist Manifesto</em> is that checklists help identify avoidable mistakes. A checklist isn't intended to describe an algorithm, but rather to make sure that crucial steps aren't forgotten. </p> <p> Another important point from <em>The Checklist Manifesto</em> is that a checklist is only effective if it's not too big. A checklist that tries to cover every eventuality isn't useful, because then people don't follow it. </p> <p> As you write, refactoring is a big topic, covered by several books. All the creativity and experience that goes into refactoring doesn't seem like something that can easily be expressed as an effective checklist. </p> <p> I don't mind being proven wrong, though, so by all means give it a go. </p> </div> <div class="comment-date">2019-10-25 21:51 UTC</div> </div> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Tautological assertion https://blog.ploeh.dk/2019/10/14/tautological-assertion 2019-10-14T18:39:00+00:00 Mark Seemann <div id="post"> <p> <em>It's surprisingly easy to write a unit test assertion that never fails.</em> </p> <p> Recently I was mob programming with a pair of <a href="https://idq.dk">IDQ</a>'s programmers. We were starting a new code base, using test-driven development (TDD). This was the first test we wrote: </p> <p> <pre>[<span style="color:#2b91af;">Fact</span>] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">async</span>&nbsp;<span style="color:#2b91af;">Task</span>&nbsp;<span style="font-weight:bold;color:#74531f;">HandleObserveUnitStatusStartsSaga</span>() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">subscribers</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">List</span>&lt;<span style="color:#2b91af;">Guid</span>&gt;&nbsp;{&nbsp;<span style="color:#2b91af;">Guid</span>.<span style="color:#74531f;">Parse</span>(<span style="color:#a31515;">&quot;{4D093799-9CCC-4135-8CB3-8661985A5853}&quot;</span>)&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">StatusPolicy</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Data&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">StatusPolicyData</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;UnitId&nbsp;=&nbsp;123, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Subscribers&nbsp;=&nbsp;<span style="font-weight:bold;color:#1f377f;">subscribers</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">subscriber</span>&nbsp;=&nbsp;<span style="color:#2b91af;">Guid</span>.<span style="color:#74531f;">Parse</span>(<span style="color:#a31515;">&quot;{003C5527-7747-4C7A-980E-67040DB738C3}&quot;</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">message</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ObserveUnitStatus</span>(123,&nbsp;<span style="font-weight:bold;color:#1f377f;">subscriber</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">context</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">TestableMessageHandlerContext</span>(); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">await</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">sut</span>.<span style="font-weight:bold;color:#74531f;">Handle</span>(<span style="font-weight:bold;color:#1f377f;">message</span>,&nbsp;<span style="font-weight:bold;color:#1f377f;">context</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">Contains</span>(<span style="font-weight:bold;color:#1f377f;">subscriber</span>,&nbsp;<span style="font-weight:bold;color:#1f377f;">sut</span>.Data.Subscribers); }</pre> </p> <p> This unit test uses <a href="https://xunit.net">xUnit.net</a> 2.4.0 and <a href="https://particular.net/nservicebus">NServiceBus</a> 7.1.10 on .NET Core 2.2. The System Under Test (SUT) is intended to be an NServiceBus Saga that monitors a resource for status changes. If a <em>unit</em> changes status, the Saga will alert its subscribers. </p> <p> The test verifies that when a new subscriber wishes to observe a unit, then its ID is added to the policy's list of subscribers. </p> <p> The test induced us to implement <code>Handle</code> like this: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Task</span>&nbsp;<span style="font-weight:bold;color:#74531f;">Handle</span>(<span style="color:#2b91af;">ObserveUnitStatus</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">message</span>,&nbsp;<span style="color:#2b91af;">IMessageHandlerContext</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">context</span>) { &nbsp;&nbsp;&nbsp;&nbsp;Data.Subscribers.<span style="font-weight:bold;color:#74531f;">Add</span>(<span style="font-weight:bold;color:#1f377f;">message</span>.SubscriberId); &nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#8f08c4;">return</span>&nbsp;<span style="color:#2b91af;">Task</span>.CompletedTask; }</pre> </p> <p> Following the <em>red-green-refactor</em> cycle of TDD, this seemed an appropriate implementation. </p> <h3 id="8182b3c72b2940b98195f41b7a1193e8"> Enter the Devil <a href="#8182b3c72b2940b98195f41b7a1193e8" title="permalink">#</a> </h3> <p> I often use the <a href="/2019/10/07/devils-advocate">Devil's advocate</a> technique to figure out what to do next, so I made this change to the <code>Handle</code> method: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Task</span>&nbsp;<span style="font-weight:bold;color:#74531f;">Handle</span>(<span style="color:#2b91af;">ObserveUnitStatus</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">message</span>,&nbsp;<span style="color:#2b91af;">IMessageHandlerContext</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">context</span>) { &nbsp;&nbsp;&nbsp;&nbsp;Data.Subscribers.<span style="font-weight:bold;color:#74531f;">Clear</span>(); &nbsp;&nbsp;&nbsp;&nbsp;Data.Subscribers.<span style="font-weight:bold;color:#74531f;">Add</span>(<span style="font-weight:bold;color:#1f377f;">message</span>.SubscriberId); &nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#8f08c4;">return</span>&nbsp;<span style="color:#2b91af;">Task</span>.CompletedTask; }</pre> </p> <p> The change is that the method first deletes all existing subscribers. This is obviously wrong, but it passes all tests. That's no surprise, since I intentionally introduced the change to make us improve the test. </p> <h3 id="55331957e1de4691bb182c2aae614f4e"> False negative <a href="#55331957e1de4691bb182c2aae614f4e" title="permalink">#</a> </h3> <p> We had to write a new test, or improve the existing test, so that the defect I just introduced would be caught. I suggested an improvement to the existing test: </p> <p> <pre>[<span style="color:#2b91af;">Fact</span>] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">async</span>&nbsp;<span style="color:#2b91af;">Task</span>&nbsp;<span style="font-weight:bold;color:#74531f;">HandleObserveUnitStatusStartsSaga</span>() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">subscribers</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">List</span>&lt;<span style="color:#2b91af;">Guid</span>&gt;&nbsp;{&nbsp;<span style="color:#2b91af;">Guid</span>.<span style="color:#74531f;">Parse</span>(<span style="color:#a31515;">&quot;{4D093799-9CCC-4135-8CB3-8661985A5853}&quot;</span>)&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">StatusPolicy</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Data&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">StatusPolicyData</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;UnitId&nbsp;=&nbsp;123, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Subscribers&nbsp;=&nbsp;<span style="font-weight:bold;color:#1f377f;">subscribers</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">subscriber</span>&nbsp;=&nbsp;<span style="color:#2b91af;">Guid</span>.<span style="color:#74531f;">Parse</span>(<span style="color:#a31515;">&quot;{003C5527-7747-4C7A-980E-67040DB738C3}&quot;</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">message</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ObserveUnitStatus</span>(123,&nbsp;<span style="font-weight:bold;color:#1f377f;">subscriber</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">context</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">TestableMessageHandlerContext</span>(); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">await</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">sut</span>.<span style="font-weight:bold;color:#74531f;">Handle</span>(<span style="font-weight:bold;color:#1f377f;">message</span>,&nbsp;<span style="font-weight:bold;color:#1f377f;">context</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">Contains</span>(<span style="font-weight:bold;color:#1f377f;">subscriber</span>,&nbsp;<span style="font-weight:bold;color:#1f377f;">sut</span>.Data.Subscribers); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">Superset</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#1f377f;">expectedSubset</span>:&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">HashSet</span>&lt;<span style="color:#2b91af;">Guid</span>&gt;(<span style="font-weight:bold;color:#1f377f;">subscribers</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#1f377f;">actual</span>:&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">HashSet</span>&lt;<span style="color:#2b91af;">Guid</span>&gt;(<span style="font-weight:bold;color:#1f377f;">sut</span>.Data.Subscribers)); }</pre> </p> <p> The only change is the addition of the last assertion. </p> <p> Smugly I asked the keyboard driver to run the tests, anticipating that it would now fail. </p> <p> It passed. </p> <p> We'd just managed to write a <a href="http://xunitpatterns.com/false%20negative.html">false negative</a>. Even though there's a defect in the code, the test still passes. I was nonplussed. None of us expected the test to pass, yet it does. </p> <p> It took us a minute to figure out what was wrong. Before you read on, try to figure it out for yourself. Perhaps it's immediately clear to you, but it took three people with decades of programming experience a few minutes to spot the problem. </p> <h3 id="6f985240963a40d8af913e251b3b86bf"> Aliasing <a href="#6f985240963a40d8af913e251b3b86bf" title="permalink">#</a> </h3> <p> The problem is <a href="https://en.wikipedia.org/wiki/Aliasing_(computing)">aliasing</a>. While named differently, <code>subscribers</code> and <code>sut.Data.Subscribers</code> is the same object. Of course one is a subset of the other, since a set is considered to be a subset of itself. </p> <p> The assertion is tautological. It can never fail. </p> <h3 id="9e8ec91f731e4f67aceefbebb76799d4"> Fixing the problem <a href="#9e8ec91f731e4f67aceefbebb76799d4" title="permalink">#</a> </h3> <p> It's surprisingly easy to write tautological assertions when working with mutable state. This regularly happens to me, perhaps a few times a month. Once you've realised that this has happened, however, it's easy to address. </p> <p> <code>subscribers</code> shouldn't change during the test, so make it immutable. </p> <p> <pre>[<span style="color:#2b91af;">Fact</span>] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">async</span>&nbsp;<span style="color:#2b91af;">Task</span>&nbsp;<span style="font-weight:bold;color:#74531f;">HandleObserveUnitStatusStartsSaga</span>() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Guid</span>&gt;&nbsp;<span style="font-weight:bold;color:#1f377f;">subscribers</span>&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;<span style="color:#2b91af;">Guid</span>.<span style="color:#74531f;">Parse</span>(<span style="color:#a31515;">&quot;{4D093799-9CCC-4135-8CB3-8661985A5853}&quot;</span>)&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">StatusPolicy</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Data&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">StatusPolicyData</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;UnitId&nbsp;=&nbsp;123, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Subscribers&nbsp;=&nbsp;<span style="font-weight:bold;color:#1f377f;">subscribers</span>.<span style="font-weight:bold;color:#74531f;">ToList</span>() &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">subscriber</span>&nbsp;=&nbsp;<span style="color:#2b91af;">Guid</span>.<span style="color:#74531f;">Parse</span>(<span style="color:#a31515;">&quot;{003C5527-7747-4C7A-980E-67040DB738C3}&quot;</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">message</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ObserveUnitStatus</span>(123,&nbsp;<span style="font-weight:bold;color:#1f377f;">subscriber</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">context</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">TestableMessageHandlerContext</span>(); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">await</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">sut</span>.<span style="font-weight:bold;color:#74531f;">Handle</span>(<span style="font-weight:bold;color:#1f377f;">message</span>,&nbsp;<span style="font-weight:bold;color:#1f377f;">context</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">Contains</span>(<span style="font-weight:bold;color:#1f377f;">subscriber</span>,&nbsp;<span style="font-weight:bold;color:#1f377f;">sut</span>.Data.Subscribers); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">Superset</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#1f377f;">expectedSubset</span>:&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">HashSet</span>&lt;<span style="color:#2b91af;">Guid</span>&gt;(<span style="font-weight:bold;color:#1f377f;">subscribers</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#1f377f;">actual</span>:&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">HashSet</span>&lt;<span style="color:#2b91af;">Guid</span>&gt;(<span style="font-weight:bold;color:#1f377f;">sut</span>.Data.Subscribers)); }</pre> </p> <p> An array strictly isn't immutable, but declaring it as <code>IEnumerable&lt;Guid&gt;</code> hides the mutation capabilities. The test now has to copy <code>subscribers</code> to a list before assigning it to the policy's data. This anti-aliases <code>subscribers</code> from <code>sut.Data.Subscribers</code>, and causes the test to fail. After all, there's a defect in the <code>Handle</code> method. </p> <p> You now have to remove the offending line: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Task</span>&nbsp;<span style="font-weight:bold;color:#74531f;">Handle</span>(<span style="color:#2b91af;">ObserveUnitStatus</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">message</span>,&nbsp;<span style="color:#2b91af;">IMessageHandlerContext</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">context</span>) { &nbsp;&nbsp;&nbsp;&nbsp;Data.Subscribers.<span style="font-weight:bold;color:#74531f;">Add</span>(<span style="font-weight:bold;color:#1f377f;">message</span>.SubscriberId); &nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#8f08c4;">return</span>&nbsp;<span style="color:#2b91af;">Task</span>.CompletedTask; }</pre> </p> <p> This makes the test pass. </p> <h3 id="fd8e8ab5e7314ebbadc4775741af11fa"> Summary <a href="#fd8e8ab5e7314ebbadc4775741af11fa" title="permalink">#</a> </h3> <p> This article shows an example where I was surprised by aliasing. An assertion that I thought would fail turned out to be a false negative. </p> <p> You can easily make the mistake of writing a test that always passes. If you haven't tried it, you may think that you're too smart to do that, but it regularly happens to me. Other TDD practitioners have told me that it also happens to them. </p> <p> This is the reason that the <em>red-green-refactor</em> process encourages you to run each new test <em>and see it fail</em>. If you haven't seen it fail, you don't know if you've avoided a tautology. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Devil's advocate https://blog.ploeh.dk/2019/10/07/devils-advocate 2019-10-07T15:00:00+00:00 Mark Seemann <div id="post"> <p> <em>How do you know when you have enough test cases. The Devil's Advocate technique can help you decide.</em> </p> <p> When I review unit tests, I often utilise a technique I call <em>Devil's Advocate</em>. I do the same whenever I consider if I have a sufficient number of test cases. The first time I explicitly named the technique was, I think, in my <a href="/outside-in-tdd">Outside-in TDD Pluralsight course</a>, in which I also discuss the so-called <em>Gollum style</em> variation. I don't think, however, that I've ever written an article explicitly about this topic. The current text attempts to rectify that omission. </p> <h3 id="a66b04a5812b4a84ba3a60a8609e58be"> Coverage <a href="#a66b04a5812b4a84ba3a60a8609e58be" title="permalink">#</a> </h3> <p> Programmers new to unit testing often struggle with identifying useful test cases. I sometimes see people writing redundant unit tests, while, on the other hand, forgetting to add important test cases. How do you know which test cases to add, and how do you know when you've added enough? </p> <p> I may return to the first question in another article, but in this, I wish to address the second question. How do you know that you have a sufficient set of test cases? </p> <p> You may think that this is a question of turning on code coverage. Surely, if you have <a href="/2015/11/16/code-coverage-is-a-useless-target-measure">100% code coverage</a>, that's sufficient? </p> <p> It's not. Consider this simple class: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">capacity</span>) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Capacity&nbsp;=&nbsp;<span style="color:#1f377f;">capacity</span>; &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:blue;">int</span>&nbsp;Capacity&nbsp;{&nbsp;<span style="color:blue;">get</span>;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;<span style="color:#74531f;">CanAccept</span>(<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Reservation</span>&gt;&nbsp;<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;<span style="color:#1f377f;">reservation</span>) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservations</span>.<span style="color:#74531f;">Sum</span>(<span style="color:#1f377f;">r</span>&nbsp;=&gt;&nbsp;<span style="color:#1f377f;">r</span>.Quantity); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">if</span>&nbsp;(Capacity&nbsp;&lt;&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;+&nbsp;<span style="color:#1f377f;">reservation</span>.Quantity) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:blue;">false</span>; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:blue;">true</span>; &nbsp;&nbsp;&nbsp;&nbsp;} }</pre> </p> <p> This class implements the (simplified) decision logic for an online restaurant reservation system. The <code>CanAccept</code> method has a cyclomatic complexity of 2, so it should be easy to cover with a pair of unit tests: </p> <p> <pre>[<span style="color:#2b91af;">Fact</span>] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptWithNoPriorReservations</span>() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;4 &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>:&nbsp;10); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>(<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>[0],&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">True</span>(<span style="color:#1f377f;">actual</span>); } [<span style="color:#2b91af;">Fact</span>] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptOnInsufficientCapacity</span>() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;4 &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>:&nbsp;10); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;{&nbsp;Quantity&nbsp;=&nbsp;7&nbsp;}&nbsp;}, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">False</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> These two tests together completely cover the <code>CanAccept</code> method: </p> <p> <img src="/content/binary/coverage-of-can-accept-method.png" alt="Screen shot showing that the CanAccept method is 100% covered."> </p> <p> You'd think that this is a sufficient number of test cases of the method, then. </p> <h3 id="0ac61ef87e3f4e738475e97179242db5"> As the Devil reads the Bible <a href="#0ac61ef87e3f4e738475e97179242db5" title="permalink">#</a> </h3> <p> In Scandinavia we have an idiom that <a href="https://www.kentbeck.com">Kent Beck</a> (who's worked with Norwegian companies) has also encountered: <blockquote> <p> "TIL: "like the devil reads the Bible"--meaning someone who carefully reads a book to subvert its intent" </p> <footer><cite><a href="https://twitter.com/kentbeck/status/651817458857320449">Kent Beck</a></cite></footer> </blockquote> We have the same saying in Danish, and the Swedes also use it. </p> <p> If you think of a unit test suite as an executable specification, you may consider if you can follow the specification to the letter while intentionally introduce a defect. You can easily do that with the above <code>CanAccept</code> method: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;<span style="color:#74531f;">CanAccept</span>(<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Reservation</span>&gt;&nbsp;<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;<span style="color:#1f377f;">reservation</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservations</span>.<span style="color:#74531f;">Sum</span>(<span style="color:#1f377f;">r</span>&nbsp;=&gt;&nbsp;<span style="color:#1f377f;">r</span>.Quantity); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">if</span>&nbsp;(Capacity&nbsp;&lt;=&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;+&nbsp;<span style="color:#1f377f;">reservation</span>.Quantity) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:blue;">false</span>; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:blue;">true</span>; }</pre> </p> <p> This still passes both tests, and still has a code coverage of 100%, yet it's 'obviously' wrong. </p> <p> Can you spot the difference? </p> <p> Instead of a <em>less-than</em> comparison, it now uses a <em>less-than-or-equal</em> comparison. You could easily, inadvertently, make such a mistake while programming. It belongs in the category of <em>off-by-one errors</em>, which is one of the most common type of bugs. </p> <p> This is, in a nutshell, the Devil's Advocate technique. The intent isn't to break the software by sneaking in defects, but to explore how effectively the test suite detects bugs. In the current (simplified) example, the effectiveness of the test suite isn't impressive. </p> <h3 id="d6e4c657adec4cc1bb6d10af351a415f"> Add test cases <a href="#d6e4c657adec4cc1bb6d10af351a415f" title="permalink">#</a> </h3> <p> The problem introduced by the Devil's Advocate is an edge case. If the reservation under consideration fits the restaurant's remaining capacity, but entirely consumes it, the <code>MaîtreD</code> class should still accept it. Currently, however, it doesn't. </p> <p> It'd seem that the obvious solution is to 'fix' the unit test: </p> <p> <pre>[<span style="color:#2b91af;">Fact</span>] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptWithNoPriorReservations</span>() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;10 &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>:&nbsp;10); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>(<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>[0],&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">True</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> Changing the requested <code>Quantity</code> to <code>10</code> does, indeed, cause the test to fail. </p> <h3 id="26be7b38248c4dcba5134eb4529d8214"> Beyond mutation testing <a href="#26be7b38248c4dcba5134eb4529d8214" title="permalink">#</a> </h3> <p> Until this point, you may think that the Devil's Advocate just looks like <em>an ad-hoc, informally-specified, error-prone, manual version of half of <a href="https://en.wikipedia.org/wiki/Mutation_testing">mutation testing</a></em>. So far, the change I made above could also have been made during mutation testing. </p> <p> What I sometimes do with the Devil's Advocate technique is to experiment with other, less heuristically driven changes. For instance, based on my knowledge of the existing test cases, it's not too difficult to come up with this change: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;<span style="color:#74531f;">CanAccept</span>(<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Reservation</span>&gt;&nbsp;<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;<span style="color:#1f377f;">reservation</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservations</span>.<span style="color:#74531f;">Sum</span>(<span style="color:#1f377f;">r</span>&nbsp;=&gt;&nbsp;<span style="color:#1f377f;">r</span>.Quantity); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">if</span>&nbsp;(<span style="color:#1f377f;">reservation</span>.Quantity&nbsp;!=&nbsp;10) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:blue;">false</span>; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:blue;">true</span>; }</pre> </p> <p> That's an even simpler implementation than the original, but obviously wrong. </p> <p> This should prompt you to add at least one other test case: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>] [<span style="color:#2b91af;">InlineData</span>(&nbsp;4)] [<span style="color:#2b91af;">InlineData</span>(10)] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptWithNoPriorReservations</span>(<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">quantity</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">quantity</span> &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>:&nbsp;10); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>(<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>[0],&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">True</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> Notice that I converted the test to a parametrised test. This breaks the Devil's latest attempt, while the original implementation passes all tests. </p> <p> The Devil, not to be outdone, now switches tactics and goes after the <code>reservations</code> instead: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;<span style="color:#74531f;">CanAccept</span>(<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Reservation</span>&gt;&nbsp;<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;<span style="color:#1f377f;">reservation</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;!<span style="color:#1f377f;">reservations</span>.<span style="color:#74531f;">Any</span>(); }</pre> </p> <p> This still passes all tests, including the new test case. This indicates that you'll need to add at least one test case with existing reservations, but where there's still enough capacity to accept another reservation: </p> <p> <pre>[<span style="color:#2b91af;">Fact</span>] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptWithOnePriorReservation</span>() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;4 &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>:&nbsp;10); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;{&nbsp;Quantity&nbsp;=&nbsp;4&nbsp;}&nbsp;}, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">True</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> This new test fails, prompting you to correct the implementation of <code>CanAccept</code>. The Devil, however, can do this: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;<span style="color:#74531f;">CanAccept</span>(<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Reservation</span>&gt;&nbsp;<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;<span style="color:#1f377f;">reservation</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservations</span>.<span style="color:#74531f;">Sum</span>(<span style="color:#1f377f;">r</span>&nbsp;=&gt;&nbsp;<span style="color:#1f377f;">r</span>.Quantity); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;!=&nbsp;7; }</pre> </p> <p> This is still not correct, but passes all tests. It does, however, look like you're getting closer to a proper implementation. </p> <h3 id="9b955ad4a2084823a9eeb668415eb696"> Reverse Transformation Priority Premise <a href="#9b955ad4a2084823a9eeb668415eb696" title="permalink">#</a> </h3> <p> If you find this process oddly familiar, it's because it resembles the <a href="https://blog.cleancoder.com/uncle-bob/2013/05/27/TheTransformationPriorityPremise.html">Transformation Priority Premise</a> (TPP), just reversed. <blockquote> <p> “As the tests get more specific, the code gets more generic.” </p> <footer><cite><a href="https://blog.cleancoder.com/uncle-bob/2013/05/27/TheTransformationPriorityPremise.html">Robert C. Martin</a></cite></footer> </blockquote> </p> <p> When I test-drive code, I often try to follow the TPP, but when I review code with tests, the code and the tests are already in place, and it's my task to assess both. </p> <p> Applying the Devil's Advocate review technique to <code>CanAccept</code>, it seems as though I'm getting closer to a proper implementation. It does, however, require more tests. As your next move you may, for instance, consider parametrising the test case that verifies what happens when capacity is insufficient: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>] [<span style="color:#2b91af;">InlineData</span>(7)] [<span style="color:#2b91af;">InlineData</span>(8)] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptOnInsufficientCapacity</span>(<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;4 &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>:&nbsp;10); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;{&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;}&nbsp;}, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">False</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> That doesn't help much, though, because this passes all tests: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;<span style="color:#74531f;">CanAccept</span>(<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Reservation</span>&gt;&nbsp;<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;<span style="color:#1f377f;">reservation</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservations</span>.<span style="color:#74531f;">Sum</span>(<span style="color:#1f377f;">r</span>&nbsp;=&gt;&nbsp;<span style="color:#1f377f;">r</span>.Quantity); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;&lt;&nbsp;7; }</pre> </p> <p> Compared to the initial, 'desired' implementation, there's at least two issues with this code: <ul> <li>It doesn't consider <code>reservation.Quantity</code></li> <li>It doesn't take into account the <code>Capacity</code> of the restaurant</li> </ul> This indicates that you're going to have to add more test cases, varying both <code>reservation.Quantity</code> and <code>Capacity</code>. The happy-path test cases already varies <code>reservation.Quantity</code> a bit, but <code>CanAcceptOnInsufficientCapacity</code> does not, so perhaps you can follow the TPP by varying <code>reservation.Quantity</code> in that method as well: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>] [<span style="color:#2b91af;">InlineData</span>(&nbsp;1,&nbsp;10)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;2,&nbsp;&nbsp;9)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;3,&nbsp;&nbsp;8)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;4,&nbsp;&nbsp;7)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;4,&nbsp;&nbsp;8)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;5,&nbsp;&nbsp;6)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;6,&nbsp;&nbsp;5)] [<span style="color:#2b91af;">InlineData</span>(10,&nbsp;&nbsp;1)] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptOnInsufficientCapacity</span>(<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">quantity</span>,&nbsp;<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">quantity</span> &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>:&nbsp;10); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;{&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;}&nbsp;}, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">False</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> This makes it harder for the Devil to come up with a malevolent implementation. Harder, but not impossible. </p> <p> It seems clear that since all test cases still use a hard-coded capacity, it ought to be possible to write an implementation that ignores the <code>Capacity</code>, but at this point I don't see a simple way to avoid looking at <code>reservation.Quantity</code>: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;<span style="color:#74531f;">CanAccept</span>(<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Reservation</span>&gt;&nbsp;<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;<span style="color:#1f377f;">reservation</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservations</span>.<span style="color:#74531f;">Sum</span>(<span style="color:#1f377f;">r</span>&nbsp;=&gt;&nbsp;<span style="color:#1f377f;">r</span>.Quantity); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;+&nbsp;<span style="color:#1f377f;">reservation</span>.Quantity&nbsp;&lt;&nbsp;11; }</pre> </p> <p> This implementation passes all the tests. The last batch of test cases forced the Devil to consider <code>reservation.Quantity</code>. This strongly implies that if you vary <code>Capacity</code> as well, the proper implementation out to emerge. </p> <h3 id="e3ed81c93d5847039ea53e7acd978d99"> Diminishing returns <a href="#e3ed81c93d5847039ea53e7acd978d99" title="permalink">#</a> </h3> <p> What happens, then, if you add just one test case with a different <code>Capacity</code>? </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>] [<span style="color:#2b91af;">InlineData</span>(&nbsp;1,&nbsp;10,&nbsp;10)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;2,&nbsp;&nbsp;9,&nbsp;10)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;3,&nbsp;&nbsp;8,&nbsp;10)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;4,&nbsp;&nbsp;7,&nbsp;10)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;4,&nbsp;&nbsp;8,&nbsp;10)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;5,&nbsp;&nbsp;6,&nbsp;10)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;6,&nbsp;&nbsp;5,&nbsp;10)] [<span style="color:#2b91af;">InlineData</span>(10,&nbsp;&nbsp;1,&nbsp;10)] [<span style="color:#2b91af;">InlineData</span>(&nbsp;1,&nbsp;&nbsp;1,&nbsp;&nbsp;1)] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptOnInsufficientCapacity</span>( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">quantity</span>, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">capacity</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">quantity</span> &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;{&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;}&nbsp;}, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">False</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> Notice that I just added one test case with a <code>Capacity</code> of <code>1</code>. </p> <p> You may think that this is about where the Devil ought to capitulate, but not so. This passes all tests: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;<span style="color:#74531f;">CanAccept</span>(<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Reservation</span>&gt;&nbsp;<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;<span style="color:#1f377f;">reservation</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;=&nbsp;0; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">foreach</span>&nbsp;(<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">r</span>&nbsp;<span style="color:#8f08c4;">in</span>&nbsp;<span style="color:#1f377f;">reservations</span>) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;=&nbsp;<span style="color:#1f377f;">r</span>.Quantity; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">break</span>; &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;+&nbsp;<span style="color:#1f377f;">reservation</span>.Quantity&nbsp;&lt;=&nbsp;Capacity; }</pre> </p> <p> Here you may feel the urge to protest. So far, all the Devil's Advocate implementations have been objectively <em>simpler</em> than the 'desired' implementation because it has involved fewer elements and has had a lower or equivalent <a href="https://en.wikipedia.org/wiki/Cyclomatic_complexity">cyclomatic complexity</a>. This new attempt to circumvent the specification seems more complex. </p> <p> It's also seems clearly ill-intentioned. Recall that the intent of the Devil's Advocate technique isn't to 'cheat' the unit tests, but rather to explore how well the test describe the desired behaviour of the system. The motivation is that it's easy to make off-by-one errors like inadvertently use <code>&lt;=</code> instead of <code>&lt;</code>. It doesn't seem quite as reasonable that a well-intentioned programmer accidentally would leave behind an implementation like the above. </p> <p> You can, however, make it <em>look</em> less complicated: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;<span style="color:#74531f;">CanAccept</span>(<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Reservation</span>&gt;&nbsp;<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;<span style="color:#1f377f;">reservation</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservations</span>.<span style="color:#74531f;">Select</span>(<span style="color:#1f377f;">r</span>&nbsp;=&gt;&nbsp;<span style="color:#1f377f;">r</span>.Quantity).<span style="color:#74531f;">FirstOrDefault</span>(); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;+&nbsp;<span style="color:#1f377f;">reservation</span>.Quantity&nbsp;&lt;=&nbsp;Capacity; }</pre> </p> <p> You could argue that this still looks intentionally wrong, but I've seen much code that looks like this. It seems to me that there's a kind of programmer who seems generally uncomfortable thinking in collections; they seem to subconsciously gravitate towards code that deals with singular objects. Code that attempts to get 'the' value out of a collection is, unfortunately, not that uncommon. </p> <p> Still, you might think that at this point, you've added enough test cases. That's reasonable. </p> <p> The Devil's Advocate technique isn't an <em>algorithm</em>; it has no deterministic exit criterion. It's just a heuristic that I use to explore the quality of tests. There comes a point where subjectively, I judge that the test cases <em>sufficiently</em> describe the desired behaviour. </p> <p> You may find that we've reached that point now. You could, for example, argue that in order to calculate <code>reservedSeats</code>, <code>reservations.Sum(r =&gt; r.Quantity)</code> is simpler than <code>reservations.Select(r =&gt; r.Quantity).FirstOrDefault()</code>. I'd be inclined to agree. </p> <p> There's diminishing returns to the Devil's Advocate technique. Once you find that the gains from insisting on intentionally pernicious implementations are smaller than the effort required to add more test cases, it's time to stop and commit to the test cases now in place. </p> <h3 id="609ddb35ae364efbbfb7965a646d857e"> Test case variability <a href="#609ddb35ae364efbbfb7965a646d857e" title="permalink">#</a> </h3> <p> Tests specify desired behaviour. If the tests contain less variability than the code they cover, then how can you be certain that the implementation code is correct? </p> <p> The discussion now moves into territory where I usually exercise a great deal of judgement. Read the following for inspiration, not as rigid instructions. My intent with the following is not to imply that you must always go to like extremes, but simply to demonstrate what you <em>can</em> do. Depending on circumstances (such as the cost of a defect in production), I may choose to do the following, and sometimes I may choose to skip it. </p> <p> If you consider the original implementation of <code>CanAccept</code> at the top of the article, notice that it works with <code>reservations</code> of indefinite size. If you think of <code>reservations</code> as a finite collection, it can contain zero, one, two, ten, or hundreds of elements. Yet, no test case goes beyond a single existing reservation. This is, I think, a disconnect. The tests come not even close to the degree of variability that the method can handle. If this is a piece of mission-critical software, that could be a cause for concern. </p> <p> You should add some test cases where there's two, three, or more existing reservations. People often don't do that because it seems that you'd now have to write a test method that exercises one or more test cases with two existing reservations: </p> <p> <pre>[<span style="color:#2b91af;">Fact</span>] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptWithTwoPriorReservations</span>() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;4 &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>:&nbsp;10); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;{&nbsp;Quantity&nbsp;=&nbsp;4&nbsp;},&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;{&nbsp;Quantity&nbsp;=&nbsp;1&nbsp;}&nbsp;}, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">True</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> While this method now covers the two-existing-reservations test case, you need one to cover the three-existing-reservations test case, and so on. This seems repetitive, and probably bothers you at more than one level: <ul> <li>It's just plain tedious to have to add that kind of variability</li> <li>It seems to violate the <a href="https://en.wikipedia.org/wiki/Don%27t_repeat_yourself">DRY principle</a></li> </ul> I don't hold the DRY principle as an absolute that must always be followed, but it often indicates a maintainability problem. I think this is the case here, because the new <code>CanAcceptWithTwoPriorReservations</code> test method looks a lot like the previous <code>CanAcceptWithOnePriorReservation</code> method. If someone makes changes to the <code>MaîtreD</code> class, they would have to go and revisit all those test methods. </p> <p> What you can do instead is to parametrise the key values of the collection(s) in question. While you can't put collections of objects in <code>[InlineData]</code> attributes, you <em>can</em> put arrays of constants. For existing reservations, the key values are the quantities, so supply an array of integers as a test argument: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>] [<span style="color:#2b91af;">InlineData</span>(&nbsp;4,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:blue;">int</span>[0])] [<span style="color:#2b91af;">InlineData</span>(10,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:blue;">int</span>[0])] [<span style="color:#2b91af;">InlineData</span>(&nbsp;4,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;4&nbsp;})] [<span style="color:#2b91af;">InlineData</span>(&nbsp;4,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;4,&nbsp;1&nbsp;})] [<span style="color:#2b91af;">InlineData</span>(&nbsp;2,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;2,&nbsp;1,&nbsp;3,&nbsp;2&nbsp;})] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptWhenCapacityIsSufficient</span>(<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">quantity</span>,&nbsp;<span style="color:blue;">int</span>[]&nbsp;<span style="color:#1f377f;">reservationQantities</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">quantity</span> &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>:&nbsp;10); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservations</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservationQantities</span>.<span style="color:#74531f;">Select</span>(<span style="color:#1f377f;">q</span>&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;{&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">q</span>&nbsp;}); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>(<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">True</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> This single test method replaces the previous three 'happy path' test methods. The first four <code>[InlineData]</code> annotations reproduce the previous test cases, whereas the fifth <code>[InlineData]</code> annotation adds a new test case with four existing reservations. </p> <p> I gave the method a new name to better reflect the more general nature of it. </p> <p> Notice that the <code>CanAcceptWhenCapacityIsSufficient</code> method uses <code>Select</code> to turn the array of integers into a collection of <code>Reservation</code> objects. </p> <p> You may think that I cheated, since I didn't supply any other values, such as the <code>Date</code> property, to the existing reservations. This is easily addressed: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>] [<span style="color:#2b91af;">InlineData</span>(&nbsp;4,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:blue;">int</span>[0])] [<span style="color:#2b91af;">InlineData</span>(10,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:blue;">int</span>[0])] [<span style="color:#2b91af;">InlineData</span>(&nbsp;4,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;4&nbsp;})] [<span style="color:#2b91af;">InlineData</span>(&nbsp;4,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;4,&nbsp;1&nbsp;})] [<span style="color:#2b91af;">InlineData</span>(&nbsp;2,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;2,&nbsp;1,&nbsp;3,&nbsp;2&nbsp;})] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptWhenCapacityIsSufficient</span>(<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">quantity</span>,&nbsp;<span style="color:blue;">int</span>[]&nbsp;<span style="color:#1f377f;">reservationQantities</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">date</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:#1f377f;">date</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">quantity</span> &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>:&nbsp;10); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservations</span>&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#1f377f;">reservationQantities</span>.<span style="color:#74531f;">Select</span>(<span style="color:#1f377f;">q</span>&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;{&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">q</span>,&nbsp;Date&nbsp;=&nbsp;<span style="color:#1f377f;">date</span>&nbsp;}); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>(<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">True</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> The only change compared to before is that <code>date</code> is now a variable assigned not only to <code>reservation</code>, but also to all the <code>Reservation</code> objects in <code>reservations</code>. </p> <h3 id="0564ebd7cafc44f4ba6ad017e3f0d0ce"> Towards property-based testing <a href="#0564ebd7cafc44f4ba6ad017e3f0d0ce" title="permalink">#</a> </h3> <p> Looking at a test method like <code>CanAcceptWhenCapacityIsSufficient</code> it should bother you that the <code>capacity</code> is still hard-coded. Why don't you make that a test argument as well? </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>] [<span style="color:#2b91af;">InlineData</span>(10,&nbsp;&nbsp;4,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:blue;">int</span>[0])] [<span style="color:#2b91af;">InlineData</span>(10,&nbsp;10,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:blue;">int</span>[0])] [<span style="color:#2b91af;">InlineData</span>(10,&nbsp;&nbsp;4,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;4&nbsp;})] [<span style="color:#2b91af;">InlineData</span>(10,&nbsp;&nbsp;4,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;4,&nbsp;1&nbsp;})] [<span style="color:#2b91af;">InlineData</span>(10,&nbsp;&nbsp;2,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;2,&nbsp;1,&nbsp;3,&nbsp;2&nbsp;})] [<span style="color:#2b91af;">InlineData</span>(20,&nbsp;10,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;2,&nbsp;2,&nbsp;2,&nbsp;2&nbsp;})] [<span style="color:#2b91af;">InlineData</span>(20,&nbsp;&nbsp;4,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;2,&nbsp;2,&nbsp;4,&nbsp;1,&nbsp;3,&nbsp;3&nbsp;})] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptWhenCapacityIsSufficient</span>( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">capacity</span>, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">quantity</span>, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">int</span>[]&nbsp;<span style="color:#1f377f;">reservationQantities</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">date</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:#1f377f;">date</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">quantity</span> &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservations</span>&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#1f377f;">reservationQantities</span>.<span style="color:#74531f;">Select</span>(<span style="color:#1f377f;">q</span>&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;{&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">q</span>,&nbsp;Date&nbsp;=&nbsp;<span style="color:#1f377f;">date</span>&nbsp;}); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>(<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">True</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> The first five <code>[InlineData]</code> annotations just reproduce the test cases that were already present, whereas the bottom two annotations are new test cases with another <code>capacity</code>. </p> <p> How do I come up with new test cases? It's easy: In the happy-path case, the sum of existing reservation quantities, plus the requested quantity, must be less than or equal to the <code>capacity</code>. </p> <p> It sometimes helps to slightly reframe the test method. If you allow the collection of existing reservations to be the most variable element in the test method, you can express the other values relative to that input. For example, instead of supplying the <code>capacity</code> as an absolute number, you can express a test case's capacity in relation to the existing reservations: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>] [<span style="color:#2b91af;">InlineData</span>(6,&nbsp;&nbsp;4,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:blue;">int</span>[0])] [<span style="color:#2b91af;">InlineData</span>(0,&nbsp;10,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:blue;">int</span>[0])] [<span style="color:#2b91af;">InlineData</span>(2,&nbsp;&nbsp;4,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;4&nbsp;})] [<span style="color:#2b91af;">InlineData</span>(1,&nbsp;&nbsp;4,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;4,&nbsp;1&nbsp;})] [<span style="color:#2b91af;">InlineData</span>(0,&nbsp;&nbsp;2,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;2,&nbsp;1,&nbsp;3,&nbsp;2&nbsp;})] [<span style="color:#2b91af;">InlineData</span>(2,&nbsp;10,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;2,&nbsp;2,&nbsp;2,&nbsp;2&nbsp;})] [<span style="color:#2b91af;">InlineData</span>(1,&nbsp;&nbsp;4,&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;2,&nbsp;2,&nbsp;4,&nbsp;1,&nbsp;3,&nbsp;3&nbsp;})] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptWhenCapacityIsSufficient</span>( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">capacitySurplus</span>, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">int</span>&nbsp;<span style="color:#1f377f;">quantity</span>, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">int</span>[]&nbsp;<span style="color:#1f377f;">reservationQantities</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">date</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:#1f377f;">date</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">quantity</span> &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservationQantities</span>.<span style="color:#74531f;">Sum</span>(); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">capacity</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;+&nbsp;<span style="color:#1f377f;">quantity</span>&nbsp;+&nbsp;<span style="color:#1f377f;">capacitySurplus</span>; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservations</span>&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#1f377f;">reservationQantities</span>.<span style="color:#74531f;">Select</span>(<span style="color:#1f377f;">q</span>&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;{&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">q</span>,&nbsp;Date&nbsp;=&nbsp;<span style="color:#1f377f;">date</span>&nbsp;}); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>(<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">True</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> Notice that the value supplied as a test argument is now named <code>capacitySurplus</code>. This represents the surplus capacity for each test case. For example, in the first test case, the <code>capacity</code> was previously supplied as the absolute number <code>10</code>. The requested quantity is <code>4</code>, and since there's no prior reservations in that test case, the capacity surplus, after accepting the reservation, is <code>6</code>. </p> <p> Likewise, in the second test case, the requested quantity is <code>10</code>, and since the absolute capacity is also <code>10</code>, when you reframe the test case, the surplus capacity, after accepting the reservation, is <code>0</code>. </p> <p> This seems odd if you aren't used to it. You'd probably intuitively think of a restaurant's <code>Capacity</code> as 'the most absolute' number, in that it's often a number that originates from physical constraints. </p> <p> When you're looking for test cases, however, you aren't looking for test cases for a particular restaurant. You're looking for test cases for an arbitrary restaurant. In other words, you're looking for test inputs that belong to the same <em>equivalence class</em>. </p> <h3 id="174e2338027e4f3ca0b84dd0fb6adc5f"> Property-based testing <a href="#174e2338027e4f3ca0b84dd0fb6adc5f" title="permalink">#</a> </h3> <p> I haven't explicitly stated this yet, but both the <code>capacity</code> and each reservation <code>Quantity</code> should be a positive number. This should really have been <a href="/2015/01/19/from-primitive-obsession-to-domain-modelling">captured as a proper domain object</a>, but I chose to keep these values as primitive integers in order to not complicate the example too much. </p> <p> If you look at the test parameters for the latest incarnation of <code>CanAcceptWhenCapacityIsSufficient</code>, you may now observe the following: <ul> <li><code>capacitySurplus</code> can be an arbitrary non-negative number</li> <li><code>quantity</code> can be an arbitrary positive number</li> <li><code>reservationQantities</code> can be an arbitrary array of positive numbers, including the empty array</li> </ul> This isn't too hard to express with, say, <a href="https://fscheck.github.io/FsCheck">FsCheck</a> (2.14.0): </p> <p> <pre>[<span style="color:#2b91af;">Property</span>] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;<span style="color:#74531f;">CanAcceptWhenCapacityIsSufficient</span>( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">NonNegativeInt</span>&nbsp;<span style="color:#1f377f;">capacitySurplus</span>, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">PositiveInt</span>&nbsp;<span style="color:#1f377f;">quantity</span>, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">PositiveInt</span>[]&nbsp;<span style="color:#1f377f;">reservationQantities</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">date</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(2018,&nbsp;8,&nbsp;30); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservation</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Date&nbsp;=&nbsp;<span style="color:#1f377f;">date</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">quantity</span>.Item &nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservationQantities</span>.<span style="color:#74531f;">Sum</span>(<span style="color:#1f377f;">x</span>&nbsp;=&gt;&nbsp;<span style="color:#1f377f;">x</span>.Item); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">capacity</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;+&nbsp;<span style="color:#1f377f;">quantity</span>.Item&nbsp;+&nbsp;<span style="color:#1f377f;">capacitySurplus</span>.Item; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">sut</span>&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">MaîtreD</span>(<span style="color:#1f377f;">capacity</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservations</span>&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#1f377f;">reservationQantities</span>.<span style="color:#74531f;">Select</span>(<span style="color:#1f377f;">q</span>&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;{&nbsp;Quantity&nbsp;=&nbsp;<span style="color:#1f377f;">q</span>.Item,&nbsp;Date&nbsp;=&nbsp;<span style="color:#1f377f;">date</span>&nbsp;}); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">actual</span>&nbsp;=&nbsp;<span style="color:#1f377f;">sut</span>.<span style="color:#74531f;">CanAccept</span>(<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#1f377f;">reservation</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.<span style="color:#74531f;">True</span>(<span style="color:#1f377f;">actual</span>); }</pre> </p> <p> This refactoring takes advantage of FsCheck's built-in wrapper types <code>NonNegativeInt</code> and <code>PositiveInt</code>. If you'd like an introduction to FsCheck, you could watch my <a href="/property-based-testing-intro">Introduction to Property-based Testing with F#</a> Pluralsight course. </p> <p> By default, FsCheck runs each property 100 times, so now, instead of seven test cases, you now have 100. </p> <h3 id="1eddf5bb91324b18880c78d1f1825ea6"> Limits to the Devil's Advocate technique <a href="#1eddf5bb91324b18880c78d1f1825ea6" title="permalink">#</a> </h3> <p> There's a limit to the Devil's Advocate technique. Unless you're working with <a href="/2015/02/23/property-based-testing-without-a-property-based-testing-framework">a problem where you can exhaust the entire domain of possible test cases</a>, your testing strategy is always going to be a sampling strategy. You run your automated tests with either hard-coded values or randomly generated values, but regardless, a test run isn't going to cover all possible input combinations. </p> <p> For example, a truly hostile Devil could make this change to the <code>CanAccept</code> method: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;<span style="color:#74531f;">CanAccept</span>(<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Reservation</span>&gt;&nbsp;<span style="color:#1f377f;">reservations</span>,&nbsp;<span style="color:#2b91af;">Reservation</span>&nbsp;<span style="color:#1f377f;">reservation</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">if</span>&nbsp;(<span style="color:#1f377f;">reservation</span>.Quantity&nbsp;==&nbsp;3953911) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:blue;">true</span>; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;=&nbsp;<span style="color:#1f377f;">reservations</span>.<span style="color:#74531f;">Sum</span>(<span style="color:#1f377f;">r</span>&nbsp;=&gt;&nbsp;<span style="color:#1f377f;">r</span>.Quantity); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#8f08c4;">return</span>&nbsp;<span style="color:#1f377f;">reservedSeats</span>&nbsp;+&nbsp;<span style="color:#1f377f;">reservation</span>.Quantity&nbsp;&lt;=&nbsp;Capacity; }</pre> </p> <p> Even if you increase the number of test cases that FsCheck generates to, say, 100,000, it's unlikely to find the poisonous branch. The chance of randomly generating a <code>quantity</code> of <em>exactly</em> <code>3953911</code> isn't that great. </p> <p> The Devil's Advocate technique doesn't guarantee that you'll have enough test cases to protect yourself against all sorts of odd defects. It does, however, still work well as an analysis tool to figure out if there's 'enough' test cases. </p> <h3 id="6ad7a48fd0c04f91af236d99f0722617"> Conclusion <a href="#6ad7a48fd0c04f91af236d99f0722617" title="permalink">#</a> </h3> <p> The Devil's Advocate technique is a heuristic you can use to evaluate whether more test cases would improve confidence in the test suite. You can use it to review existing (test) code, but you can also use it as inspiration for new test cases that you should consider adding. </p> <p> The technique is to deliberately implement the system under test incorrectly. The more incorrect you can make it, the more test cases you'll be likely to have to add. </p> <p> When there's only a few test cases, you can probably get away with a decidedly unsound implementation that still passes all tests. These are often simpler than the 'intended' implementation. In this phase of applying the heuristic, this clearly demonstrates the need for more test cases. </p> <p> At a later stage, you'll have to go deliberately out of your way to produce a wrong implementation that still passes all tests. When that happens, it may be time to stop. </p> <p> The intent of the technique is to uncover how many test cases you need to protect against common defects in the future. Thus, it's not a measure of <em>current</em> code coverage. </p> </div> <div id="comments"> <hr> <h2 id="comments-header"> Comments </h2> <div class="comment" id="fd53c72c360b42999b87c87649460e78"> <div class="comment-author">Tyson Williams</div> <div class="comment-content"> <blockquote> <p> When there's only a few test cases, you can probably get away with a decidedly unsound implementation that still passes all tests. These are often simpler than the 'intended' implementation. In this phase of applying the heuristic, this clearly demonstrates the need for more test cases. </p> <p> At a later stage, you'll have to go deliberately out of your way to produce a wrong implementation that still passes all tests. When that happens, it may be time to stop. </p> </blockquote> <p> I like to think of this behavior as a phrase transition. </p> <blockquote> Unless you're working with a problem where you can exhaust the entire domain of possible test cases, your testing strategy is always going to be a sampling strategy. </blockquote> <p> I agree with this in practice, but it is not always true in theory. A counter eaxample is <a href="https://en.wikipedia.org/wiki/Polynomial_interpolation">polynomial interpolation</a>. </p> <p> Normally we think of a polynomial in an indeterminate <code>x</code> of degree <code>n</code> as being specified by a list of <code>n + 1</code> coefficients, where the <code>i</code>th coefficient is the coefficient of <code>x<sup>i</sup></code>. Evaluating this polynomial given a value for <code>x</code> is easy; it just involves exponentiation, multiplication, and addition. Polynomial evaluation has a conceptual inverse called polynomial interpolation. In this direction, the input is evaluations at <code>n + 1</code> points in "general position" and the output is the <code>n + 1</code> coefficients. For example, a line is a polynomial of degree <code>1</code> and two points are in general position if they are not the same point. This is commonly expressed the phrase "Any two (distinct) points defines a line." Three points are in general position if they are not co-linear, where co-linear means that all three points are on the same line. In general, <code>n + 1</code> points are in general position if they are not all on the same polynomial of degree <code>n</code>. <p> <p> Anyway, here is the point. If a pure function is known to implement some polynomial of degree (at most) <code>n</code>, then even if the domain is infinite, there exists <code>n + 1</code> inputs such that it is sufficient to test this function for correctness on those inputs. </p> <p> This is why I think the phrase transition in the Devil's advocate testing is critical. There is some objective measure of complexity of the function under test (such as cyclomatic complexity), and we have an intuitive sense that a certain number of tests is sufficient for testing functions with that complexity. If the Devil is allowed to add monomials to the polynomial (or, heaven forbid, modify the implementation so that it is not a polynomial), then any finite number of tests can be circumvented. If instead the Devil is only allowed to modify the coefficients of the polynomial, then we have a winning strategy. </p> <blockquote> Here you may feel the urge to protest. So far, all the Devil's Advocate implementations have been objectively simpler than the 'desired' implementation because it has involved fewer elements and has had a lower or equivalent cyclomatic complexity. This new attempt to circumvent the specification seems more complex. </blockquote> <p> I think it would be exceedingly intersting if you can formally define what you mean here by "objectively". In the case of a polynomial (and speaking slightly roughly), changing the "first" nonzero coefficient to <code>0</code> decreases the complexity (i.e. the degree of the polynomial) while any other change to that coefficient or any change to any other coefficient maintains the complexity. </p> </div> <div class="comment-date">2019-10-25 01:32 UTC</div> </div> <div class="comment" id="cb15452b8c96429998efd50b67373da3"> <div class="comment-author"><a href="/">Mark Seemann</a></div> <div class="comment-content"> <p> Tyson, thank you for writing. What I meant by <em>objectively simpler</em> I partially explain in the same paragraph. I consider cyclomatic complexity one of hardly any useful measurements in software development. As I also imply in the article, I consider Robert C. Martin's <em>Transformation Priority Premise</em> to include a good ranking of code constructs, e.g. that using a constant is simpler than using a variable, and so on. </p> <p> I don't think you need to reach for polynomial interpolation in order to make your point. Just consider a function that returns a constant value, like this one: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">string</span>&nbsp;<span style="color:#74531f;">Foo</span>(<span style="color:blue;">int</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">i</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#8f08c4;">return</span>&nbsp;<span style="color:#a31515;">&quot;foo&quot;</span>; }</pre> </p> <p> You can make a similar argument about this function: You only need a single test value in order to demonstrate that it works as intended. I suppose you could view that as a zero-degree polynomial. </p> <p> Beyond what you think of as the <em>phase transition</em> I sometimes try to see what happens if I slightly <em>increase</em> the complexity of a function. For the <code>Foo</code> function, it could be a change like this: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">string</span>&nbsp;<span style="color:#74531f;">Foo</span>(<span style="color:blue;">int</span>&nbsp;<span style="font-weight:bold;color:#1f377f;">i</span>) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#8f08c4;">if</span>&nbsp;(<span style="font-weight:bold;color:#1f377f;">i</span>&nbsp;&lt;&nbsp;-1000) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#8f08c4;">return</span>&nbsp;<span style="color:#a31515;">&quot;bar&quot;</span>; &nbsp;&nbsp;&nbsp;&nbsp;<span style="font-weight:bold;color:#8f08c4;">return</span>&nbsp;<span style="color:#a31515;">&quot;foo&quot;</span>; }</pre> </p> <p> Unless you just happened to pick a number less than <code>-1000</code> for your test value, your test will not discover such a change. </p> <p> Your argument attempts to guard against that sort of change by assuming that we can somehow 'forbid' a change from a polynomial to something irregular. Real code doesn't work that way. Real code is rarely a continuous function, but rather discrete. That's the reason we have a concept such as <em>edge case</em>, because code branches at discrete values. </p> <p> A polynomial is a single function, regardless of degree. Implemented in code, it'll have a cyclomatic complexity of 1. That may not even be worth testing, because you'd essentially only be reproducing the implementation code in your test. </p> <p> The purpose of the Devil's Advocate technique isn't to demonstrate correctness; that's what unit tests are for. The purpose of the Devil's Advocate technique is to critique the tests. </p> <p> In reality, I never imagine that some malicious developer gains access to the source code. On the other hand, we all make mistakes, and I try to imagine what a likely mistake might look like. </p> </div> <div class="comment-date">2019-10-26 3:57 UTC</div> </div> </div> <hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. 10x developers https://blog.ploeh.dk/2019/09/30/10x-developers 2019-09-30T06:56:00+00:00 Mark Seemann <div id="post"> <p> <em>Do 10x developers exist? I believe that they do, but not like you may think.</em> </p> <p> The notion that some software developers are ten times (10x) as productive as 'normal' developers is decades old. Once in a while, the discussion resurfaces. It's a controversial subject, but something I've been thinking about for years, so I thought that I'd share my perspective because I don't see anyone else arguing from this position. </p> <p> While I'll try to explain my reasoning, I'll make no attempt at passing this off as anything but my current, subjective viewpoint. Please leave a comment if you have something to add. </p> <h3 id="11f6ec960f694758892bff549eb79d59"> Perspective <a href="#11f6ec960f694758892bff549eb79d59" title="permalink">#</a> </h3> <p> Meet Yohan. You've probably had a colleague like him. He's one of those software developers who gets things done, who never says no when the business asks him to help them out, who always respond with a smile to any request. </p> <p> I've had a few colleagues like Yohan in my career. It can be enlightening overhearing non-technical stakeholders discuss software developers: </p> <p> <strong>Alice:</strong> Yohan is such a dear; he helped me out with that feature on the web site, you know... </p> <p> <strong>Bob:</strong> Yes, he's a real go-getter. All the other programmers just say no and look pissed when I approach them about anything. </p> <p> <strong>Alice:</strong> Yohan always says yes, and he gets things done. He's a real 10x developer. </p> <p> <strong>Bob:</strong> We're so lucky we have him... </p> <p> Overhearing such a conversation can be frustrating. Yohan is your colleague, and you've just about had enough of him. Yohan is one of those developers who'll surround all code with a <code>try-catch</code> block, because then there'll be no exceptions in production. Yohan will make changes directly to the production system and tell no-one. Yohan will copy and paste code. Yohan will put business logic in database triggers, or rewrite logs, or use email as a messaging system, or call, parse, and run HTML-embedded JavaScript code on back-end servers. All 'because it's faster and provides more business value.' </p> <p> Yohan is a 10x developer. </p> <p> You, and the rest of your team, get nothing done. </p> <p> You get nothing done because you waste all your time cleaning up the trail of garbage and technical debt Yohan leaves in his wake. </p> <p> Business stakeholders may view Yohan as being orders of magnitude more productive than other developers, because most programming work is invisible and intangible. Whether or not someone is a 10x developer is highly subjective, and depends on perspective. </p> <h3 id="57c36b775bca4e848cadf50182028191"> Context <a href="#57c36b775bca4e848cadf50182028191" title="permalink">#</a> </h3> <p> The notion that some people are orders of magnitude more productive than the 'baseline' programmer has other problems. It implicitly assumes that a 'baseline' programmer exists in the first place. Modern software development, however, is specialised. </p> <p> As an example, I've been doing test-driven, ASP.NET-based C# server-side enterprise development for decades. Drop me into a project with my favourite stack and watch me go. On the other hand, try asking me to develop a game for the Sony PlayStation, and watch me stall. </p> <p> Clearly, then, I'm a 10x developer, for the tautological reason that I'm much better at the things that I'm good at than the things I'm not good at. </p> <p> Even the greatest <a href="https://en.wikipedia.org/wiki/R_(programming_language)">R</a> developer is unlikely to be of much help on your next <a href="https://en.wikipedia.org/wiki/COBOL">COBOL</a> project. </p> <p> As always, context matters. You can be a great programmer in a particular context, and suck in another. </p> <p> This isn't limited to technology stacks. Some people prefer co-location, while others work best by themselves. Some people are detail-oriented, while others like to look at the big picture. Some people do their best work early in the morning, and others late at night. </p> <p> And some teams of 'mediocre' programmers outperform all-star teams. (This, incidentally, is a phenomenon also <a href="https://en.wikipedia.org/wiki/UEFA_Euro_1992">sometimes seen</a> in professional <a href="https://en.wikipedia.org/wiki/Association_football">Soccer</a>.) </p> <h3 id="34e113a19c8040b0bcb38f7abf1b532d"> Evidence <a href="#34e113a19c8040b0bcb38f7abf1b532d" title="permalink">#</a> </h3> <p> Unfortunately, as I explain in my <a href="https://cleancoders.com/video-details/humane-code-real-episode-1">Humane Code</a> video, I believe that you can't measure software development productivity. Thus, the notion of a 10x developer is subjective. </p> <p> The original idea, however, is decades old, and seems, at first glance, to originate in a 'study'. If you're curious about its origins, I can't recommend <a href="http://bit.ly/leprechauns-of-software-engineering">The Leprechauns of Software Engineering</a> enough. In that book, Laurent Bossavit explains just how insubstantial the evidence is. </p> <p> If the evidence is so weak, then why does the idea that 10x developers exist keep coming back? </p> <h3 id="6988798f60bf4622808b58a3e7f55bce"> 0x developers <a href="#6988798f60bf4622808b58a3e7f55bce" title="permalink">#</a> </h3> <p> I think that the reason that the belief is recurring is that (subjectively) <em>it seems so evident</em>. Barring confirmation bias, I'm sure everyone has encountered a team member that never seemed to get anything done. </p> <p> I know that I've certainly had that experience from time to time. </p> <p> The first job I had, I hated. I just couldn't muster any enthusiasm for the work, and I'd postpone and drag out as long as possible even the simplest task. That wasn't very mature, but I was 25 and it was my first job, and I didn't know how to handle the situation I found myself in. I'm sure that my colleagues back then found that I didn't pull my part. I didn't, and I'm not proud of it, but it's true. </p> <p> I believe now that I was in the wrong context. It wasn't that I was incapable of doing the job, but at that time in my career, I absolutely loathed it, and for that reason, I wasn't productive. </p> <p> Another time, I had a colleague who seemed incapable of producing anything that helped us achieve our goals. I was concerned that I'd <a href="https://en.wikipedia.org/wiki/Bozo_bit">flipped the bozo bit</a> on that colleague, so I started to collect evidence. Our Git repository had few commits from that colleague, and the few that I could find I knew had been made in collaboration with another team member. We shared an office, and I had a pretty good idea about who worked together with whom when. </p> <p> This colleague spent a lot of time talking to other people. Us, other stakeholders, or any hapless victim who didn't escape in time. Based on these meetings and discussions, we'd hear about all sorts of ideas for improvements for our code or development process, but nothing would be implemented, and rarely did it have any relevance to what we were trying to accomplish. </p> <p> I've met programmers who get nothing done more than once. Sometimes, like the above story, they're boisterous bluffs, but most often, they just sit quietly in their corner and fidget with who knows what. </p> <p> Based on the above, mind you, I'm not saying that these people are necessarily incompetent (although I suspect that some are). They might also just find themselves in a wrong context, like I did in my first job. </p> <p> It seems clear to me, then, that there's such a thing as a <em>0x developer</em>. This is a developer who gets zero times (0x) as much done as the 'average' developer. </p> <p> For that reason it seems evident to me that 10x developers exist. Any developer who regularly manages to get code deployed to production is not only ten times, but infinitely more productive than 0x developers. </p> <p> It gets worse, though. </p> <h3 id="fe07476743704027acab9c7b949bc3a4"> −nx developers <a href="#fe07476743704027acab9c7b949bc3a4" title="permalink">#</a> </h3> <p> Not only is it my experience that 0x developers exist, I also believe that I've met more than one <em>−nx developer</em>. These are developers who are <em>minus n times</em> 'more' productive than the 'baseline' developer. In other words, they are software developers who have negative productivity. </p> <p> I've never met anyone who I suspected of deliberately sabotaging our efforts; they always seem well-meaning, but some people can produce more mess than three colleagues can clean up. Yohan, above, is such an archetype. </p> <p> One colleague I had, long ago, was so bad that the rest of the team deliberately compartmentalised him/her. We'd ask him/her to work on an isolated piece of the system, knowing that (s)he would be assigned to another project after four months. We then secretly planned to throw away the code once (s)he was gone, and rewrite it. I don't know if that was the right decision, but since we had padded all other estimates accordingly, we made our deadlines without more than the usual overruns. </p> <p> If you accept the assertion that −nx developers exist, then clearly, anyone who gets anything done at all is an <em>∞x developer</em>. </p> <h3 id="8257729b743641d0afa1c2ddbfe1b0df"> Summary <a href="#8257729b743641d0afa1c2ddbfe1b0df" title="permalink">#</a> </h3> <p> 10x developers exist, but not in the way that people normally interpret the term. </p> <p> 10x developers exist because there's great variability in (perceived) productivity. Much of the variability is context-dependent, so it's less clear if some people are just 'better at programming' than others. Still, when we consider that people like <a href="https://en.wikipedia.org/wiki/Linus_Torvalds">Linus Torvalds</a> exist, it seems compelling that this might be the case. </p> <p> Most of the variability, however, I think correlates with environment. Are you working in a technology stack with which you're comfortable? Do you like what you're doing? Do you like your colleagues? Do you like your hours? Do you like your working environment? </p> <p> Still, even if we could control for all of those variables, we might still find that some people get stuff done, and some people don't. The people who get anything done are ∞x developers. </p> <p> Employers and non-technical start-up founders sometimes look for the 10x unicorns, just like they look for <em>rock star developers</em>. <blockquote> <p> "To really confuse recruiters, someone should make a programming language called Rockstar." </p> <footer><cite><a href="https://twitter.com/paulstovell/status/1013960369465782273">Paul Stovell</a></cite></footer> </blockquote> The above tweet inspired <a href="http://www.dylanbeattie.net">Dylan Beattie</a> to create <a href="https://codewithrockstar.com">the Rockstar programming language</a>. </p> <p> Perhaps we should also create a <em>10x</em> programming language, so that we could put <em>certified Rockstar programmer, 10x developer</em> on our resumes. </p> </div> <hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Unit testing wai applications https://blog.ploeh.dk/2019/09/23/unit-testing-wai-applications 2019-09-23T06:35:00+00:00 Mark Seemann <div id="post"> <p> <em>One way to unit test a wai application with the API provided by Network.Wai.Test.</em> </p> <p> I'm currently developing a REST API in <a href="https://www.haskell.org">Haskell</a> using <a href="https://haskell-servant.readthedocs.io">Servant</a>, and I'd like to test the HTTP API as well as the functions that I use to compose it. The Servant documentation, as well as the <em>servant</em> <a href="https://haskellstack.org">Stack</a> template, uses <a href="http://hackage.haskell.org/package/hspec">hspec</a> to drive the tests. </p> <p> I tried to develop my code with hspec, but I found it confusing and inflexible. It's possible that I only found it inflexible because I didn't understand it well enough, but I don't think you can argue with my experience of finding it confusing. </p> <p> I prefer a combination of <a href="https://hackage.haskell.org/package/HUnit">HUnit</a> and <a href="http://hackage.haskell.org/package/QuickCheck">QuickCheck</a>. It turns out that it's possible to test a <a href="http://hackage.haskell.org/package/wai">wai</a> application (including Servant) using only those test libraries. </p> <h3 id="1c7a7365bd0c425e85691625d00adcd0"> Testable HTTP requests <a href="#1c7a7365bd0c425e85691625d00adcd0" title="permalink">#</a> </h3> <p> When testing against the HTTP API itself, you want something that can simulate the HTTP traffic. That capability is provided by <a href="http://hackage.haskell.org/package/wai-extra/docs/Network-Wai-Test.html">Network.Wai.Test</a>. At first, however, it wasn't entirely clear to me how that library works, but I could see that the Servant-recommended <a href="http://hackage.haskell.org/package/hspec-wai/docs/Test-Hspec-Wai.html">Test.Hspec.Wai</a> is just a thin wrapper over <em>Network.Wai.Test</em> (notice how <a href="/2019/07/01/yes-silver-bullet">open source makes such research much easier</a>). </p> <p> It turns out that <em>Network.Wai.Test</em> enables you to run your tests in a <code>Session</code> monad. You can, for example, define a simple HTTP GET request like this: </p> <p> <pre><span style="color:blue;">import</span>&nbsp;<span style="color:blue;">qualified</span>&nbsp;Data.ByteString&nbsp;<span style="color:blue;">as</span>&nbsp;BS <span style="color:blue;">import</span>&nbsp;<span style="color:blue;">qualified</span>&nbsp;Data.ByteString.Lazy&nbsp;<span style="color:blue;">as</span>&nbsp;LBS <span style="color:blue;">import</span>&nbsp;Network.HTTP.Types <span style="color:blue;">import</span>&nbsp;Network.Wai <span style="color:blue;">import</span>&nbsp;Network.Wai.Test <span style="color:#2b91af;">get</span>&nbsp;::&nbsp;<span style="color:blue;">BS</span>.<span style="color:blue;">ByteString</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Session</span>&nbsp;<span style="color:blue;">SResponse</span> get&nbsp;url&nbsp;=&nbsp;request&nbsp;$&nbsp;setPath&nbsp;defaultRequest&nbsp;{&nbsp;requestMethod&nbsp;=&nbsp;methodGet&nbsp;}&nbsp;url </pre> </p> <p> This <code>get</code> function takes a <code>url</code> and returns a <code>Session SResponse</code>. It uses the <code>defaultRequest</code>, so it doesn't set any specific HTTP headers. </p> <p> For HTTP POST requests, I needed a function that'd POST a JSON document to a particular URL. For that purpose, I had to do a little more work: </p> <p> <pre><span style="color:#2b91af;">postJSON</span>&nbsp;::&nbsp;<span style="color:blue;">BS</span>.<span style="color:blue;">ByteString</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">LBS</span>.<span style="color:blue;">ByteString</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Session</span>&nbsp;<span style="color:blue;">SResponse</span> postJSON&nbsp;url&nbsp;json&nbsp;=&nbsp;srequest&nbsp;$&nbsp;SRequest&nbsp;req&nbsp;json &nbsp;&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;&nbsp;&nbsp;req&nbsp;=&nbsp;setPath&nbsp;defaultRequest &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{&nbsp;requestMethod&nbsp;=&nbsp;methodPost &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;,&nbsp;requestHeaders&nbsp;=&nbsp;[(hContentType,&nbsp;<span style="color:#a31515;">&quot;application/json&quot;</span>)]}&nbsp;url</pre> </p> <p> This is a little more involved than the <code>get</code> function, because it also has to supply the <code>Content-Type</code> HTTP header. If you don't supply that header with the <code>application/json</code> value, your API is going to reject the request when you attempt to post a string with a JSON object. </p> <p> Apart from that, it works the same way as the <code>get</code> function. </p> <h3 id="d726d432f53b4817b5dc9716a2fabc36"> Running a test session <a href="#d726d432f53b4817b5dc9716a2fabc36" title="permalink">#</a> </h3> <p> The <code>get</code> and <code>postJSON</code> functions both return <code>Session</code> values, so a test must run in the <code>Session</code> monad. This is easily done with Haskell's <code>do</code> notation; you'll see an example of that later in the article. </p> <p> First, however, you'll need a way to run a <code>Session</code>. <em>Network.Wai.Test</em> provides a function for that, called <code>runSession</code>. Besides a <code>Session a</code> value, though, it also requires an <code>Application</code> value. </p> <p> In my test library, I already have an <code>Application</code>, although it's running in <code>IO</code> (for reasons that'll take another article to explain): </p> <p> <pre><span style="color:#2b91af;">app</span>&nbsp;::&nbsp;<span style="color:#2b91af;">IO</span>&nbsp;<span style="color:blue;">Application</span></pre> </p> <p> With this value, you can easily convert any <code>Session a</code> to <code>IO a</code>: </p> <p> <pre><span style="color:#2b91af;">runSessionWithApp</span>&nbsp;::&nbsp;<span style="color:blue;">Session</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">IO</span>&nbsp;a runSessionWithApp&nbsp;s&nbsp;=&nbsp;app&nbsp;&gt;&gt;=&nbsp;runSession&nbsp;s</pre> </p> <p> The next step is to figure out how to turn an <code>IO a</code> into a test. </p> <h3 id="febab39aa10d4d78bfd0bc4d3d45ca8a"> Running a property <a href="#febab39aa10d4d78bfd0bc4d3d45ca8a" title="permalink">#</a> </h3> <p> You can turn an <code>IO a</code> into a <code>Property</code> with either <code>ioProperty</code> or <code>idempotentIOProperty</code>. I admit that the documentation doesn't make the distinction between the two entirely clear, but <code>ioProperty</code> sounds like the safer choice, so that's what I went for here. </p> <p> With <code>ioProperty</code> you now have a <code>Property</code> that you can turn into a <code>Test</code> using <code>testProperty</code> from <a href="http://hackage.haskell.org/package/test-framework-quickcheck2/docs/Test-Framework-Providers-QuickCheck2.html">Test.Framework.Providers.QuickCheck2</a>: </p> <p> <pre><span style="color:#2b91af;">appProperty</span>&nbsp;::&nbsp;(<span style="color:blue;">Functor</span>&nbsp;f,&nbsp;<span style="color:blue;">Testable</span>&nbsp;prop,&nbsp;<span style="color:blue;">Testable</span>&nbsp;(f&nbsp;<span style="color:blue;">Property</span>)) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=&gt;&nbsp;TestName&nbsp;-&gt;&nbsp;f&nbsp;(Session&nbsp;prop)&nbsp;-&gt;&nbsp;Test appProperty&nbsp;name&nbsp;= &nbsp;&nbsp;testProperty&nbsp;name&nbsp;.&nbsp;<span style="color:blue;">fmap</span>&nbsp;(ioProperty&nbsp;.&nbsp;runSessionWithApp)</pre> </p> <p> The type of this function seems more cryptic than strictly necessary. What's that <code>Functor f</code> doing there? </p> <p> The way I've written the tests, each property receives input from QuickCheck in the form of function arguments. I could have given the <code>appProperty</code> function a more restricted type, to make it clearer what's going on: </p> <p> <pre><span style="color:#2b91af;">appProperty</span>&nbsp;::&nbsp;(<span style="color:blue;">Arbitrary</span>&nbsp;a,&nbsp;<span style="color:blue;">Show</span>&nbsp;a,&nbsp;<span style="color:blue;">Testable</span>&nbsp;prop) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=&gt;&nbsp;TestName&nbsp;-&gt;&nbsp;(a&nbsp;-&gt;&nbsp;Session&nbsp;prop)&nbsp;-&gt;&nbsp;Test appProperty&nbsp;name&nbsp;= &nbsp;&nbsp;testProperty&nbsp;name&nbsp;.&nbsp;<span style="color:blue;">fmap</span>&nbsp;(ioProperty&nbsp;.&nbsp;runSessionWithApp)</pre> </p> <p> This is the same function, just with a more restricted type. It states that for any <code>Arbitrary a, Show a</code>, a test is a function that takes <code>a</code> as input and returns a <code>Session prop</code>. This restricts tests to take a single input value, which means that you'll have to write all those properties in tupled, uncurried form. You could relax that requirement by introducing a <code>newtype</code> and a type class with an instance that recursively enables curried functions. That's what <a href="http://hackage.haskell.org/package/hspec-wai/docs/Test-Hspec-Wai-QuickCheck.html">Test.Hspec.Wai.QuickCheck</a> does. I decided not to add that extra level of indirection, and instead living with having to write all my properties in tupled form. </p> <p> The <code>Functor f</code> in the above, relaxed type, then, is in actual use the Reader functor. You'll see some examples next. </p> <h3 id="0ebc8724e39149e88e5e71763b03d499"> Properties <a href="#0ebc8724e39149e88e5e71763b03d499" title="permalink">#</a> </h3> <p> You can now define some properties. Here's a simple example: </p> <p> <pre>appProperty&nbsp;<span style="color:#a31515;">&quot;responds&nbsp;with&nbsp;404&nbsp;when&nbsp;no&nbsp;reservation&nbsp;exists&quot;</span>&nbsp;$&nbsp;\rid&nbsp;-&gt;&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;actual&nbsp;&lt;-&nbsp;get&nbsp;$&nbsp;<span style="color:#a31515;">&quot;/reservations/&quot;</span>&nbsp;&lt;&gt;&nbsp;toASCIIBytes&nbsp;rid &nbsp;&nbsp;assertStatus&nbsp;404&nbsp;actual</pre> </p> <p> This is an inlined property, similar to how <a href="/2018/05/07/inlined-hunit-test-lists">I inline HUnit tests in test lists</a>. </p> <p> First, notice that the property is written as a lambda expression, which means that it fits the mould of <code>a -&gt; Session prop</code>. The input value <code>rid</code> (<em>reservationID</em>) is a <a href="http://hackage.haskell.org/package/uuid/docs/Data-UUID.html">UUID</a> value (for which an <code>Arbitrary</code> instance exists via <a href="http://hackage.haskell.org/package/quickcheck-instances">quickcheck-instances</a>). </p> <p> While the test runs in the <code>Session</code> monad, the <code>do</code> notation makes <code>actual</code> an <code>SResponse</code> value that you can then assert with <code>assertStatus</code> (from <em>Network.Wai.Test</em>). </p> <p> This property reproduces an interaction like this: </p> <p> <pre>&amp; curl -v http://localhost:8080/reservations/db38ac75-9ccd-43cc-864a-ce13e90a71d8 * Trying ::1:8080... * TCP_NODELAY set * Trying 127.0.0.1:8080... * TCP_NODELAY set * Connected to localhost (127.0.0.1) port 8080 (#0) &gt; GET /reservations/db38ac75-9ccd-43cc-864a-ce13e90a71d8 HTTP/1.1 &gt; Host: localhost:8080 &gt; User-Agent: curl/7.65.1 &gt; Accept: */* &gt; * Mark bundle as not supporting multiuse &lt; HTTP/1.1 404 Not Found &lt; Transfer-Encoding: chunked &lt; Date: Tue, 02 Jul 2019 18:09:51 GMT &lt; Server: Warp/3.2.27 &lt; * Connection #0 to host localhost left intact</pre> </p> <p> The important result is that the status code is <code>404 Not Found</code>, which is also what the property asserts. </p> <p> If you need more than one input value to your property, you have to write the property in tupled form: </p> <p> <pre>appProperty&nbsp;<span style="color:#a31515;">&quot;fails&nbsp;when&nbsp;reservation&nbsp;is&nbsp;POSTed&nbsp;with&nbsp;invalid&nbsp;quantity&quot;</span>&nbsp;$&nbsp;\ &nbsp;&nbsp;(ValidReservation&nbsp;r,&nbsp;NonNegative&nbsp;q)&nbsp;-&gt;&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;invalid&nbsp;=&nbsp;r&nbsp;{&nbsp;reservationQuantity&nbsp;=&nbsp;<span style="color:blue;">negate</span>&nbsp;q&nbsp;} &nbsp;&nbsp;actual&nbsp;&lt;-&nbsp;postJSON&nbsp;<span style="color:#a31515;">&quot;/reservations&quot;</span>&nbsp;$&nbsp;encode&nbsp;invalid &nbsp;&nbsp;assertStatus&nbsp;400&nbsp;actual</pre> </p> <p> This property still takes a single input, but that input is a tuple where the first element is a <code>ValidReservation</code> and the second element a <code>NonNegative Int</code>. The <a href="/2019/09/02/naming-newtypes-for-quickcheck-arbitraries">ValidReservation newtype wrapper</a> ensures that <code>r</code> is a valid reservation record. This ensures that the property only exercises the path where the reservation quantity is zero or negative. It accomplishes this by negating <code>q</code> and replacing the <code>reservationQuantity</code> with that negative (or zero) number. </p> <p> It then encodes (with <a href="http://hackage.haskell.org/package/aeson">aeson</a>) the <code>invalid</code> reservation and posts it using the <code>postJSON</code> function. </p> <p> Finally it asserts that the HTTP status code is <code>400 Bad Request</code>. </p> <h3 id="ae5b3f7b07634799ad2af0d9a2ac668c"> Summary <a href="#ae5b3f7b07634799ad2af0d9a2ac668c" title="permalink">#</a> </h3> <p> After having tried using <em>Test.Hspec.Wai</em> for some time, I decided to refactor my tests to QuickCheck and HUnit. Once I figured out how <em>Network.Wai.Test</em> works, the remaining work wasn't too difficult. While there's little written documentation for the modules, the types (as usual) act as documentation. Using the types, and looking a little at the underlying code, I was able to figure out how to use the test API. </p> <p> You write tests against <em>wai</em> applications in the <code>Session</code> monad. You can then use <code>runSession</code> to turn the <code>Session</code> into an <code>IO</code> value. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Picture archivist in F# https://blog.ploeh.dk/2019/09/16/picture-archivist-in-f 2019-09-16T05:59:00+00:00 Mark Seemann <div id="post"> <p> <em>A comprehensive code example showing how to implement a functional architecture in F#.</em> </p> <p> This article shows how to implement the <a href="/2019/08/26/functional-file-system">picture archivist architecture described in a previous article</a>. In short, the task is to move some image files to directories based on their date-taken metadata. The architectural idea is to load a directory structure from disk into an in-memory tree, manipulate that tree, and use the resulting tree to perform the desired actions: </p> <p> <img src="/content/binary/functional-file-system-interaction.png" alt="A functional program typically loads data, transforms it, and stores it again."> </p> <p> Much of the program will manipulate the tree data, which is immutable. </p> <p> The previous article showed how to implement the <a href="/2019/09/09/picture-archivist-in-haskell">picture archivist architecture in Haskell</a>. In this article, you'll see how to do it in <a href="https://fsharp.org">F#</a>. This is essentially a port of the <a href="https://www.haskell.org">Haskell</a> code. </p> <h3 id="949a876ffec843e09d4faa5ae1c1b4c5"> Tree <a href="#949a876ffec843e09d4faa5ae1c1b4c5" title="permalink">#</a> </h3> <p> You can start by defining a <a href="https://en.wikipedia.org/wiki/Rose_tree">rose tree</a>: </p> <p> <pre><span style="color:blue;">type</span>&nbsp;Tree&lt;&#39;a,&nbsp;&#39;b&gt;&nbsp;=&nbsp;Node&nbsp;<span style="color:blue;">of</span>&nbsp;&#39;a&nbsp;*&nbsp;Tree&lt;&#39;a,&nbsp;&#39;b&gt;&nbsp;list&nbsp;|&nbsp;Leaf&nbsp;<span style="color:blue;">of</span>&nbsp;&#39;b</pre> </p> <p> If you wanted to, you could put all the <code>Tree</code> code in a reusable library, because none of it is coupled to a particular application, such as <a href="https://amzn.to/2V06Kji">moving pictures</a>. You could also write a comprehensive test suite for the following functions, but in this article, I'll skip that. </p> <p> Notice that this sort of tree explicitly distinguishes between internal and leaf nodes. This is necessary because you'll need to keep track of the directory names (the internal nodes), while at the same time you'll want to enrich the leaves with additional data - data that you can't meaningfully add to the internal nodes. You'll see this later in the article. </p> <p> While I typically tend to define F# types outside of modules (so that you don't have to, say, prefix the type name with the module name - <code>Tree.Tree</code> is so awkward), the rest of the tree code goes into a module, including two helper functions: </p> <p> <pre><span style="color:blue;">module</span>&nbsp;Tree&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:green;">//&nbsp;&#39;b&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,&#39;b&gt;</span> &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;leaf&nbsp;=&nbsp;Leaf &nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:green;">//&nbsp;&#39;a&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,&#39;b&gt;&nbsp;list&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,&#39;b&gt;</span> &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;node&nbsp;x&nbsp;xs&nbsp;=&nbsp;Node&nbsp;(x,&nbsp;xs)</pre> </p> <p> The <code>leaf</code> function doesn't add much value, but the <code>node</code> function offers a curried alternative to the <code>Node</code> case constructor. That's occasionally useful. </p> <p> The rest of the code related to trees is also defined in the <code>Tree</code> module, but I'm going to present it formatted as free-standing functions. If you're confused about the layout of the code, the entire code base is <a href="https://github.com/ploeh/picture-archivist">available on GitHub</a>. </p> <p> The <a href="/2019/08/05/rose-tree-catamorphism">rose tree catamorphism</a> is this <code>cata</code> function: </p> <p> <pre><span style="color:green;">//&nbsp;(&#39;a&nbsp;-&gt;&nbsp;&#39;c&nbsp;list&nbsp;-&gt;&nbsp;&#39;c)&nbsp;-&gt;&nbsp;(&#39;b&nbsp;-&gt;&nbsp;&#39;c)&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,&#39;b&gt;&nbsp;-&gt;&nbsp;&#39;c</span> <span style="color:blue;">let</span>&nbsp;<span style="color:blue;">rec</span>&nbsp;cata&nbsp;fd&nbsp;ff&nbsp;=&nbsp;<span style="color:blue;">function</span> &nbsp;&nbsp;&nbsp;&nbsp;|&nbsp;Leaf&nbsp;x&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;ff&nbsp;x &nbsp;&nbsp;&nbsp;&nbsp;|&nbsp;Node&nbsp;(x,&nbsp;xs)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;xs&nbsp;|&gt;&nbsp;List.map&nbsp;(cata&nbsp;fd&nbsp;ff)&nbsp;|&gt;&nbsp;fd&nbsp;x</pre> </p> <p> In the corresponding Haskell implementation of this architecture, I called this function <code>foldTree</code>, so why not retain that name? The short answer is that the naming conventions differ between Haskell and F#, and while I favour learning from Haskell, I still want my F# code to be as <a href="/2015/08/03/idiomatic-or-idiosyncratic">idiomatic</a> as possible. </p> <p> While I don't enforce that client code <em>must</em> use the <code>Tree</code> module name to access the functions within, I prefer to name the functions so that they make sense when used with qualified access. Having to write <code>Tree.foldTree</code> seems redundant. A more idiomatic name would be <code>fold</code>, so that you could write <code>Tree.fold</code>. The problem with that name, though, is that <code>fold</code> usually implies a list-biased <em>fold</em> (corresponding to <code>foldl</code> in Haskell), and I'll actually need that name for that particular purpose later. </p> <p> So, <code>cata</code> it is. </p> <p> In this article, tree functionality is (with one exception) directly or transitively implemented with <code>cata</code>. </p> <h3 id="3f30722983ad47bd83c88cec4ba80983"> Filtering trees <a href="#3f30722983ad47bd83c88cec4ba80983" title="permalink">#</a> </h3> <p> It'll be useful to be able to filter the contents of a tree. For example, the picture archivist program will only move image files with valid metadata. This means that it'll need to filter out all files that aren't image files, as well as image files without valid metadata. </p> <p> It turns out that it'll be useful to supply a function that throws away <code>None</code> values from a tree of <code>option</code> leaves. This is similar to <a href="https://msdn.microsoft.com/en-us/visualfsharpdocs/conceptual/list.choose%5B't%2C'u%5D-function-%5Bfsharp%5D">List.choose</a>, so I call it <code>Tree.choose</code>: </p> <p> <pre><span style="color:green;">//&nbsp;(&#39;a&nbsp;-&gt;&nbsp;&#39;b&nbsp;option)&nbsp;-&gt;&nbsp;Tree&lt;&#39;c,&#39;a&gt;&nbsp;-&gt;&nbsp;Tree&lt;&#39;c,&#39;b&gt;&nbsp;option</span> <span style="color:blue;">let</span>&nbsp;choose&nbsp;f&nbsp;=&nbsp;cata&nbsp;(<span style="color:blue;">fun</span>&nbsp;x&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;List.choose&nbsp;id&nbsp;&gt;&gt;&nbsp;node&nbsp;x&nbsp;&gt;&gt;&nbsp;Some)&nbsp;(f&nbsp;&gt;&gt;&nbsp;Option.map&nbsp;Leaf)</pre> </p> <p> You may find the type of the function surprising. Why does it return a <code>Tree option</code>, instead of simply a <code>Tree</code>? </p> <p> While <code>List.choose</code> simply returns a list, it can do this because lists can be empty. This <code>Tree</code> type, on the other hand, can't be empty. If the purpose of <code>Tree.choose</code> is to throw away all <code>None</code> values, then how do you return a tree from <code>Leaf None</code>? </p> <p> You can't return a <code>Leaf</code> because you have no value to put in the leaf. Similarly, you can't return a <code>Node</code> because, again, you have no value to put in the node. </p> <p> In order to handle this edge case, then, you'll have to return <code>None</code>: </p> <p> <pre>&gt; let l : Tree&lt;string, int option&gt; = Leaf None;; val l : Tree&lt;string,int option&gt; = Leaf None &gt; Tree.choose id l;; val it : Tree&lt;string,int&gt; option = None</pre> </p> <p> If you have anything other than a <code>None</code> leaf, though, you'll get a proper tree, but wrapped in an <code>option</code>: </p> <p> <pre>&gt; Tree.node "Foo" [Leaf (Some 42); Leaf None; Leaf (Some 2112)] |&gt; Tree.choose id;; val it : Tree&lt;string,int&gt; option = Some (Node ("Foo",[Leaf 42; Leaf 2112]))</pre> </p> <p> While the resulting tree is wrapped in a <code>Some</code> case, the leaves contain unwrapped values. </p> <h3 id="32f46f2c16cf428abc39c3d79433caa6"> Bifunctor, functor, and folds <a href="#32f46f2c16cf428abc39c3d79433caa6" title="permalink">#</a> </h3> <p> Through its type class language feature, Haskell has formal definitions of <a href="/2018/03/22/functors">functors</a>, <a href="/2018/12/24/bifunctors">bifunctors</a>, and other types of <em>folds</em> (list-biased <a href="/2019/04/29/catamorphisms">catamorphisms</a>). F# doesn't have a similar degree of formalism, which means that while you can still implement the corresponding functionality, you'll have to rely on conventions to make the functions recognisable. </p> <p> It's straighforward to start with the bifunctor functionality: </p> <p> <pre><span style="color:green;">//&nbsp;(&#39;a&nbsp;-&gt;&nbsp;&#39;b)&nbsp;-&gt;&nbsp;(&#39;c&nbsp;-&gt;&nbsp;&#39;d)&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,&#39;c&gt;&nbsp;-&gt;&nbsp;Tree&lt;&#39;b,&#39;d&gt;</span> <span style="color:blue;">let</span>&nbsp;bimap&nbsp;f&nbsp;g&nbsp;=&nbsp;cata&nbsp;(f&nbsp;&gt;&gt;&nbsp;node)&nbsp;(g&nbsp;&gt;&gt;&nbsp;leaf)</pre> </p> <p> This is, apart from the syntax differences, the same implementation as in Haskell. Based on <code>bimap</code>, you can also trivially implement <code>mapNode</code> and <code>mapLeaf</code> functions if you'd like, but you're not going to need those for the code in this article. You do need, however, a function that we could consider an alias of a hypothetical <code>mapLeaf</code> function: </p> <p> <pre><span style="color:green;">//&nbsp;(&#39;b&nbsp;-&gt;&nbsp;&#39;c)&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,&#39;b&gt;&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,&#39;c&gt;</span> <span style="color:blue;">let</span>&nbsp;map&nbsp;f&nbsp;=&nbsp;bimap&nbsp;id&nbsp;f</pre> </p> <p> This makes <code>Tree</code> a functor. </p> <p> It'll also be useful to reduce a tree to a potentially more compact value, so you can add some specialised folds: </p> <p> <pre><span style="color:green;">//&nbsp;(&#39;c&nbsp;-&gt;&nbsp;&#39;a&nbsp;-&gt;&nbsp;&#39;c)&nbsp;-&gt;&nbsp;(&#39;c&nbsp;-&gt;&nbsp;&#39;b&nbsp;-&gt;&nbsp;&#39;c)&nbsp;-&gt;&nbsp;&#39;c&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,&#39;b&gt;&nbsp;-&gt;&nbsp;&#39;c</span> <span style="color:blue;">let</span>&nbsp;bifold&nbsp;f&nbsp;g&nbsp;z&nbsp;t&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;flip&nbsp;f&nbsp;x&nbsp;y&nbsp;=&nbsp;f&nbsp;y&nbsp;x &nbsp;&nbsp;&nbsp;&nbsp;cata&nbsp;(<span style="color:blue;">fun</span>&nbsp;x&nbsp;xs&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;flip&nbsp;f&nbsp;x&nbsp;&gt;&gt;&nbsp;List.fold&nbsp;(&gt;&gt;)&nbsp;id&nbsp;xs)&nbsp;(flip&nbsp;g)&nbsp;t&nbsp;z <span style="color:green;">//&nbsp;(&#39;a&nbsp;-&gt;&nbsp;&#39;c&nbsp;-&gt;&nbsp;&#39;c)&nbsp;-&gt;&nbsp;(&#39;b&nbsp;-&gt;&nbsp;&#39;c&nbsp;-&gt;&nbsp;&#39;c)&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,&#39;b&gt;&nbsp;-&gt;&nbsp;&#39;c&nbsp;-&gt;&nbsp;&#39;c</span> <span style="color:blue;">let</span>&nbsp;bifoldBack&nbsp;f&nbsp;g&nbsp;t&nbsp;z&nbsp;=&nbsp;cata&nbsp;(<span style="color:blue;">fun</span>&nbsp;x&nbsp;xs&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;List.foldBack&nbsp;(&lt;&lt;)&nbsp;xs&nbsp;id&nbsp;&gt;&gt;&nbsp;f&nbsp;x)&nbsp;g&nbsp;t&nbsp;z</pre> </p> <p> In an attempt to emulate the F# naming conventions, I named the functions as I did. There are similar functions in the <code>List</code> and <code>Option</code> modules, for instance. If you're comparing the F# code with the Haskell code in the previous article, <code>Tree.bifold</code> corresponds to <code>bifoldl</code>, and <code>Tree.bifoldBack</code> corresponds to <code>bifoldr</code>. </p> <p> These enable you to implement folds over leaves only: </p> <p> <pre><span style="color:green;">//&nbsp;(&#39;c&nbsp;-&gt;&nbsp;&#39;b&nbsp;-&gt;&nbsp;&#39;c)&nbsp;-&gt;&nbsp;&#39;c&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,&#39;b&gt;&nbsp;-&gt;&nbsp;&#39;c</span> <span style="color:blue;">let</span>&nbsp;fold&nbsp;f&nbsp;=&nbsp;bifold&nbsp;(<span style="color:blue;">fun</span>&nbsp;x&nbsp;_&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;x)&nbsp;f <span style="color:green;">//&nbsp;(&#39;b&nbsp;-&gt;&nbsp;&#39;c&nbsp;-&gt;&nbsp;&#39;c)&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,&#39;b&gt;&nbsp;-&gt;&nbsp;&#39;c&nbsp;-&gt;&nbsp;&#39;c</span> <span style="color:blue;">let</span>&nbsp;foldBack&nbsp;f&nbsp;=&nbsp;bifoldBack&nbsp;(<span style="color:blue;">fun</span>&nbsp;_&nbsp;x&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;x)&nbsp;f</pre> </p> <p> These, again, enable you to implement another function that'll turn out to be useful in this article: </p> <p> <pre><span style="color:green;">//&nbsp;(&#39;b&nbsp;-&gt;&nbsp;unit)&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,&#39;b&gt;&nbsp;-&gt;&nbsp;unit</span> <span style="color:blue;">let</span>&nbsp;iter&nbsp;f&nbsp;=&nbsp;fold&nbsp;(<span style="color:blue;">fun</span>&nbsp;()&nbsp;x&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;f&nbsp;x)&nbsp;()</pre> </p> <p> The picture archivist program isn't going to explicitly need all of these, but transitively, it will. </p> <h3 id="8a9a50c69a2d461cac5bb87fa4cf3cd9"> Moving pictures <a href="#8a9a50c69a2d461cac5bb87fa4cf3cd9" title="permalink">#</a> </h3> <p> So far, all the code shown here could be in a general-purpose reusable library, since it contains no functionality specifically related to image files. The rest of the code in this article, however, will be specific to the program. I'll put the domain model code in another module that I call <code>Archive</code>. Later in the article, we'll look at how to load a tree from the file system, but for now, we'll just pretend that we have such a tree. </p> <p> The major logic of the program is to create a destination tree based on a source tree. The leaves of the tree will have to carry some extra information apart from a file path, so you can introduce a specific type to capture that information: </p> <p> <pre><span style="color:blue;">type</span>&nbsp;PhotoFile&nbsp;=&nbsp;{&nbsp;File&nbsp;:&nbsp;FileInfo;&nbsp;TakenOn&nbsp;:&nbsp;DateTime&nbsp;}</pre> </p> <p> A <code>PhotoFile</code> not only contains the file path for an image file, but also the date the photo was taken. This date can be extracted from the file's metadata, but that's an impure operation, so we'll delegate that work to the start of the program. We'll return to that later. </p> <p> Given a source tree of <code>PhotoFile</code> leaves, though, the program must produce a destination tree of files: </p> <p> <pre><span style="color:green;">//&nbsp;string&nbsp;-&gt;&nbsp;Tree&lt;&#39;a,PhotoFile&gt;&nbsp;-&gt;&nbsp;Tree&lt;string,FileInfo&gt;</span> <span style="color:blue;">let</span>&nbsp;moveTo&nbsp;destination&nbsp;t&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;dirNameOf&nbsp;(dt&nbsp;:&nbsp;DateTime)&nbsp;=&nbsp;sprintf&nbsp;<span style="color:#a31515;">&quot;%d-%02d&quot;</span>&nbsp;dt.Year&nbsp;dt.Month &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;groupByDir&nbsp;pf&nbsp;m&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;key&nbsp;=&nbsp;dirNameOf&nbsp;pf.TakenOn &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;dir&nbsp;=&nbsp;Map.tryFind&nbsp;key&nbsp;m&nbsp;|&gt;&nbsp;Option.defaultValue&nbsp;[] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Map.add&nbsp;key&nbsp;(pf.File&nbsp;::&nbsp;dir)&nbsp;m &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;addDir&nbsp;name&nbsp;files&nbsp;dirs&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Tree.node&nbsp;name&nbsp;(List.map&nbsp;Leaf&nbsp;files)&nbsp;::&nbsp;dirs &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;m&nbsp;=&nbsp;Tree.foldBack&nbsp;groupByDir&nbsp;t&nbsp;Map.empty &nbsp;&nbsp;&nbsp;&nbsp;Map.foldBack&nbsp;addDir&nbsp;m&nbsp;[]&nbsp;|&gt;&nbsp;Tree.node&nbsp;destination</pre> </p> <p> This <code>moveTo</code> function looks, perhaps, overwhelming, but it's composed of three conceptual steps: <ol> <li>Create a map of destination folders (<code>m</code>).</li> <li>Create a list of branches from the map (<code>Map.foldBack addDir m []</code>).</li> <li>Create a tree from the list (<code>Tree.node destination</code>).</li> </ol> The <code>moveTo</code> function starts by folding the input data into a map <code>m</code>. The map is keyed by the directory name, which is formatted by the <code>dirNameOf</code> function. This function takes a <code>DateTime</code> as input and formats it to a <code>YYYY-MM</code> format. For example, December 20, 2018 becomes <code>"2018-12"</code>. </p> <p> The entire mapping step groups the <code>PhotoFile</code> values into a map of the type <code>Map&lt;string,FileInfo list&gt;</code>. All the image files taken in April 2014 are added to the list with the <code>"2014-04"</code> key, all the image files taken in July 2011 are added to the list with the <code>"2011-07"</code> key, and so on. </p> <p> In the next step, the <code>moveTo</code> function converts the map to a list of trees. This will be the branches (or sub-directories) of the <code>destination</code> directory. Because of the desired structure of the destination tree, this is a list of shallow branches. Each node contains only leaves. </p> <p> <img src="/content/binary/shallow-photo-destination-directories.png" alt="Shallow photo destination directories."> </p> <p> The only remaining step is to add that list of branches to a <code>destination</code> node. This is done by piping (<code>|&gt;</code>) the list of sub-directories into <code>Tree.node destination</code>. </p> <p> Since this is a <a href="https://en.wikipedia.org/wiki/Pure_function">pure function</a>, it's <a href="/2015/05/07/functional-design-is-intrinsically-testable">easy to unit test</a>. Just create some test cases and call the function. First, the test cases. </p> <p> In this code base, I'm using <a href="https://xunit.github.io">xUnit.net</a> 2.4.1, so I'll first create a set of test cases as a test-specific class: </p> <p> <pre><span style="color:blue;">type</span>&nbsp;MoveToDestinationTestData&nbsp;()&nbsp;<span style="color:blue;">as</span>&nbsp;this&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">inherit</span>&nbsp;TheoryData&lt;Tree&lt;string,&nbsp;PhotoFile&gt;,&nbsp;string,&nbsp;Tree&lt;string,&nbsp;string&gt;&gt;&nbsp;() &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;photoLeaf&nbsp;name&nbsp;(y,&nbsp;mth,&nbsp;d,&nbsp;h,&nbsp;m,&nbsp;s)&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;{&nbsp;File&nbsp;=&nbsp;FileInfo&nbsp;name;&nbsp;TakenOn&nbsp;=&nbsp;DateTime&nbsp;(y,&nbsp;mth,&nbsp;d,&nbsp;h,&nbsp;m,&nbsp;s)&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">do</span>&nbsp;this.Add&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;photoLeaf&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>&nbsp;(2018,&nbsp;11,&nbsp;9,&nbsp;11,&nbsp;47,&nbsp;17), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#a31515;">&quot;D&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(&nbsp;<span style="color:#a31515;">&quot;D&quot;</span>,&nbsp;[Node&nbsp;(<span style="color:#a31515;">&quot;2018-11&quot;</span>,&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>])])) &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">do</span>&nbsp;this.Add&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;S&quot;</span>,&nbsp;[photoLeaf&nbsp;<span style="color:#a31515;">&quot;4&quot;</span>&nbsp;(1972,&nbsp;6,&nbsp;6,&nbsp;16,&nbsp;15,&nbsp;0)]), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#a31515;">&quot;D&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;D&quot;</span>,&nbsp;[Node&nbsp;(<span style="color:#a31515;">&quot;1972-06&quot;</span>,&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;4&quot;</span>])])) &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">do</span>&nbsp;this.Add&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;S&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;photoLeaf&nbsp;<span style="color:#a31515;">&quot;L&quot;</span>&nbsp;(2002,&nbsp;10,&nbsp;12,&nbsp;17,&nbsp;16,&nbsp;15); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;photoLeaf&nbsp;<span style="color:#a31515;">&quot;J&quot;</span>&nbsp;(2007,&nbsp;4,&nbsp;21,&nbsp;17,&nbsp;18,&nbsp;19)]), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#a31515;">&quot;D&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;D&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;2002-10&quot;</span>,&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;L&quot;</span>]); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;2007-04&quot;</span>,&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;J&quot;</span>])])) &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">do</span>&nbsp;this.Add&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;1&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;photoLeaf&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;(2010,&nbsp;1,&nbsp;12,&nbsp;17,&nbsp;16,&nbsp;15); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;photoLeaf&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>&nbsp;(2010,&nbsp;3,&nbsp;12,&nbsp;17,&nbsp;16,&nbsp;15); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;photoLeaf&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>&nbsp;(2010,&nbsp;1,&nbsp;21,&nbsp;17,&nbsp;18,&nbsp;19)]), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#a31515;">&quot;2&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;2&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;2010-01&quot;</span>,&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>;&nbsp;Leaf&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>]); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;2010-03&quot;</span>,&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>])])) &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">do</span>&nbsp;this.Add&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;foo&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;bar&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;photoLeaf&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;(2010,&nbsp;1,&nbsp;12,&nbsp;17,&nbsp;16,&nbsp;15); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;photoLeaf&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>&nbsp;(2010,&nbsp;3,&nbsp;12,&nbsp;17,&nbsp;16,&nbsp;15); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;photoLeaf&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>&nbsp;(2010,&nbsp;1,&nbsp;21,&nbsp;17,&nbsp;18,&nbsp;19)]); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;baz&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;photoLeaf&nbsp;<span style="color:#a31515;">&quot;d&quot;</span>&nbsp;(2010,&nbsp;3,&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;4); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;photoLeaf&nbsp;<span style="color:#a31515;">&quot;e&quot;</span>&nbsp;(2011,&nbsp;3,&nbsp;4,&nbsp;3,&nbsp;2,&nbsp;1)])]), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#a31515;">&quot;qux&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;qux&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;2010-01&quot;</span>,&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>;&nbsp;Leaf&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>]); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;2010-03&quot;</span>,&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>;&nbsp;Leaf&nbsp;<span style="color:#a31515;">&quot;d&quot;</span>]); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;2011-03&quot;</span>,&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;e&quot;</span>])]))</pre> </p> <p> That looks like a lot of code, but is really just a list of test cases. Each test case is a triple of a source tree, a destination directory name, and an expected result (another tree). </p> <p> The test itself, on the other hand, is compact: </p> <p> <pre>[&lt;Theory;&nbsp;ClassData(typeof&lt;MoveToDestinationTestData&gt;)&gt;] <span style="color:blue;">let</span>&nbsp;``Move&nbsp;to&nbsp;destination``&nbsp;source&nbsp;destination&nbsp;expected&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;actual&nbsp;=&nbsp;Archive.moveTo&nbsp;destination&nbsp;source &nbsp;&nbsp;&nbsp;&nbsp;expected&nbsp;=!&nbsp;Tree.map&nbsp;string&nbsp;actual</pre> </p> <p> The <code>=!</code> operator comes from <a href="https://github.com/SwensenSoftware/unquote">Unquote</a> and means something like <em>must equal</em>. It's an assertion that will throw an exception if <code>expected</code> isn't equal to <code>Tree.map string actual</code>. </p> <p> The reason that the assertion maps <code>actual</code> to a tree of strings is that <code>actual</code> is a <code>Tree&lt;string,FileInfo&gt;</code>, but <code>FileInfo</code> doesn't have structural equality. So either I had to implement a test-specific equality comparer for <code>FileInfo</code> (and for <code>Tree&lt;string,FileInfo&gt;</code>), or map the tree to something with proper equality, such as a <code>string</code>. I chose the latter. </p> <h3 id="abe95ba6865745bc9df8004079d8a250"> Calculating moves <a href="#abe95ba6865745bc9df8004079d8a250" title="permalink">#</a> </h3> <p> One pure step remains. The result of calling the <code>moveTo</code> function is a tree with the desired structure. In order to actually move the files, though, for each file you'll need to keep track of both the source path and the destination path. To make that explicit, you can define a type for that purpose: </p> <p> <pre><span style="color:blue;">type</span>&nbsp;Move&nbsp;=&nbsp;{&nbsp;Source&nbsp;:&nbsp;FileInfo;&nbsp;Destination&nbsp;:&nbsp;FileInfo&nbsp;}</pre> </p> <p> A <code>Move</code> is simply a data structure. Contrast this with typical object-oriented design, where it would be a (possibly polymorphic) method on an object. In functional programming, you'll regularly model <em>intent</em> with a data structure. As long as intents remain data, you can easily manipulate them, and once you're done with that, you can run an interpreter over your data structure to perform the work you want accomplished. </p> <p> The unit test cases for the <code>moveTo</code> function suggest that file names are local file names like <code>"L"</code>, <code>"J"</code>, <code>"a"</code>, and so on. That was only to make the tests as compact as possible, since the function actually doesn't manipulate the specific <code>FileInfo</code> objects. </p> <p> In reality, the file names will most likely be longer, and they could also contain the full path, instead of the local path: <code>"C:\foo\bar\a.jpg"</code>. </p> <p> If you call <code>moveTo</code> with a tree where each leaf has a fully qualified path, the output tree will have the desired structure of the destination tree, but the leaves will still contain the full path to each source file. That means that you can calculate a <code>Move</code> for each file: </p> <p> <pre><span style="color:green;">//&nbsp;Tree&lt;string,FileInfo&gt;&nbsp;-&gt;&nbsp;Tree&lt;string,Move&gt;</span> <span style="color:blue;">let</span>&nbsp;calculateMoves&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;replaceDirectory&nbsp;(f&nbsp;:&nbsp;FileInfo)&nbsp;d&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;FileInfo&nbsp;(Path.Combine&nbsp;(d,&nbsp;f.Name)) &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;<span style="color:blue;">rec</span>&nbsp;imp&nbsp;path&nbsp;=&nbsp;<span style="color:blue;">function</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|&nbsp;Leaf&nbsp;x&nbsp;<span style="color:blue;">-&gt;</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;{&nbsp;Source&nbsp;=&nbsp;x;&nbsp;Destination&nbsp;=&nbsp;replaceDirectory&nbsp;x&nbsp;path&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|&nbsp;Node&nbsp;(x,&nbsp;xs)&nbsp;<span style="color:blue;">-&gt;</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;newNPath&nbsp;=&nbsp;Path.Combine&nbsp;(path,&nbsp;x) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Tree.node&nbsp;newNPath&nbsp;(List.map&nbsp;(imp&nbsp;newNPath)&nbsp;xs) &nbsp;&nbsp;&nbsp;&nbsp;imp&nbsp;<span style="color:#a31515;">&quot;&quot;</span></pre> </p> <p> This function takes as input a <code>Tree&lt;string,FileInfo&gt;</code>, which is compatible with the output of <code>moveTo</code>. It returns a <code>Tree&lt;string,Move&gt;</code>, i.e. a tree where the leaves are <code>Move</code> values. </p> <p> Earlier, I wrote that you can implement desired <code>Tree</code> functionality with the <code>cata</code> function, but that was a simplification. If you can implement the functionality of <code>calculateMoves</code> with <code>cata</code>, I don't know how. You can, however, implement it using explicit pattern matching and simple recursion. </p> <p> The <code>imp</code> function builds up a file path as it recursively negotiates the tree. All <code>Leaf</code> nodes are converted to a <code>Move</code> value using the leaf node's current <code>FileInfo</code> value as the <code>Source</code>, and the <code>path</code> to figure out the desired <code>Destination</code>. </p> <p> This code is still easy to unit test. First, test cases: </p> <p> <pre><span style="color:blue;">type</span>&nbsp;CalculateMovesTestData&nbsp;()&nbsp;<span style="color:blue;">as</span>&nbsp;this&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">inherit</span>&nbsp;TheoryData&lt;Tree&lt;string,&nbsp;FileInfo&gt;,&nbsp;Tree&lt;string,&nbsp;(string&nbsp;*&nbsp;string)&gt;&gt;&nbsp;() &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">do</span>&nbsp;this.Add&nbsp;(Leaf&nbsp;(FileInfo&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>),&nbsp;Leaf&nbsp;(<span style="color:#a31515;">&quot;1&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>)) &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">do</span>&nbsp;this.Add&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;[Leaf&nbsp;(FileInfo&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>)]), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;[Leaf&nbsp;(<span style="color:#a31515;">&quot;1&quot;</span>,&nbsp;Path.Combine&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>))])) &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">do</span>&nbsp;this.Add&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;[Leaf&nbsp;(FileInfo&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>);&nbsp;Leaf&nbsp;(FileInfo&nbsp;<span style="color:#a31515;">&quot;2&quot;</span>)]), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;(<span style="color:#a31515;">&quot;1&quot;</span>,&nbsp;Path.Combine&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>)); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;(<span style="color:#a31515;">&quot;2&quot;</span>,&nbsp;Path.Combine&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;2&quot;</span>))])) &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">do</span>&nbsp;this.Add&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;b&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;(FileInfo&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;(FileInfo&nbsp;<span style="color:#a31515;">&quot;2&quot;</span>)]); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;c&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;(FileInfo&nbsp;<span style="color:#a31515;">&quot;3&quot;</span>)])]), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(Path.Combine&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>),&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;(<span style="color:#a31515;">&quot;1&quot;</span>,&nbsp;Path.Combine&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>)); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;(<span style="color:#a31515;">&quot;2&quot;</span>,&nbsp;Path.Combine&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;2&quot;</span>))]); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(Path.Combine&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>),&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;(<span style="color:#a31515;">&quot;3&quot;</span>,&nbsp;Path.Combine&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;3&quot;</span>))])]))</pre> </p> <p> The test cases in this parametrised test are tuples of an input tree and the expected tree. For each test case, the test calls the <code>Archive.calculateMoves</code> function with <code>tree</code> and asserts that the <code>actual</code> tree is equal to the <code>expected</code> tree: </p> <p> <pre>[&lt;Theory;&nbsp;ClassData(typeof&lt;CalculateMovesTestData&gt;)&gt;] <span style="color:blue;">let</span>&nbsp;``Calculate&nbsp;moves``&nbsp;tree&nbsp;expected&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;actual&nbsp;=&nbsp;Archive.calculateMoves&nbsp;tree &nbsp;&nbsp;&nbsp;&nbsp;expected&nbsp;=!&nbsp;Tree.map&nbsp;(<span style="color:blue;">fun</span>&nbsp;m&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;(m.Source.ToString&nbsp;(),&nbsp;m.Destination.ToString&nbsp;()))&nbsp;actual</pre> </p> <p> Again, the test maps <code>FileInfo</code> objects to <code>strings</code> to support easy comparison. </p> <p> That's all the pure code you need in order to implement the desired functionality. Now you only need to write some code that loads a tree from disk, and imprints a destination tree to disk, as well as the code that composes it all. </p> <h3 id="bac6be79cf8c44a7b47923e2ec90d99f"> Loading a tree from disk <a href="#bac6be79cf8c44a7b47923e2ec90d99f" title="permalink">#</a> </h3> <p> The remaining code in this article is impure. You could put it in dedicated modules, but for this program, you're only going to need three functions and a bit of composition code, so you could also just put it all in the <code>Program</code> module. That's what I did. </p> <p> To load a tree from disk, you'll need a root directory, under which you load the entire tree. Given a directory path, you read a tree using a recursive function like this: </p> <p> <pre><span style="color:green;">//&nbsp;string&nbsp;-&gt;&nbsp;Tree&lt;string,string&gt;</span> <span style="color:blue;">let</span>&nbsp;<span style="color:blue;">rec</span>&nbsp;readTree&nbsp;path&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;File.Exists&nbsp;path &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">then</span>&nbsp;Leaf&nbsp;path &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">else</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;dirsAndFiles&nbsp;=&nbsp;Directory.EnumerateFileSystemEntries&nbsp;path &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;branches&nbsp;=&nbsp;Seq.map&nbsp;readTree&nbsp;dirsAndFiles&nbsp;|&gt;&nbsp;Seq.toList &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(path,&nbsp;branches)</pre> </p> <p> This recursive function starts by checking whether the <code>path</code> is a file that exists. If it does, the path is a file, so it creates a new <code>Leaf</code> with that path. </p> <p> If <code>path</code> isn't a file, it's a directory. In that case, use <code>Directory.EnumerateFileSystemEntries</code> to enumerate all the directories and files in that directory, and map all those directory entries recursively. That produces all the <code>branches</code> for the current node. Finally, return a new <code>Node</code> with the <code>path</code> and the <code>branches</code>. </p> <h3 id="7f5e06eb61024264ad214d41b63a8a74"> Loading metadata <a href="#7f5e06eb61024264ad214d41b63a8a74" title="permalink">#</a> </h3> <p> The <code>readTree</code> function only produces a tree with <code>string</code> leaves, while the program requires a tree with <code>PhotoFile</code> leaves. You'll need to read the <a href="https://en.wikipedia.org/wiki/Exif">Exif</a> metadata from each file and enrich the tree with the <em>date-taken</em> data. </p> <p> In this code base, I've written a little <code>Photo</code> module to extract the desired metadata from an image file. I'm not going to list all the code here; if you're interested, the code is <a href="https://github.com/ploeh/picture-archivist">available on GitHub</a>. The <code>Photo</code> module enables you to write an impure operation like this: </p> <p> <pre><span style="color:green;">//&nbsp;FileInfo&nbsp;-&gt;&nbsp;PhotoFile&nbsp;option</span> <span style="color:blue;">let</span>&nbsp;readPhoto&nbsp;file&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;Photo.extractDateTaken&nbsp;file &nbsp;&nbsp;&nbsp;&nbsp;|&gt;&nbsp;Option.map&nbsp;(<span style="color:blue;">fun</span>&nbsp;dateTaken&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;{&nbsp;File&nbsp;=&nbsp;file;&nbsp;TakenOn&nbsp;=&nbsp;dateTaken&nbsp;})</pre> </p> <p> This operation can fail for various reasons: <ul> <li>The file may not exist.</li> <li>The file exists, but has no metadata.</li> <li>The file has metadata, but no <em>date-taken</em> metadata.</li> <li>The <em>date-taken</em> metadata string is malformed.</li> </ul> When you traverse a <code>Tree&lt;string,string&gt;</code> with <code>readPhoto</code>, you'll get a <code>Tree&lt;string,PhotoFile option&gt;</code>. That's when you'll need <code>Tree.choose</code>. You'll see this soon. </p> <h3 id="59159ef499884e10ae92e5ef6e666c36"> Writing a tree to disk <a href="#59159ef499884e10ae92e5ef6e666c36" title="permalink">#</a> </h3> <p> The above <code>calculateMoves</code> function creates a <code>Tree&lt;string,Move&gt;</code>. The final piece of impure code you'll need to write is an operation that traverses such a tree and executes each <code>Move</code>. </p> <p> <pre><span style="color:green;">//&nbsp;Tree&lt;&#39;a,Move&gt;&nbsp;-&gt;&nbsp;unit</span> <span style="color:blue;">let</span>&nbsp;writeTree&nbsp;t&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;copy&nbsp;m&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Directory.CreateDirectory&nbsp;m.Destination.DirectoryName&nbsp;|&gt;&nbsp;ignore &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;m.Source.CopyTo&nbsp;m.Destination.FullName&nbsp;|&gt;&nbsp;ignore &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;printfn&nbsp;<span style="color:#a31515;">&quot;Copied&nbsp;to&nbsp;%s&quot;</span>&nbsp;m.Destination.FullName &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;compareFiles&nbsp;m&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;sourceStream&nbsp;=&nbsp;File.ReadAllBytes&nbsp;m.Source.FullName &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;destinationStream&nbsp;=&nbsp;File.ReadAllBytes&nbsp;m.Destination.FullName &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;sourceStream&nbsp;=&nbsp;destinationStream &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;move&nbsp;m&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;copy&nbsp;m &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;compareFiles&nbsp;m&nbsp;<span style="color:blue;">then</span>&nbsp;m.Source.Delete&nbsp;() &nbsp;&nbsp;&nbsp;&nbsp;Tree.iter&nbsp;move&nbsp;t</pre> </p> <p> The <code>writeTree</code> function traverses the input tree, and for each <code>Move</code>, it first copies the file, then it verifies that the copy was successful, and finally, if that's the case, it deletes the source file. </p> <h3 id="f30093164b184bbf877f307fa4cf4c63"> Composition <a href="#f30093164b184bbf877f307fa4cf4c63" title="permalink">#</a> </h3> <p> You can now compose an <em>impure-pure-impure sandwich</em> from all the Lego pieces: </p> <p> <pre><span style="color:green;">//&nbsp;string&nbsp;-&gt;&nbsp;string&nbsp;-&gt;&nbsp;unit</span> <span style="color:blue;">let</span>&nbsp;movePhotos&nbsp;source&nbsp;destination&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;sourceTree&nbsp;=&nbsp;readTree&nbsp;source&nbsp;|&gt;&nbsp;Tree.map&nbsp;FileInfo &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;photoTree&nbsp;=&nbsp;Tree.choose&nbsp;readPhoto&nbsp;sourceTree &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;destinationTree&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Option.map&nbsp;(Archive.moveTo&nbsp;destination&nbsp;&gt;&gt;&nbsp;Archive.calculateMoves)&nbsp;photoTree &nbsp;&nbsp;&nbsp;&nbsp;Option.iter&nbsp;writeTree&nbsp;destinationTree</pre> </p> <p> First, you load the <code>sourceTree</code> using the <code>readTree</code> operation. This returns a <code>Tree&lt;string,string&gt;</code>, so map the leaves to <code>FileInfo</code> objects. You then load the image metatadata by traversing <code>sourceTree</code> with <code>Tree.choose readPhoto</code>. Each call to <code>readPhoto</code> produces a <code>PhotoFile option</code>, so this is where you want to use <code>Tree.choose</code> to throw all the <code>None</code> values away. </p> <p> Those two lines of code is the initial impure step of the sandwich (yes: mixed metaphors, I know). </p> <p> The pure part of the sandwich is the composition of the pure functions <code>moveTo</code> and <code>calculateMoves</code>. Since <code>photoTree</code> is a <code>Tree&lt;string,PhotoFile&gt; option</code>, you'll need to perform that transformation inside of <code>Option.map</code>. The resulting <code>destinationTree</code> is a <code>Tree&lt;string,Move&gt; option</code>. </p> <p> The final, impure step of the sandwich, then, is to apply all the moves with <code>writeTree</code>. </p> <h3 id="ab0013f79c184586a10aa014db496bef"> Execution <a href="#ab0013f79c184586a10aa014db496bef" title="permalink">#</a> </h3> <p> The <code>movePhotos</code> operation takes <code>source</code> and <code>destination</code> arguments. You could hypothetically call it from a rich client or a background process, but here I'll just call if from a command-line program. The <code>main</code> operation will have to parse the input arguments and call <code>movePhotos</code>: </p> <p> <pre>[&lt;EntryPoint&gt;] <span style="color:blue;">let</span>&nbsp;main&nbsp;argv&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">match</span>&nbsp;argv&nbsp;<span style="color:blue;">with</span> &nbsp;&nbsp;&nbsp;&nbsp;|&nbsp;[|source;&nbsp;destination|]&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;movePhotos&nbsp;source&nbsp;destination &nbsp;&nbsp;&nbsp;&nbsp;|&nbsp;_&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;printfn&nbsp;<span style="color:#a31515;">&quot;Please&nbsp;provide&nbsp;source&nbsp;and&nbsp;destination&nbsp;directories&nbsp;as&nbsp;arguments.&quot;</span> &nbsp;&nbsp;&nbsp;&nbsp;0&nbsp;<span style="color:green;">//&nbsp;return&nbsp;an&nbsp;integer&nbsp;exit&nbsp;code</span></pre> </p> <p> You could write more sophisticated parsing of the program arguments, but that's not the topic of this article, so I only wrote the bare minimum required to get the program working. </p> <p> You can now compile and run the program: </p> <p> <pre>$ ./ArchivePictures "C:\Users\mark\Desktop\Test" "C:\Users\mark\Desktop\Test-Out" Copied to C:\Users\mark\Desktop\Test-Out\2003-04\2003-04-29 15.11.50.jpg Copied to C:\Users\mark\Desktop\Test-Out\2011-07\2011-07-10 13.09.36.jpg Copied to C:\Users\mark\Desktop\Test-Out\2014-04\2014-04-18 14.05.02.jpg Copied to C:\Users\mark\Desktop\Test-Out\2014-04\2014-04-17 17.11.40.jpg Copied to C:\Users\mark\Desktop\Test-Out\2014-05\2014-05-23 16.07.20.jpg Copied to C:\Users\mark\Desktop\Test-Out\2014-06\2014-06-21 16.48.40.jpg Copied to C:\Users\mark\Desktop\Test-Out\2014-06\2014-06-30 15.44.52.jpg Copied to C:\Users\mark\Desktop\Test-Out\2016-05\2016-05-01 09.25.23.jpg Copied to C:\Users\mark\Desktop\Test-Out\2017-08\2017-08-22 19.53.28.jpg</pre> </p> <p> This does indeed produce the expected destination directory structure. </p> <p> <img src="/content/binary/picture-archivist-destination-directory.png" alt="Seven example directories with pictures."> </p> <p> It's always nice when something turns out to work in practice, as well as in theory. </p> <h3 id="3e4503b89d8f4b81b8b9cac9d1f39021"> Summary <a href="#3e4503b89d8f4b81b8b9cac9d1f39021" title="permalink">#</a> </h3> <p> <a href="/2018/11/19/functional-architecture-a-definition">Functional software architecture</a> involves separating pure from impure code so that no pure functions invoke impure operations. Often, you can achieve that with what I call the <em>impure-pure-impure sandwich</em> architecture. In this example, you saw how to model the file system as a tree. This enables you to separate the impure file interactions from the pure program logic. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Picture archivist in Haskell https://blog.ploeh.dk/2019/09/09/picture-archivist-in-haskell 2019-09-09T08:19:00+00:00 Mark Seemann <div id="post"> <p> <em>A comprehensive code example showing how to implement a functional architecture in Haskell.</em> </p> <p> This article shows how to implement the <a href="/2019/08/26/functional-file-system">picture archivist architecture described in the previous article</a>. In short, the task is to move some image files to directories based on their date-taken metadata. The architectural idea is to load a directory structure from disk into an in-memory tree, manipulate that tree, and use the resulting tree to perform the desired actions: </p> <p> <img src="/content/binary/functional-file-system-interaction.png" alt="A functional program typically loads data, transforms it, and stores it again."> </p> <p> Much of the program will manipulate the tree data, which is immutable. </p> <h3 id="770cf37f0e3c457782ea20b53257f2d1"> Tree <a href="#770cf37f0e3c457782ea20b53257f2d1" title="permalink">#</a> </h3> <p> You can start by defining a <a href="https://en.wikipedia.org/wiki/Rose_tree">rose tree</a>: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;Tree&nbsp;a&nbsp;b&nbsp;=&nbsp;Node&nbsp;a&nbsp;[Tree&nbsp;a&nbsp;b]&nbsp;|&nbsp;Leaf&nbsp;b&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> If you wanted to, you could put all the <code>Tree</code> code in a reusable library, because none of it is coupled to a particular application, such as <a href="https://amzn.to/2V06Kji">moving pictures</a>. You could also write a comprehensive test suite for the following functions, but in this article, I'll skip that. </p> <p> Notice that this sort of tree explicitly distinguishes between internal and leaf nodes. This is necessary because you'll need to keep track of the directory names (the internal nodes), while at the same time you'll want to enrich the leaves with additional data - data that you can't meaningfully add to the internal nodes. You'll see this later in the article. </p> <p> The <a href="/2019/08/05/rose-tree-catamorphism">rose tree catamorphism</a> is this <code>foldTree</code> function: </p> <p> <pre><span style="color:#2b91af;">foldTree</span>&nbsp;::&nbsp;(a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;[c]&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;(b&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Tree</span>&nbsp;a&nbsp;b&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c foldTree&nbsp;&nbsp;_&nbsp;fl&nbsp;(Leaf&nbsp;x)&nbsp;=&nbsp;fl&nbsp;x foldTree&nbsp;fn&nbsp;fl&nbsp;(Node&nbsp;x&nbsp;xs)&nbsp;=&nbsp;fn&nbsp;x&nbsp;$&nbsp;foldTree&nbsp;fn&nbsp;fl&nbsp;&lt;$&gt;&nbsp;xs</pre> </p> <p> Sometimes I name the catamorphism <code>cata</code>, sometimes something like <code>tree</code>, but using a library like <code>Data.Tree</code> as another source of inspiration, in this article I chose to name it <code>foldTree</code>. </p> <p> In this article, tree functionality is (with one exception) directly or transitively implemented with <code>foldTree</code>. </p> <h3 id="f5541d8a36b04cf9a455824c5f3a21c7"> Filtering trees <a href="#f5541d8a36b04cf9a455824c5f3a21c7" title="permalink">#</a> </h3> <p> It'll be useful to be able to filter the contents of a tree. For example, the picture archivist program will only move image files with valid metadata. This means that it'll need to filter out all files that aren't image files, as well as image files without valid metadata. </p> <p> It turns out that it'll be useful to supply a function that throws away <code>Nothing</code> values from a tree of <code>Maybe</code> leaves. This is similar to the <code>catMaybes</code> function from <code>Data.Maybe</code>, so I call it <code>catMaybeTree</code>: </p> <p> <pre><span style="color:#2b91af;">catMaybeTree</span>&nbsp;::&nbsp;<span style="color:blue;">Tree</span>&nbsp;a&nbsp;(<span style="color:#2b91af;">Maybe</span>&nbsp;b)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&nbsp;(<span style="color:blue;">Tree</span>&nbsp;a&nbsp;b) catMaybeTree&nbsp;=&nbsp;foldTree&nbsp;(\x&nbsp;-&gt;&nbsp;Just&nbsp;.&nbsp;Node&nbsp;x&nbsp;.&nbsp;catMaybes)&nbsp;(<span style="color:blue;">fmap</span>&nbsp;Leaf)</pre> </p> <p> You may find the type of the function surprising. Why does it return a <code>Maybe Tree</code>, instead of simply a <code>Tree</code>? And if you accept the type as given, isn't this simply the <code>sequence</code> function? </p> <p> While <code>catMaybes</code> simply returns a list, it can do this because lists can be empty. This <code>Tree</code> type, on the other hand, can't be empty. If the purpose of <code>catMaybeTree</code> is to throw away all <code>Nothing</code> values, then how do you return a tree from <code>Leaf Nothing</code>? </p> <p> You can't return a <code>Leaf</code> because you have no value to put in the leaf. Similarly, you can't return a <code>Node</code> because, again, you have no value to put in the node. </p> <p> In order to handle this edge case, then, you'll have to return <code>Nothing</code>: </p> <p> <pre>Prelude Tree&gt; catMaybeTree $ Leaf Nothing Nothing</pre> </p> <p> Isn't this the same as <code>sequence</code>, then? It's not, because <code>sequence</code> short-circuits all data, as this list example shows: </p> <p> <pre>Prelude&gt; sequence [Just 42, Nothing, Just 2112] Nothing</pre> </p> <p> Contrast this with the behaviour of <code>catMaybes</code>: </p> <p> <pre>Prelude Data.Maybe&gt; catMaybes [Just 42, Nothing, Just 2112] [42,2112]</pre> </p> <p> You've yet to see the <code>Traversable</code> instance for <code>Tree</code>, but it behaves in the same way: </p> <p> <pre>Prelude Tree&gt; sequence $ Node "Foo" [Leaf (Just 42), Leaf Nothing, Leaf (Just 2112)] Nothing</pre> </p> <p> The <code>catMaybeTree</code> function, on the other hand, returns a filtered tree: </p> <p> <pre>Prelude Tree&gt; catMaybeTree $ Node "Foo" [Leaf (Just 42), Leaf Nothing, Leaf (Just 2112)] Just (Node "Foo" [Leaf 42,Leaf 2112])</pre> </p> <p> While the resulting tree is wrapped in a <code>Just</code> case, the leaves contain unwrapped values. </p> <h3 id="5f0287c6d6fe42f3ad73a8e31ba9b3c4"> Instances <a href="#5f0287c6d6fe42f3ad73a8e31ba9b3c4" title="permalink">#</a> </h3> <p> The <a href="/2019/08/05/rose-tree-catamorphism">article about the rose tree catamorphism</a> already covered how to add instances of <code>Bifunctor</code>, <code>Bifoldable</code>, and <code>Bitraversable</code>, so I'll give this only cursory treatment. Refer to that article for a more detailed treatment. The code that accompanies that article also has <a href="http://hackage.haskell.org/package/QuickCheck">QuickCheck</a> properties that verify the various laws associated with those instances. Here, I'll just list the instances without further comment: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Bifunctor</span>&nbsp;<span style="color:blue;">Tree</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;bimap&nbsp;f&nbsp;s&nbsp;=&nbsp;foldTree&nbsp;(Node&nbsp;.&nbsp;f)&nbsp;(Leaf&nbsp;.&nbsp;s) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Bifoldable</span>&nbsp;<span style="color:blue;">Tree</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;bifoldMap&nbsp;f&nbsp;=&nbsp;foldTree&nbsp;(\x&nbsp;xs&nbsp;-&gt;&nbsp;f&nbsp;x&nbsp;&lt;&gt;&nbsp;mconcat&nbsp;xs) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Bitraversable</span>&nbsp;<span style="color:blue;">Tree</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;bitraverse&nbsp;f&nbsp;s&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;foldTree&nbsp;(\x&nbsp;xs&nbsp;-&gt;&nbsp;Node&nbsp;&lt;$&gt;&nbsp;f&nbsp;x&nbsp;&lt;*&gt;&nbsp;sequenceA&nbsp;xs)&nbsp;(<span style="color:blue;">fmap</span>&nbsp;Leaf&nbsp;.&nbsp;s) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;(<span style="color:blue;">Tree</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;=&nbsp;second <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Foldable</span>&nbsp;(<span style="color:blue;">Tree</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;foldMap&nbsp;=&nbsp;bifoldMap&nbsp;mempty <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Traversable</span>&nbsp;(<span style="color:blue;">Tree</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;sequenceA&nbsp;=&nbsp;bisequenceA&nbsp;.&nbsp;first&nbsp;pure</pre> </p> <p> The picture archivist program isn't going to explicitly need all of these, but transitively, it will. </p> <h3 id="d1bbd6ef895f45619822126f44bf6bfb"> Moving pictures <a href="#d1bbd6ef895f45619822126f44bf6bfb" title="permalink">#</a> </h3> <p> So far, all the code shown here could be in a general-purpose reusable library, since it contains no functionality specifically related to image files. The rest of the code in this article, however, will be specific to the program. I'll put the domain model code in another module and import some functionality: </p> <p> <pre><span style="color:blue;">module</span>&nbsp;Archive&nbsp;<span style="color:blue;">where</span> <span style="color:blue;">import</span>&nbsp;Data.Time <span style="color:blue;">import</span>&nbsp;Text.Printf <span style="color:blue;">import</span>&nbsp;System.FilePath <span style="color:blue;">import</span>&nbsp;<span style="color:blue;">qualified</span>&nbsp;Data.Map.Strict&nbsp;<span style="color:blue;">as</span>&nbsp;Map <span style="color:blue;">import</span>&nbsp;Tree</pre> </p> <p> Notice that <code>Tree</code> is one of the imported modules. </p> <p> Later, we'll look at how to load a tree from the file system, but for now, we'll just pretend that we have such a tree. </p> <p> The major logic of the program is to create a destination tree based on a source tree. The leaves of the tree will have to carry some extra information apart from a file path, so you can introduce a specific type to capture that information: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;PhotoFile&nbsp;= &nbsp;&nbsp;PhotoFile&nbsp;{&nbsp;photoFileName&nbsp;::&nbsp;FilePath,&nbsp;takenOn&nbsp;::&nbsp;LocalTime&nbsp;} &nbsp;&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> A <code>PhotoFile</code> not only contains the file path for an image file, but also the date the photo was taken. This date can be extracted from the file's metadata, but that's an impure operation, so we'll delegate that work to the start of the program. We'll return to that later. </p> <p> Given a source tree of <code>PhotoFile</code> leaves, though, the program must produce a destination tree of files: </p> <p> <pre><span style="color:#2b91af;">moveTo</span>&nbsp;::&nbsp;(<span style="color:blue;">Foldable</span>&nbsp;t,&nbsp;<span style="color:blue;">Ord</span>&nbsp;a,&nbsp;<span style="color:blue;">PrintfType</span>&nbsp;a)&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;t&nbsp;<span style="color:blue;">PhotoFile</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Tree</span>&nbsp;a&nbsp;<span style="color:#2b91af;">FilePath</span> moveTo&nbsp;destination&nbsp;= &nbsp;&nbsp;Node&nbsp;destination&nbsp;.&nbsp;Map.foldrWithKey&nbsp;addDir&nbsp;<span style="color:blue;">[]</span>&nbsp;.&nbsp;<span style="color:blue;">foldr</span>&nbsp;groupByDir&nbsp;Map.empty &nbsp;&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;&nbsp;&nbsp;dirNameOf&nbsp;(LocalTime&nbsp;d&nbsp;_)&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;(y,&nbsp;m,&nbsp;_)&nbsp;=&nbsp;toGregorian&nbsp;d&nbsp;<span style="color:blue;">in</span>&nbsp;printf&nbsp;<span style="color:#a31515;">&quot;%d-%02d&quot;</span>&nbsp;y&nbsp;m &nbsp;&nbsp;&nbsp;&nbsp;groupByDir&nbsp;(PhotoFile&nbsp;fileName&nbsp;t)&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Map.insertWith&nbsp;<span style="color:#2b91af;">(++)</span>&nbsp;(dirNameOf&nbsp;t)&nbsp;[fileName] &nbsp;&nbsp;&nbsp;&nbsp;addDir&nbsp;name&nbsp;files&nbsp;dirs&nbsp;=&nbsp;Node&nbsp;name&nbsp;(Leaf&nbsp;&lt;$&gt;&nbsp;files)&nbsp;:&nbsp;dirs</pre> </p> <p> This <code>moveTo</code> function looks, perhaps, overwhelming, but it's composed of only three steps: <ol> <li>Create a map of destination folders (<code>foldr groupByDir Map.empty</code>).</li> <li>Create a list of branches from the map (<code>Map.foldrWithKey addDir []</code>).</li> <li>Create a tree from the list (<code>Node destination</code>).</li> </ol> Recall that when Haskell functions are composed with the <code>.</code> operator, you'll have to read the composition from right to left. </p> <p> Notice that this function works with any <code>Foldable</code> data container, so it'd work with lists and other data structures besides trees. </p> <p> The <code>moveTo</code> function starts by folding the input data into a map. The map is keyed by the directory name, which is formatted by the <code>dirNameOf</code> function. This function takes a <code>LocalTime</code> as input and formats it to a <code>YYYY-MM</code> format. For example, December 20, 2018 becomes <code>"2018-12"</code>. </p> <p> The entire mapping step groups the <code>PhotoFile</code> values into a map of the type <code>Map a [FilePath]</code>. All the image files taken in April 2014 are added to the list with the <code>"2014-04"</code> key, all the image files taken in July 2011 are added to the list with the <code>"2011-07"</code> key, and so on. </p> <p> In the next step, the <code>moveTo</code> function converts the map to a list of trees. This will be the branches (or sub-directories) of the <code>destination</code> directory. Because of the desired structure of the destination tree, this is a list of shallow branches. Each node contains only leaves. </p> <p> <img src="/content/binary/shallow-photo-destination-directories.png" alt="Shallow photo destination directories."> </p> <p> The only remaining step is to add that list of branches to a <code>destination</code> node. </p> <p> Since this is a <a href="https://en.wikipedia.org/wiki/Pure_function">pure function</a>, it's <a href="/2015/05/07/functional-design-is-intrinsically-testable">easy to unit test</a>. Just create some input values and call the function: </p> <p> <pre><span style="color:#a31515;">&quot;Move&nbsp;to&nbsp;destination&quot;</span>&nbsp;~:&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;(source,&nbsp;destination,&nbsp;expected)&nbsp;&lt;- &nbsp;&nbsp;&nbsp;&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(&nbsp;Leaf&nbsp;$&nbsp;PhotoFile&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>&nbsp;$&nbsp;lt&nbsp;2018&nbsp;11&nbsp;9&nbsp;11&nbsp;47&nbsp;17 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;,&nbsp;<span style="color:#a31515;">&quot;D&quot;</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;,&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;D&quot;</span>&nbsp;[Node&nbsp;<span style="color:#a31515;">&quot;2018-11&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>]]) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;S&quot;</span>&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;PhotoFile&nbsp;<span style="color:#a31515;">&quot;4&quot;</span>&nbsp;$&nbsp;lt&nbsp;1972&nbsp;6&nbsp;6&nbsp;16&nbsp;15&nbsp;00] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;,&nbsp;<span style="color:#a31515;">&quot;D&quot;</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;,&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;D&quot;</span>&nbsp;[Node&nbsp;<span style="color:#a31515;">&quot;1972-06&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;4&quot;</span>]]) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;S&quot;</span>&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;PhotoFile&nbsp;<span style="color:#a31515;">&quot;L&quot;</span>&nbsp;$&nbsp;lt&nbsp;2002&nbsp;10&nbsp;12&nbsp;17&nbsp;16&nbsp;15, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;PhotoFile&nbsp;<span style="color:#a31515;">&quot;J&quot;</span>&nbsp;$&nbsp;lt&nbsp;2007&nbsp;4&nbsp;21&nbsp;17&nbsp;18&nbsp;19] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;,&nbsp;<span style="color:#a31515;">&quot;D&quot;</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;,&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;D&quot;</span>&nbsp;[Node&nbsp;<span style="color:#a31515;">&quot;2002-10&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;L&quot;</span>],&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;2007-04&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;J&quot;</span>]]) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;PhotoFile&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;$&nbsp;lt&nbsp;2010&nbsp;1&nbsp;12&nbsp;17&nbsp;16&nbsp;15, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;PhotoFile&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>&nbsp;$&nbsp;lt&nbsp;2010&nbsp;3&nbsp;12&nbsp;17&nbsp;16&nbsp;15, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;PhotoFile&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>&nbsp;$&nbsp;lt&nbsp;2010&nbsp;1&nbsp;21&nbsp;17&nbsp;18&nbsp;19] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;,&nbsp;<span style="color:#a31515;">&quot;2&quot;</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;,&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;2&quot;</span>&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;2010-01&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;Leaf&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>], &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;2010-03&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>]]) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;foo&quot;</span>&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;bar&quot;</span>&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;PhotoFile&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;$&nbsp;lt&nbsp;2010&nbsp;1&nbsp;12&nbsp;17&nbsp;16&nbsp;15, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;PhotoFile&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>&nbsp;$&nbsp;lt&nbsp;2010&nbsp;3&nbsp;12&nbsp;17&nbsp;16&nbsp;15, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;PhotoFile&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>&nbsp;$&nbsp;lt&nbsp;2010&nbsp;1&nbsp;21&nbsp;17&nbsp;18&nbsp;19], &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;baz&quot;</span>&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;PhotoFile&nbsp;<span style="color:#a31515;">&quot;d&quot;</span>&nbsp;$&nbsp;lt&nbsp;2010&nbsp;3&nbsp;1&nbsp;2&nbsp;3&nbsp;4, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;PhotoFile&nbsp;<span style="color:#a31515;">&quot;e&quot;</span>&nbsp;$&nbsp;lt&nbsp;2011&nbsp;3&nbsp;4&nbsp;3&nbsp;2&nbsp;1 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;,&nbsp;<span style="color:#a31515;">&quot;qux&quot;</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;,&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;qux&quot;</span>&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;2010-01&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>,&nbsp;Leaf&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>], &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;2010-03&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>,&nbsp;Leaf&nbsp;<span style="color:#a31515;">&quot;d&quot;</span>], &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;2011-03&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;e&quot;</span>]]) &nbsp;&nbsp;&nbsp;&nbsp;] &nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;actual&nbsp;=&nbsp;moveTo&nbsp;destination&nbsp;source &nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;$&nbsp;expected&nbsp;~=?&nbsp;actual</pre> </p> <p> This is an <a href="/2018/05/07/inlined-hunit-test-lists">inlined</a> <a href="/2018/04/30/parametrised-unit-tests-in-haskell">parametrised HUnit test</a>. While it looks like a big unit test, it still follows my <a href="/2013/06/24/a-heuristic-for-formatting-code-according-to-the-aaa-pattern">test formatting heuristic</a>. There's only three expressions, but the <em>arrange</em> expression is big because it creates a list of test cases. </p> <p> Each test case is a triple of a <code>source</code> tree, a <code>destination</code> directory name, and an <code>expected</code> result. In order to make the test data code more compact, it utilises this test-specific helper function: </p> <p> <pre>lt&nbsp;y&nbsp;mth&nbsp;d&nbsp;h&nbsp;m&nbsp;s&nbsp;=&nbsp;LocalTime&nbsp;(fromGregorian&nbsp;y&nbsp;mth&nbsp;d)&nbsp;(TimeOfDay&nbsp;h&nbsp;m&nbsp;s)</pre> </p> <p> For each test case, the test calls the <code>moveTo</code> function with the <code>destination</code> directory name and the <code>source</code> tree. It then asserts that the <code>expected</code> value is equal to the <code>actual</code> value. </p> <h3 id="bcf9e8fd9d1b42bbb47b811be75385d0"> Calculating moves <a href="#bcf9e8fd9d1b42bbb47b811be75385d0" title="permalink">#</a> </h3> <p> One pure step remains. The result of calling the <code>moveTo</code> function is a tree with the desired structure. In order to actually move the files, though, for each file you'll need to keep track of both the source path and the destination path. To make that explicit, you can define a type for that purpose: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;Move&nbsp;= &nbsp;&nbsp;Move&nbsp;{&nbsp;sourcePath&nbsp;::&nbsp;FilePath,&nbsp;destinationPath&nbsp;::&nbsp;FilePath&nbsp;} &nbsp;&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> A <code>Move</code> is simply a data structure. Contrast this with typical object-oriented design, where it would be a (possibly polymorphic) method on an object. In functional programming, you'll regularly model <em>intent</em> with a data structure. As long as intents remain data, you can easily manipulate them, and once you're done with that, you can run an interpreter over your data structure to perform the work you want accomplished. </p> <p> The unit test cases for the <code>moveTo</code> function suggest that file names are local file names like <code>"L"</code>, <code>"J"</code>, <code>"a"</code>, and so on. That was only to make the tests as compact as possible, since the function actually doesn't manipulate the specific <code>FilePath</code> values. </p> <p> In reality, the file names will most likely be longer, and they could also contain the full path, instead of the local path: <code>"C:\foo\bar\a.jpg"</code>. </p> <p> If you call <code>moveTo</code> with a tree where each leaf has a fully qualified path, the output tree will have the desired structure of the destination tree, but the leaves will still contain the full path to each source file. That means that you can calculate a <code>Move</code> for each file: </p> <p> <pre><span style="color:#2b91af;">calculateMoves</span>&nbsp;::&nbsp;<span style="color:blue;">Tree</span>&nbsp;<span style="color:#2b91af;">FilePath</span>&nbsp;<span style="color:#2b91af;">FilePath</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Tree</span>&nbsp;<span style="color:#2b91af;">FilePath</span>&nbsp;<span style="color:blue;">Move</span> calculateMoves&nbsp;=&nbsp;imp&nbsp;<span style="color:#a31515;">&quot;&quot;</span> &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;imp&nbsp;path&nbsp;&nbsp;&nbsp;&nbsp;(Leaf&nbsp;x)&nbsp;=&nbsp;Leaf&nbsp;$&nbsp;Move&nbsp;x&nbsp;$&nbsp;replaceDirectory&nbsp;x&nbsp;path &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;imp&nbsp;path&nbsp;(Node&nbsp;x&nbsp;xs)&nbsp;=&nbsp;Node&nbsp;(path&nbsp;&lt;/&gt;&nbsp;x)&nbsp;$&nbsp;imp&nbsp;(path&nbsp;&lt;/&gt;&nbsp;x)&nbsp;&lt;$&gt;&nbsp;xs</pre> </p> <p> This function takes as input a <code>Tree FilePath FilePath</code>, which is compatible with the output of <code>moveTo</code>. It returns a <code>Tree FilePath Move</code>, i.e. a tree where the leaves are <code>Move</code> values. </p> <p> To be fair, returning a tree is overkill. A <code>[Move]</code> (list of moves) would have been just as useful, but in this article, I'm trying to describe how to write code with a <a href="/2018/11/19/functional-architecture-a-definition">functional architecture</a>. In the overview article, I explained how you can model a file system using a rose tree, and in order to emphasise that point, I'll stick with that model a little while longer. </p> <p> Earlier, I wrote that you can implement desired <code>Tree</code> functionality with the <code>foldTree</code> function, but that was a simplification. If you can implement the functionality of <code>calculateMoves</code> with <code>foldTree</code>, I don't know how. You can, however, implement it using explicit pattern matching and simple recursion. </p> <p> The <code>imp</code> function builds up a file path (using the <code>&lt;/&gt;</code> path combinator) as it recursively negotiates the tree. All <code>Leaf</code> nodes are converted to a <code>Move</code> value using the leaf node's current <code>FilePath</code> value as the <code>sourcePath</code>, and the <code>path</code> to figure out the desired <code>destinationPath</code>. </p> <p> This code is still easy to unit test: </p> <p> <pre><span style="color:#a31515;">&quot;Calculate&nbsp;moves&quot;</span>&nbsp;~:&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;(tree,&nbsp;expected)&nbsp;&lt;- &nbsp;&nbsp;&nbsp;&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(Leaf&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>,&nbsp;Leaf&nbsp;$&nbsp;Move&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(Node&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>],&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;[Leaf&nbsp;$&nbsp;Move&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>&nbsp;$&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;&lt;/&gt;&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>]), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(Node&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>,&nbsp;Leaf&nbsp;<span style="color:#a31515;">&quot;2&quot;</span>],&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;Move&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>&nbsp;$&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;&lt;/&gt;&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;Move&nbsp;<span style="color:#a31515;">&quot;2&quot;</span>&nbsp;$&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;&lt;/&gt;&nbsp;<span style="color:#a31515;">&quot;2&quot;</span>]), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(Node&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;[Node&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>,&nbsp;Leaf&nbsp;<span style="color:#a31515;">&quot;2&quot;</span>],&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>&nbsp;[Leaf&nbsp;<span style="color:#a31515;">&quot;3&quot;</span>]], &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;&lt;/&gt;&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>)&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;Move&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>&nbsp;$&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;&lt;/&gt;&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>&nbsp;&lt;/&gt;&nbsp;<span style="color:#a31515;">&quot;1&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;Move&nbsp;<span style="color:#a31515;">&quot;2&quot;</span>&nbsp;$&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;&lt;/&gt;&nbsp;<span style="color:#a31515;">&quot;b&quot;</span>&nbsp;&lt;/&gt;&nbsp;<span style="color:#a31515;">&quot;2&quot;</span>], &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Node&nbsp;(<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;&lt;/&gt;&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>)&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Leaf&nbsp;$&nbsp;Move&nbsp;<span style="color:#a31515;">&quot;3&quot;</span>&nbsp;$&nbsp;<span style="color:#a31515;">&quot;a&quot;</span>&nbsp;&lt;/&gt;&nbsp;<span style="color:#a31515;">&quot;c&quot;</span>&nbsp;&lt;/&gt;&nbsp;<span style="color:#a31515;">&quot;3&quot;</span>]]) &nbsp;&nbsp;&nbsp;&nbsp;] &nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;actual&nbsp;=&nbsp;calculateMoves&nbsp;tree &nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;$&nbsp;expected&nbsp;~=?&nbsp;actual</pre> </p> <p> The test cases in this parametrised test are tuples of an input <code>tree</code> and the <code>expected</code> tree. For each test case, the test calls the <code>calculateMoves</code> function with <code>tree</code> and asserts that the <code>actual</code> tree is equal to the <code>expected</code> tree. </p> <p> That's all the pure code you need in order to implement the desired functionality. Now you only need to write some code that loads a tree from disk, and imprints a destination tree to disk, as well as the code that composes it all. </p> <h3 id="062fff475b2b47e188dbd2bc930aa882"> Loading a tree from disk <a href="#062fff475b2b47e188dbd2bc930aa882" title="permalink">#</a> </h3> <p> The remaining code in this article is impure. You could put it in dedicated modules, but for this program, you're only going to need three functions and a bit of composition code, so you could also just put it all in the <code>Main</code> module. That's what I did. </p> <p> To load a tree from disk, you'll need a root directory, under which you load the entire tree. Given a directory path, you read a tree using a recursive function like this: </p> <p> <pre><span style="color:#2b91af;">readTree</span>&nbsp;::&nbsp;<span style="color:#2b91af;">FilePath</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">IO</span>&nbsp;(<span style="color:blue;">Tree</span>&nbsp;<span style="color:#2b91af;">FilePath</span>&nbsp;<span style="color:#2b91af;">FilePath</span>) readTree&nbsp;path&nbsp;=&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;isFile&nbsp;&lt;-&nbsp;doesFileExist&nbsp;path &nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;isFile &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">then</span>&nbsp;<span style="color:blue;">return</span>&nbsp;$&nbsp;Leaf&nbsp;path &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">else</span>&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;dirsAndfiles&nbsp;&lt;-&nbsp;listDirectory&nbsp;path &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;paths&nbsp;=&nbsp;<span style="color:blue;">fmap</span>&nbsp;(path&nbsp;&lt;/&gt;)&nbsp;dirsAndfiles &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;branches&nbsp;&lt;-&nbsp;traverse&nbsp;readTree&nbsp;paths &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;$&nbsp;Node&nbsp;path&nbsp;branches</pre> </p> <p> This recursive function starts by checking whether the <code>path</code> is a file or a directory. If it's a file, it creates a new <code>Leaf</code> with that <code>FilePath</code>. </p> <p> If <code>path</code> isn't a file, it's a directory. In that case, use <code>listDirectory</code> to enumerate all the directories and files in that directory. These are only local names, so prefix them with <code>path</code> to create full paths, then <code>traverse</code> all those directory entries recursively. That produces all the <code>branches</code> for the current node. Finally, return a new <code>Node</code> with the <code>path</code> and the <code>branches</code>. </p> <h3 id="5ba31d6e6e7f4eee942e39349a45e1ed"> Loading metadata <a href="#5ba31d6e6e7f4eee942e39349a45e1ed" title="permalink">#</a> </h3> <p> The <code>readTree</code> function only produces a tree with <code>FilePath</code> leaves, while the program requires a tree with <code>PhotoFile</code> leaves. You'll need to read the <a href="https://en.wikipedia.org/wiki/Exif">Exif</a> metadata from each file and enrich the tree with the <em>date-taken</em> data. </p> <p> In this code base, I've used the <a href="http://hackage.haskell.org/package/hsexif">hsexif</a> library for this. That enables you to write an impure operation like this: </p> <p> <pre><span style="color:#2b91af;">readPhoto</span>&nbsp;::&nbsp;<span style="color:#2b91af;">FilePath</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">IO</span>&nbsp;(<span style="color:#2b91af;">Maybe</span>&nbsp;<span style="color:blue;">PhotoFile</span>) readPhoto&nbsp;path&nbsp;=&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;exifData&nbsp;&lt;-&nbsp;parseFileExif&nbsp;path &nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;dateTaken&nbsp;=&nbsp;either&nbsp;(<span style="color:blue;">const</span>&nbsp;Nothing)&nbsp;Just&nbsp;exifData&nbsp;&gt;&gt;=&nbsp;getDateTimeOriginal &nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;$&nbsp;PhotoFile&nbsp;path&nbsp;&lt;$&gt;&nbsp;dateTaken</pre> </p> <p> This operation can fail for various reasons: <ul> <li>The file may not exist.</li> <li>The file exists, but has no metadata.</li> <li>The file has metadata, but no <em>date-taken</em> metadata.</li> <li>The <em>date-taken</em> metadata string is malformed.</li> </ul> The program is just going to skip all files from which it can't extract <em>date-taken</em> metadata, so <code>readPhoto</code> converts the <code>Either</code> value returned by <code>parseFileExif</code> to <code>Maybe</code> and binds the result with <code>getDateTimeOriginal</code>. </p> <p> When you <code>traverse</code> a <code>Tree FilePath FilePath</code> with <code>readPhoto</code>, you'll get a <code>Tree FilePath (Maybe PhotoFile)</code>. That's when you'll need <code>catMaybeTree</code>. You'll see this soon. </p> <h3 id="8b8d1709f9ed4fe2bc78e4ea9b2a2508"> Writing a tree to disk <a href="#8b8d1709f9ed4fe2bc78e4ea9b2a2508" title="permalink">#</a> </h3> <p> The above <code>calculateMoves</code> function creates a <code>Tree FilePath Move</code>. The final piece of impure code you'll need to write is an operation that traverses such a tree and executes each <code>Move</code>. </p> <p> <pre><span style="color:#2b91af;">applyMoves</span>&nbsp;::&nbsp;<span style="color:blue;">Foldable</span>&nbsp;t&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;t&nbsp;<span style="color:blue;">Move</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">IO</span>&nbsp;() applyMoves&nbsp;=&nbsp;traverse_&nbsp;move &nbsp;&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;&nbsp;&nbsp;move&nbsp;m&nbsp;=&nbsp;copy&nbsp;m&nbsp;&gt;&gt;&nbsp;compareFiles&nbsp;m&nbsp;&gt;&gt;=&nbsp;deleteSource &nbsp;&nbsp;&nbsp;&nbsp;copy&nbsp;(Move&nbsp;s&nbsp;d)&nbsp;=&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;createDirectoryIfMissing&nbsp;True&nbsp;$&nbsp;takeDirectory&nbsp;d &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;copyFileWithMetadata&nbsp;s&nbsp;d &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">putStrLn</span>&nbsp;$&nbsp;<span style="color:#a31515;">&quot;Copied&nbsp;to&nbsp;&quot;</span>&nbsp;++&nbsp;<span style="color:blue;">show</span>&nbsp;d &nbsp;&nbsp;&nbsp;&nbsp;compareFiles&nbsp;m@(Move&nbsp;s&nbsp;d)&nbsp;=&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;sourceBytes&nbsp;&lt;-&nbsp;B.<span style="color:blue;">readFile</span>&nbsp;s &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;destinationBytes&nbsp;&lt;-&nbsp;B.<span style="color:blue;">readFile</span>&nbsp;d &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;$&nbsp;<span style="color:blue;">if</span>&nbsp;sourceBytes&nbsp;==&nbsp;destinationBytes&nbsp;<span style="color:blue;">then</span>&nbsp;Just&nbsp;m&nbsp;<span style="color:blue;">else</span>&nbsp;Nothing &nbsp;&nbsp;&nbsp;&nbsp;deleteSource&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Nothing&nbsp;=&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">()</span> &nbsp;&nbsp;&nbsp;&nbsp;deleteSource&nbsp;(Just&nbsp;(Move&nbsp;s&nbsp;_))&nbsp;=&nbsp;removeFile&nbsp;s</pre> </p> <p> As I wrote above, a tree of <code>Move</code> values is, to be honest, overkill. Any <code>Foldable</code> container will do, as the <code>applyMoves</code> operation demonstrates. It traverses the data structure, and for each <code>Move</code>, it first copies the file, then it verifies that the copy was successful, and finally, if that's the case, it deletes the source file. </p> <p> All of the operations invoked by these three steps are defined in various libraries part of the base GHC installation. You're welcome to peruse <a href="https://github.com/ploeh/picture-archivist">the source code repository</a> if you're interested in the details. </p> <h3 id="d336cf55dc9746c08cbed32041803173"> Composition <a href="#d336cf55dc9746c08cbed32041803173" title="permalink">#</a> </h3> <p> You can now compose an <em>impure-pure-impure sandwich</em> from all the Lego pieces: </p> <p> <pre><span style="color:#2b91af;">movePhotos</span>&nbsp;::&nbsp;<span style="color:#2b91af;">FilePath</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">FilePath</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">IO</span>&nbsp;() movePhotos&nbsp;source&nbsp;destination&nbsp;=&nbsp;<span style="color:blue;">fmap</span>&nbsp;fold&nbsp;$&nbsp;runMaybeT&nbsp;$&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;sourceTree&nbsp;&lt;-&nbsp;lift&nbsp;$&nbsp;readTree&nbsp;source &nbsp;&nbsp;photoTree&nbsp;&lt;-&nbsp;MaybeT&nbsp;$&nbsp;catMaybeTree&nbsp;&lt;$&gt;&nbsp;traverse&nbsp;readPhoto&nbsp;sourceTree &nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;destinationTree&nbsp;=&nbsp;calculateMoves&nbsp;$&nbsp;moveTo&nbsp;destination&nbsp;photoTree &nbsp;&nbsp;lift&nbsp;$&nbsp;applyMoves&nbsp;destinationTree</pre> </p> <p> First, you load the <code>sourceTree</code> using the <code>readTree</code> operation. This is a <code>Tree FilePath FilePath</code> value, because the code is written in <code>do</code> notation, and the context is <code>MaybeT IO ()</code>. You then load the image metatadata by traversing <code>sourceTree</code> with <code>readPhoto</code>. This produces a <code>Tree FilePath (Maybe PhotoFile)</code> that you then filter with <code>catMaybeTree</code>. Again, because of <code>do</code> notation and monad transformer shenanigans, <code>photoTree</code> is a <code>Tree FilePath PhotoFile</code> value. </p> <p> Those two lines of code is the initial impure step of the sandwich (yes: mixed metaphors, I know). </p> <p> The pure part of the sandwich is the composition of the pure functions <code>moveTo</code> and <code>calculateMoves</code>. The result is a <code>Tree FilePath Move</code> value. </p> <p> The final, impure step of the sandwich, then, is to <code>applyMoves</code>. </p> <h3 id="8b44f4d2cd2241e18bff6d40c1ad9ee9"> Execution <a href="#8b44f4d2cd2241e18bff6d40c1ad9ee9" title="permalink">#</a> </h3> <p> The <code>movePhotos</code> operation takes <code>source</code> and <code>destination</code> arguments. You could hypothetically call it from a rich client or a background process, but here I'll just call if from a command-line program. The <code>main</code> operation will have to parse the input arguments and call <code>movePhotos</code>: </p> <p> <pre><span style="color:#2b91af;">main</span>&nbsp;::&nbsp;<span style="color:#2b91af;">IO</span>&nbsp;() main&nbsp;=&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;args&nbsp;&lt;-&nbsp;getArgs &nbsp;&nbsp;<span style="color:blue;">case</span>&nbsp;args&nbsp;<span style="color:blue;">of</span> &nbsp;&nbsp;&nbsp;&nbsp;[source,&nbsp;destination]&nbsp;-&gt;&nbsp;movePhotos&nbsp;source&nbsp;destination &nbsp;&nbsp;&nbsp;&nbsp;_&nbsp;-&gt;&nbsp;<span style="color:blue;">putStrLn</span>&nbsp;<span style="color:#a31515;">&quot;Please&nbsp;provide&nbsp;source&nbsp;and&nbsp;destination&nbsp;directories&nbsp;as&nbsp;arguments.&quot;</span></pre> </p> <p> You could write more sophisticated parsing of the program arguments, but that's not the topic of this article, so I only wrote the bare minimum required to get the program working. </p> <p> You can now compile and run the program: </p> <p> <pre>$ ./archpics "C:\Users\mark\Desktop\Test" "C:\Users\mark\Desktop\Test-Out" Copied to "C:\\Users\\mark\\Desktop\\Test-Out\\2003-04\\2003-04-29 15.11.50.jpg" Copied to "C:\\Users\\mark\\Desktop\\Test-Out\\2011-07\\2011-07-10 13.09.36.jpg" Copied to "C:\\Users\\mark\\Desktop\\Test-Out\\2014-04\\2014-04-17 17.11.40.jpg" Copied to "C:\\Users\\mark\\Desktop\\Test-Out\\2014-04\\2014-04-18 14.05.02.jpg" Copied to "C:\\Users\\mark\\Desktop\\Test-Out\\2014-05\\2014-05-23 16.07.20.jpg" Copied to "C:\\Users\\mark\\Desktop\\Test-Out\\2014-06\\2014-06-30 15.44.52.jpg" Copied to "C:\\Users\\mark\\Desktop\\Test-Out\\2014-06\\2014-06-21 16.48.40.jpg" Copied to "C:\\Users\\mark\\Desktop\\Test-Out\\2016-05\\2016-05-01 09.25.23.jpg" Copied to "C:\\Users\\mark\\Desktop\\Test-Out\\2017-08\\2017-08-22 19.53.28.jpg"</pre> </p> <p> This does indeed produce the expected destination directory structure. </p> <p> <img src="/content/binary/picture-archivist-destination-directory.png" alt="Seven example directories with pictures."> </p> <p> It's always nice when something turns out to work in practice, as well as in theory. </p> <h3 id="c50c7ac1276146d79715a5e7ddadfe6d"> Summary <a href="#c50c7ac1276146d79715a5e7ddadfe6d" title="permalink">#</a> </h3> <p> Functional software architecture involves separating pure from impure code so that no pure functions invoke impure operations. Often, you can achieve that with what I call the <em>impure-pure-impure sandwich</em> architecture. In this example, you saw how to model the file system as a tree. This enables you to separate the impure file interactions from the pure program logic. </p> <p> The Haskell type system enforces the <em>functional interaction law</em>, which implies that the architecture is, indeed, properly functional. Other languages, like <a href="https://fsharp.org">F#</a>, don't enforce the law via the compiler, but that doesn't prevent you doing functional programming. Now that we've verified that the architecture is, indeed, functional, we can port it to F#. </p> <p> <strong>Next:</strong> <a href="/2019/09/16/picture-archivist-in-f">Picture archivist in F#</a>. </p> </div> <div id="comments"> <hr> <h2 id="comments-header"> Comments </h2> <div class="comment" id="f237d98d453a4bcb9a3d58a05bf21d34"> <div class="comment-author"><a href="https://majiehong.com">Jiehong</a></div> <div class="comment-content"> <p> This seems a fair architecture. </p> <p> However, at first glance it does not seem very memory efficient, because everything might be loaded in RAM, and that poses a strict limit. </p> <p> But then, I remember that Haskell does lazy evaluation, so is it the case here? Are path and the tree lazily loaded and processed? </p> <p> In "traditional" architectures, IO would be scattered inside the program, and as each file might be read one at a time, and handled. This sandwich of purity with impure buns forces not to do that. </p> </div> <div class="comment-date">2019-09-09 11:47 UTC</div> </div> <div class="comment" id="ca660cdc1f094bfb8cc9896bb1084460"> <div class="comment-author"><a href="/">Mark Seemann</a></div> <div class="comment-content"> <p> Jiehong, thank you for writing. It's true that Haskell is lazily evaluated, but some strictness rules apply to <code>IO</code>, so it's not so simple. </p> <p> Just running a quick experiment with the code base shown here, when I try to move thousands of files, the program sits and thinks for quite some time before it starts to output progress. This indicates to me that it does, indeed, load at least the <em>structure</em> of the tree into memory before it starts moving the files. Once it does that, though, it looks like it runs at constant memory. </p> <p> There's an interplay of laziness and <code>IO</code> in Haskell that I still don't sufficiently master. When I publish the port to F#, however, it should be clear that you could replace all the nodes of the tree with explicitly lazy values. I'd be surprised if something like that isn't possible in Haskell as well, but here I'll solicit help from readers more well-versed in these matters than I am. </p> </div> <div class="comment-date">2019-09-09 19:16 UTC</div> </div> <div class="comment" id="dd26f6d047b5492b8a012b30d96ad18b"> <div class="comment-author">André Cardoso</div> <div class="comment-content"> <p> I really like your posts and I'm really liking this series. But I struggle with Haskell syntax, specially the difference between the operators $, &lt;$&gt;, &lt;&gt;, &lt;*&gt;. Is there a cheat sheet explaining these operators? </p> </div> <div class="comment-date">2019-09-12 13:51 UTC</div> </div> <div class="comment" id="2e71f695ed9f4cfa8467df818f072da8"> <div class="comment-author"><a href="/">Mark Seemann</a></div> <div class="comment-content"> <p> André, thank you for writing. I've written about why <a href="/2018/07/02/terse-operators-make-business-code-more-readable">I think that terse operators make the code overall more readable</a>, but that's obviously not an explanation of any of those operators. </p> <p> I'm not aware of any cheat sheets for Haskell, although a Google search seems to indicate that many exist. I'm not sure that a cheat sheet will help much if one doesn't know Haskell, and if one does know Haskell, one is likely to also know those operators. </p> <p> <a href="https://hackage.haskell.org/package/base/docs/Prelude.html#v:-36-">$</a> is a sort of delimiter that often saves you from having to nest other function calls in brackets. </p> <p> <a href="https://hackage.haskell.org/package/base/docs/Prelude.html#v:-60--36--62-">&lt;$&gt;</a> is just an infix alias for <code>fmap</code>. In C#, that <a href="/2018/03/22/functors">corresponds to the <code>Select</code> method</a>. </p> <p> <code>&lt;&gt;</code> is a generalised associative binary operation as defined by <a href="http://hackage.haskell.org/package/base/docs/Data-Semigroup.html">Data.Semigroup</a> or <a href="http://hackage.haskell.org/package/base/docs/Data-Monoid.html">Data.Monoid</a>. You can <a href="/2017/10/05/monoids-semigroups-and-friends">read more about monoids and semigroups here on the blog</a>. </p> <p> <a href="http://hackage.haskell.org/package/base/docs/Control-Applicative.html">&lt;*&gt;</a> is part of the <code>Applicative</code> type class. It's hard to translate to other languages, but <a href="/2018/10/01/applicative-functors">when I make the attempt</a>, I usually call it <code>Apply</code>. </p> </div> <div class="comment-date">2019-09-12 15:45 UTC</div> </div> </div> <hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Naming newtypes for QuickCheck Arbitraries https://blog.ploeh.dk/2019/09/02/naming-newtypes-for-quickcheck-arbitraries 2019-09-02T13:07:00+00:00 Mark Seemann <div id="post"> <p> <em>A simple naming scheme for newtypes to add Arbitrary instances.</em> </p> <p> Naming is one of those recurring difficult problems in software development. How do you come up with good names? </p> <p> I'm not aware of any <em>general</em> heuristic for that, but sometimes, in specific contexts, a naming scheme presents itself. Here's one. </p> <h3 id="c7391ad662e943f1bbe2b52d6b8bde59"> Orphan instances <a href="#c7391ad662e943f1bbe2b52d6b8bde59" title="permalink">#</a> </h3> <p> When you write <a href="http://hackage.haskell.org/package/QuickCheck">QuickCheck</a> properties that involve your own custom types, you'll have to add <code>Arbitrary</code> instances for those types. As an example, here's a restaurant reservation record type: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;Reservation&nbsp;=&nbsp;Reservation &nbsp;&nbsp;{&nbsp;reservationId&nbsp;::&nbsp;UUID &nbsp;&nbsp;,&nbsp;reservationDate&nbsp;::&nbsp;LocalTime &nbsp;&nbsp;,&nbsp;reservationName&nbsp;::&nbsp;String &nbsp;&nbsp;,&nbsp;reservationEmail&nbsp;::&nbsp;String &nbsp;&nbsp;,&nbsp;reservationQuantity&nbsp;::&nbsp;Int &nbsp;&nbsp;}&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Read</span>,&nbsp;<span style="color:#2b91af;">Generic</span>)</pre> </p> <p> You can easily add an Arbitrary instance to such a type: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Arbitrary</span>&nbsp;<span style="color:blue;">Reservation</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;arbitrary&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;liftM5&nbsp;Reservation&nbsp;arbitrary&nbsp;arbitrary&nbsp;arbitrary&nbsp;arbitrary&nbsp;arbitrary</pre> </p> <p> The type itself is part of your domain model, while the <code>Arbitrary</code> instance only belongs to your test code. You shouldn't add the <code>Arbitrary</code> instance to the domain model, but that means that you'll have to define the instance apart from the type definition. That, however, is an orphan instance, and the compiler will complain: </p> <p> <pre>test\ReservationAPISpec.hs:31:1: <span style="color:red;">warning:</span> [<span style="color:red;">-Worphans</span>] Orphan instance: instance Arbitrary Reservation To avoid this move the instance declaration to the module of the class or of the type, or wrap the type with a newtype and declare the instance on the new type. <span style="color:blue;">|</span> <span style="color:blue;">31 |</span> <span style="color:red;">instance Arbitrary Reservation where</span> <span style="color:blue;">|</span> <span style="color:red;">^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^...</span></pre> </p> <p> Technically, this isn't a difficult problem to solve. The warning even suggests remedies. Moving the instance to the module that declares the type is, however, inappropriate, since test-specific instances don't belong in the domain model. Wrapping the type in a <code>newtype</code> is more appropriate, but what should you call the type? </p> <h3 id="c192d6524b4b4444a35121443f9a61a8"> Suppress the warning <a href="#c192d6524b4b4444a35121443f9a61a8" title="permalink">#</a> </h3> <p> I had trouble coming up with good names for such <code>newtype</code> wrappers, so at first I decided to just suppress that particular compiler warning. I simply added the <code>-fno-warn-orphans</code> flag <em>exclusively to my test code</em>. </p> <p> That solved the immediate problem, but I felt a little dirty. It's okay, though, because you're not supposed to reuse test libraries anyway, so the usual problems with orphan instances don't apply. </p> <p> After having worked a little like this, however, it dawned on me that I needed more than one <code>Arbitrary</code> instance, and a naming scheme presented itself. </p> <h3 id="a946b2c622c6403cb69a3f224551514c"> Naming scheme <a href="#a946b2c622c6403cb69a3f224551514c" title="permalink">#</a> </h3> <p> For some of the properties I wrote, I needed a <em>valid</em> <code>Reservation</code> value. In this case, <em>valid</em> means that the <code>reservationQuantity</code> is a positive number, and that the <code>reservationDate</code> is in the future. It seemed natural to signify these constraints with a <code>newtype</code>: </p> <p> <pre><span style="color:blue;">newtype</span>&nbsp;ValidReservation&nbsp;=&nbsp;ValidReservation&nbsp;Reservation&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Show</span>) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Arbitrary</span>&nbsp;<span style="color:blue;">ValidReservation</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;arbitrary&nbsp;=&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;&nbsp;&nbsp;rid&nbsp;&lt;-&nbsp;arbitrary &nbsp;&nbsp;&nbsp;&nbsp;d&nbsp;&lt;-&nbsp;(\dt&nbsp;-&gt;&nbsp;addLocalTime&nbsp;(getPositive&nbsp;dt)&nbsp;now2019)&nbsp;&lt;$&gt;&nbsp;arbitrary &nbsp;&nbsp;&nbsp;&nbsp;n&nbsp;&lt;-&nbsp;arbitrary &nbsp;&nbsp;&nbsp;&nbsp;e&nbsp;&lt;-&nbsp;arbitrary &nbsp;&nbsp;&nbsp;&nbsp;(Positive&nbsp;q)&nbsp;&lt;-&nbsp;arbitrary &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;$&nbsp;ValidReservation&nbsp;$&nbsp;Reservation&nbsp;rid&nbsp;d&nbsp;n&nbsp;e&nbsp;q</pre> </p> <p> The <code>newtype</code> is, naturally, called <code>ValidReservation</code> and can, for example, be used like this: </p> <p> <pre>it&nbsp;<span style="color:#a31515;">&quot;responds&nbsp;with&nbsp;200&nbsp;after&nbsp;reservation&nbsp;is&nbsp;added&quot;</span>&nbsp;$&nbsp;WQC.property&nbsp;$&nbsp;\ &nbsp;&nbsp;(ValidReservation&nbsp;r)&nbsp;-&gt;&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;_&nbsp;&lt;-&nbsp;postJSON&nbsp;<span style="color:#a31515;">&quot;/reservations&quot;</span>&nbsp;$&nbsp;encode&nbsp;r &nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;actual&nbsp;=&nbsp;get&nbsp;$&nbsp;<span style="color:#a31515;">&quot;/reservations/&quot;</span>&nbsp;&lt;&gt;&nbsp;toASCIIBytes&nbsp;(reservationId&nbsp;r) &nbsp;&nbsp;actual&nbsp;`shouldRespondWith`&nbsp;200</pre> </p> <p> For the few properties where <em>any</em> <code>Reservation</code> goes, a name for a <code>newtype</code> now also suggests itself: </p> <p> <pre><span style="color:blue;">newtype</span>&nbsp;AnyReservation&nbsp;=&nbsp;AnyReservation&nbsp;Reservation&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Show</span>) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Arbitrary</span>&nbsp;<span style="color:blue;">AnyReservation</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;arbitrary&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;AnyReservation&nbsp;&lt;$&gt; &nbsp;&nbsp;&nbsp;&nbsp;liftM5&nbsp;Reservation&nbsp;arbitrary&nbsp;arbitrary&nbsp;arbitrary&nbsp;arbitrary&nbsp;arbitrary</pre> </p> <p> The only use I've had for that particular instance so far, though, is to ensure that any <code>Reservation</code> correctly serialises to, and deserialises from, JSON: </p> <p> <pre>it&nbsp;<span style="color:#a31515;">&quot;round-trips&quot;</span>&nbsp;$&nbsp;property&nbsp;$&nbsp;\(AnyReservation&nbsp;r)&nbsp;-&gt;&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;json&nbsp;=&nbsp;encode&nbsp;r &nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;actual&nbsp;=&nbsp;decode&nbsp;json &nbsp;&nbsp;actual&nbsp;`shouldBe`&nbsp;Just&nbsp;r</pre> </p> <p> With those two <code>newtype</code> wrappers, I no longer have any orphan instances. </p> <h3 id="758fef8609784b998c3fad65b2fe6e2f"> Summary <a href="#758fef8609784b998c3fad65b2fe6e2f" title="permalink">#</a> </h3> <p> A simple naming scheme for <code>newtype</code> wrappers for QuickCheck <code>Arbitrary</code> instances, then, is: <ul> <li>If the instance is truly unbounded, prefix the wrapper name with <em>Any</em></li> <li>If the instance only produces valid values, prefix the wrapper name with <em>Valid</em></li> </ul> This strikes me as a practical naming scheme. Other variations seem natural. If, for example, you need an <em>invalid</em> value, you can prefix the wrapper name with <em>Invalid</em>. Why you'd need that, though, I'm not sure. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Functional file system https://blog.ploeh.dk/2019/08/26/functional-file-system 2019-08-26T06:00:00+00:00 Mark Seemann <div id="post"> <p> <em>How do you model file systems in a functional manner, so that unit testing is enabled? An overview.</em> </p> <p> One of the many reasons that I like functional programming is that it's <a href="/2015/05/07/functional-design-is-intrinsically-testable">intrinsically testable</a>. In object-oriented programming, you often have to jump through hoops to enable testing. This is also the case whenever you need to interact with the computer's file system. Just try to search the web for <em>file system interface</em>, or <em>mock file system</em>. I'm not going to give you any links, because I think such questions are <a href="https://en.wikipedia.org/wiki/XY_problem">XY problems</a>. I don't think that the most common suggestions are proper solutions. </p> <p> In functional programming, anyway, <a href="/2017/01/30/partial-application-is-dependency-injection">Dependency Injection isn't functional, because it makes everything impure</a>. How, then, do you model the file system in such a way that it's pure, decoupled from the logic you'd like to add on top of it, and still has enough fidelity that you can perform most tasks? </p> <p> You model the file system as a tree, or a forest. </p> <h3 id="4920bedd948d4f7487a13fa96f836371"> File systems are hierarchies <a href="#4920bedd948d4f7487a13fa96f836371" title="permalink">#</a> </h3> <p> It should come as no surprise that file systems are hierarchies, or trees. Each logical drive is the root of a tree. Files are leaves, and directories are internal nodes. Does that sound familiar? That sounds like a <a href="/2019/07/29/church-encoded-rose-tree">rose tree</a>. </p> <p> Rose trees are immutable data structures. It doesn't get much more functional than that. Why not using a rose tree (or a forest) to model the file system? </p> <p> What about interaction with the actual file system? Usually, when you encounter object-oriented attempts at decoupling an abstraction from the actual file system, you'll find polymorphic operations such as <code>WriteAllText</code>, <code>GetFileSystemEntries</code>, <code>CreateDirectory</code>, and so on. These would be the (mockable) methods that you have to implement, usually as <a href="http://xunitpatterns.com/Humble%20Object.html">Humble Objects</a>. </p> <p> If you, instead of a set of interfaces, model the file system as a forest, interacting with the actual file system is not even part of the abstraction. That's a typical shift of perspective from object-oriented design to functional programming. </p> <p> <img src="/content/binary/ood-and-fp-views-on-fily-system-abstraction.png" alt="Object-oriented and functional ways to abstractly model file systems."> </p> <p> In object-oriented design, you typically attempt to model <em>data with behaviour</em>. Sometimes that fits the underlying reality well, but in this case it doesn't. While you have file and directory objects with behaviour, the actual structure of a file system is implicit. It's hidden in the interactions between the objects. </p> <p> By modelling the file system as a tree, you explicitly use the structure of the data. How you load a tree into program memory, or how you imprint a tree unto the file system isn't part of the abstraction. When it comes to input and output, you're free to do what you want. </p> <p> Once you have a model of a directory structure in memory, you can manipulate it to your heart's content. Since <a href="/2019/08/19/a-rose-tree-functor">rose trees are functors</a>, you know that all transformations are structure-preserving. That means that you don't even need to write tests for those parts of your application. </p> <p> You'll appreciate an example, I'm sure. </p> <h3 id="5e19438122b94e059c155509e96c964f"> Picture archivist example <a href="#5e19438122b94e059c155509e96c964f" title="permalink">#</a> </h3> <p> As an example, I'll attempt to answer <a href="https://codereview.stackexchange.com/q/99271/3878">an old Code Review question</a>. I already gave <a href="https://codereview.stackexchange.com/a/99290/3878">an answer</a> in 2015, but I'm not so happy with it today as I was back then. The question is great, though, because it explicitly demonstrates how people have a hard time escaping the notion that abstraction is only available via interfaces or abstract base classes. In 2015, I had long since figured out that <a href="/2009/05/28/DelegatesAreAnonymousInterfaces">delegates (and thus functions) are anonymous interfaces</a>, but I still hadn't figured out how to separate pure from impure behaviour. </p> <p> The question's scenario is how to implement a small program that can inspect a collection of image files, extract the date-taken metadata from each file, and move the files to a new directory structure based on that information. </p> <p> For example, you could have files organised in various directories according to motive. </p> <p> <img src="/content/binary/picture-archivist-source-directory.png" alt="Three example directories with pictures."> </p> <p> You soon realise, however, that that archiving strategy is untenable, because what do you do if there's more than one type of motive in a picture? Instead, you decide to organise the files according to month and year. </p> <p> <img src="/content/binary/picture-archivist-destination-directory.png" alt="Seven example directories with pictures."> </p> <p> Clearly, there's some input and output involved in this application, but there's also some logic that you'd like to unit test. You need to parse the metadata, figure out where to move each image file, filter out files that are not images, and so on. </p> <h3 id="e3bc8b23a3494628a44348749a0369ca"> Object-oriented picture archivist <a href="#e3bc8b23a3494628a44348749a0369ca" title="permalink">#</a> </h3> <p> If you were to implement such a picture archivist program with an object-oriented design, you may use Dependency Injection so that you can 'mock' the file system during unit testing. A typical program might then work like this at run time: </p> <p> <img src="/content/binary/object-oriented-file-system-interaction.png" alt="An object-oriented program typically has busy interaction with the file system."> </p> <p> The program has fine-grained, busy interaction with the file system (through a polymorphic interface). It'll typically read one file, load its metadata, decide where to put the file, and copy it there. Then it'll move on to the next file, although it might also do this in parallel. Throughout the program execution, there's input and output going on, which makes it difficult to isolate the pure from the impure code. </p> <p> Even if you write a program like that in <a href="https://fsharp.org">F#</a>, it's hardly a <a href="/2018/11/19/functional-architecture-a-definition">functional architecture</a>. </p> <p> Such an architecture is, in theory, testable, but my experience is that if you attempt to reproduce such busy, fine-grained interaction with mocks and stubs, you're likely to end up with brittle tests. </p> <h3 id="6cddf0e7ca3549c49a87006bfba5d349"> Functional picture archivist <a href="#6cddf0e7ca3549c49a87006bfba5d349" title="permalink">#</a> </h3> <p> In functional programming, you'll have to <a href="/2017/02/02/dependency-rejection">reject the notion of dependencies</a>. Instead, you can often resort to the simple architecture I call an <em>impure-pure-impure sandwich</em>; here, specifically: <ol> <li>Load data from disk (impure)</li> <li>Transform the data (pure)</li> <li>Write data to disk (impure)</li> </ol> A typical program might then work like this at run time: </p> <p> <img src="/content/binary/functional-file-system-interaction.png" alt="A functional program typically loads data, transforms it, and stores it again."> </p> <p> When the program starts, it loads data from disk into a tree. It then manipulates the in-memory model of the files in question, and once it's done, it traverses the entire tree and applies the changes. </p> <p> This gives you a much clearer separation between the pure and impure parts of the code base. The pure part is bigger, and easier to unit test. </p> <h3 id="09d2184be64a428d85b4f01f1149ea7a"> Example code <a href="#09d2184be64a428d85b4f01f1149ea7a" title="permalink">#</a> </h3> <p> This article gave you an overview of the functional architecture. In the next two articles, you'll see how to do this in practice. First, I'll implement the above architecture in <a href="https://www.haskell.org">Haskell</a>, so that we know that if it works there, the architecture does, indeed, respect <a href="/2018/11/19/functional-architecture-a-definition">the functional interaction law</a>. </p> <p> Based on the Haskell implementation, you'll then see a port to F#. <ul> <li><a href="/2019/09/09/picture-archivist-in-haskell">Picture archivist in Haskell</a></li> <li><a href="/2019/09/16/picture-archivist-in-f">Picture archivist in F#</a></li> </ul> These two articles share the same architecture. You can read both, or one of them, as you like. The source code is available on GitHub. </p> <h3 id="09e32b681b7a48aa808965bd66c4794b"> Summary <a href="#09e32b681b7a48aa808965bd66c4794b" title="permalink">#</a> </h3> <p> One of the hardest problems in transitioning from object-oriented programming to functional programming is that the design approach is so different. Many well-understood design patterns and principles don't translate easily. Dependency Injection is one of those. Often, you'll have to flip the model on its head, so to speak, before you can take it on in a functional manner. </p> <p> While most object-oriented programmers would say that object-oriented design involves focusing on 'the nouns', in practice, it often revolves around interactions and behaviour. Sometimes, that's appropriate, but often, it's not. </p> <p> Functional programming, in contrast, tends to take a more data-oriented perspective. Load some data, manipulate it, and publish it. If you can come up with an appropriate data structure for the data, you're probably on your way to implementing a functional architecture. </p> <p> <strong>Next:</strong> <a href="/2019/09/09/picture-archivist-in-haskell">Picture archivist in Haskell</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. A rose tree functor https://blog.ploeh.dk/2019/08/19/a-rose-tree-functor 2019-08-19T08:08:00+00:00 Mark Seemann <div id="post"> <p> <em>Rose trees form normal functors. A place-holder article for object-oriented programmers.</em> </p> <p> This article is an instalment in <a href="/2018/03/22/functors">an article series about functors</a>. As another article explains, <a href="/2019/08/12/rose-tree-bifunctor">a rose tree is a bifunctor</a>. This makes it trivially a functor. As such, this article is mostly a place-holder to fit the spot in the <em>functor table of contents</em>, thereby indicating that rose trees are functors. </p> <p> Since a rose tree is a bifunctor, it's actually not one, but two, functors. Many languages, C# included, are best equipped to deal with unambiguous functors. This is also true in <a href="https://haskell.org">Haskell</a>, where you'd usally define the <code>Functor</code> instance over a bifunctor's right, or second, side. Likewise, in C#, you can make <code>IRoseTree&lt;N, L&gt;</code> a functor by implementing <code>Select</code>: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L1</span>&gt;&nbsp;Select&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>,&nbsp;<span style="color:#2b91af;">L1</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;source, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">L</span>,&nbsp;<span style="color:#2b91af;">L1</span>&gt;&nbsp;selector) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;source.SelectLeaf(selector); }</pre> </p> <p> This method simply delegates all implementation to the <code>SelectLeaf</code> method; it's just <code>SelectLeaf</code> by another name. It obeys the functor laws, since these are just specializations of the bifunctor laws, and we know that a rose tree is a proper bifunctor. </p> <p> It would have been technically possible to instead implement a <code>Select</code> method by calling <code>SelectNode</code>, but it seems marginally more useful to enable syntactic sugar for mapping over the leaves. </p> <h3 id="134b75d98069421e9fe70a8630ac140f"> Menu example <a href="#134b75d98069421e9fe70a8630ac140f" title="permalink">#</a> </h3> <p> As an example, imagine that you're defining part of a menu bar for an old-fashioned desktop application. Perhaps you're even loading the structure of the menu from a text file. Doing so, you could create a simple tree that represents the <em>edit</em> menu: </p> <p> <pre><span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;editMenuTemplate&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(<span style="color:#a31515;">&quot;Edit&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(<span style="color:#a31515;">&quot;Find&nbsp;and&nbsp;Replace&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;Find&quot;</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;Replace&quot;</span>)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(<span style="color:#a31515;">&quot;Case&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;Upper&quot;</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;Lower&quot;</span>)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;Cut&quot;</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;Copy&quot;</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;Paste&quot;</span>));</pre> </p> <p> At this point, you have an <code>IRoseTree&lt;string, string&gt;</code>, so you might as well have used a <a href="/2018/08/06/a-tree-functor">'normal' tree</a> instead of a rose tree. The above template, however, is only a first step, because you have this <a href="https://en.wikipedia.org/wiki/Command_pattern">Command</a> class: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">Command</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;Command(<span style="color:blue;">string</span>&nbsp;name) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Name&nbsp;=&nbsp;name; &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:blue;">string</span>&nbsp;Name&nbsp;{&nbsp;<span style="color:blue;">get</span>;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:blue;">virtual</span>&nbsp;<span style="color:blue;">void</span>&nbsp;Execute() &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;} }</pre> </p> <p> Apart from this base class, you also have classes that derive from it: <code>FindCommand</code>, <code>ReplaceCommand</code>, and so on. These classes override the <code>Execute</code> method by implenting <em>find</em>, <em>replace</em>, etc. functionality. Imagine that you also have a store or dictionary of these derived objects. This enables you to transform the template tree into a useful user menu: </p> <p> <pre><span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:#2b91af;">Command</span>&gt;&nbsp;editMenu&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">from</span>&nbsp;name&nbsp;<span style="color:blue;">in</span>&nbsp;editMenuTemplate &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">select</span>&nbsp;commandStore.Lookup(name);</pre> </p> <p> Notice how this transforms only the leaves, using the command store's <code>Lookup</code> method. This example uses C# query syntax, because this is what the <code>Select</code> method enables, but you could also have written the translation by just calling the <code>Select</code> method. </p> <p> The internal nodes in a menu have no behavious, so it makes little sense to attempt to turn them into <code>Command</code> objects as well. They're only there to provide structure to the menu. With a 'normal' tree, you wouldn't have been able to enrich only the leaves, while leaving the internal nodes untouched, but with a rose tree you can. </p> <p> The above example uses the <code>Select</code> method (via query syntax) to translate the nodes, thereby providing a demonstration of how to use the rose tree as the functor it is. </p> <h3 id="c77f1f9491b246f1bdb7c75d93eaa4ff"> Summary <a href="#c77f1f9491b246f1bdb7c75d93eaa4ff" title="permalink">#</a> </h3> <p> The <code>Select</code> doesn't implement any behaviour not already provided by <code>SelectLeaf</code>, but it enables C# query syntax. The C# compiler understands functors, but not bifunctors, so when you have a bifunctor, you might as well light up that language feature as well by adding a <code>Select</code> method. </p> <p> <strong>Next:</strong> <a href="/2018/08/13/a-visitor-functor">A Visitor functor</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Rose tree bifunctor https://blog.ploeh.dk/2019/08/12/rose-tree-bifunctor 2019-08-12T10:33:00+00:00 Mark Seemann <div id="post"> <p> <em>A rose tree forms a bifunctor. An article for object-oriented developers.</em> </p> <p> This article is an instalment in <a href="/2018/12/24/bifunctors">an article series about bifunctors</a>. While the overview article explains that there's essentially two practically useful bifunctors, here's a third one. <a href="https://en.wikipedia.org/wiki/Rose_tree">rose trees</a>. </p> <h3 id="985e3bc5291c4f8ba98ce258e78f4ec8"> Mapping both dimensions <a href="#985e3bc5291c4f8ba98ce258e78f4ec8" title="permalink">#</a> </h3> <p> Like in the <a href="/2019/01/07/either-bifunctor">previous article on the Either bifunctor</a>, I'll start by implementing the simultaneous two-dimensional translation <code>SelectBoth</code>: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N1</span>,&nbsp;<span style="color:#2b91af;">L1</span>&gt;&nbsp;SelectBoth&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">N1</span>,&nbsp;<span style="color:#2b91af;">L</span>,&nbsp;<span style="color:#2b91af;">L1</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;source, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">N1</span>&gt;&nbsp;selectNode, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">L</span>,&nbsp;<span style="color:#2b91af;">L1</span>&gt;&nbsp;selectLeaf) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;source.Cata( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;node:&nbsp;(n,&nbsp;branches)&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseNode</span>&lt;<span style="color:#2b91af;">N1</span>,&nbsp;<span style="color:#2b91af;">L1</span>&gt;(selectNode(n),&nbsp;branches), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;leaf:&nbsp;l&nbsp;=&gt;&nbsp;(<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N1</span>,&nbsp;<span style="color:#2b91af;">L1</span>&gt;)<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:#2b91af;">N1</span>,&nbsp;<span style="color:#2b91af;">L1</span>&gt;(selectLeaf(l))); }</pre> </p> <p> This article uses the previously shown <a href="/2019/07/29/church-encoded-rose-tree">Church-encoded rose tree</a> and <a href="/2019/08/05/rose-tree-catamorphism">its catamorphism</a> <code>Cata</code>. </p> <p> In the <code>leaf</code> case, the <code>l</code> argument received by the lambda expression is an object of the type <code>L</code>, since the <code>source</code> tree is an <code>IRoseTree&lt;N, L&gt;</code> object; i.e. a tree with leaves of the type <code>L</code> and nodes of the type <code>N</code>. The <code>selectLeaf</code> argument is a function that converts an <code>L</code> object to an <code>L1</code> object. Since <code>l</code> is an <code>L</code> object, you can call <code>selectLeaf</code> with it to produce an <code>L1</code> object. You can use this resulting object to create a new <code>RoseLeaf&lt;N1, L1&gt;</code>. Keep in mind that while the <code>RoseLeaf</code> class requires two type arguments, it never requires an object of its <code>N</code> type argument, which means that you can create an object with any <em>node</em> type argument, including <code>N1</code>, even if you don't have an object of that type. </p> <p> In the <code>node</code> case, the lambda expression receives two objects: <code>n</code> and <code>branches</code>. The <code>n</code> object has the type <code>N</code>, while the <code>branches</code> object has the type <code>IEnumerable&lt;IRoseTree&lt;N1, L1&gt;&gt;</code>. In other words, the <code>branches</code> have already been translated to the desired result type. That's how the catamorphism works. This means that you only have to figure out how to translate the <code>N</code> object <code>n</code> to an <code>N1</code> object. The <code>selectNode</code> function argument can do that, so you can then create a new <code>RoseNode&lt;N1, L1&gt;</code> and return it. </p> <p> This works as expected: </p> <p> <pre>&gt; <span style="color:blue;">var</span>&nbsp;tree&nbsp;=&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(<span style="color:#a31515;">&quot;foo&quot;</span>,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">int</span>&gt;(42),&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">int</span>&gt;(1337)); &gt; tree RoseNode&lt;string, int&gt;("foo", IRoseTree&lt;string, int&gt;[2] { 42, 1337 }) &gt; tree.SelectBoth(s&nbsp;=&gt;&nbsp;s.Length,&nbsp;i&nbsp;=&gt;&nbsp;i.ToString()) RoseNode&lt;int, string&gt;(3, IRoseTree&lt;int, string&gt;[2] { "42", "1337" })</pre> </p> <p> This <em>C# Interactive</em> example shows how to convert a tree with internal string nodes and integer leaves to a tree of internal integer nodes and string leaves. The strings are converted to strings by counting their <code>Length</code>, while the integers are turned into strings using the standard <code>ToString</code> method available on all objects. </p> <h3 id="c0ea04cfe7d3412c86b9ba3953812025"> Mapping nodes <a href="#c0ea04cfe7d3412c86b9ba3953812025" title="permalink">#</a> </h3> <p> When you have <code>SelectBoth</code>, you can trivially implement the translations for each dimension in isolation. For <a href="/2018/12/31/tuple-bifunctor">tuple bifunctors</a>, I called these methods <code>SelectFirst</code> and <code>SelectSecond</code>, while for <a href="/2019/01/07/either-bifunctor">Either bifunctors</a>, I chose to name them <code>SelectLeft</code> and <code>SelectRight</code>. Continuing the trend of naming the translations after what they translate, instead of their positions, I'll name the corresponding methods here <code>SelectNode</code> and <code>SelectLeaf</code>. In <a href="https://www.haskell.org">Haskell</a>, the functions associated with <code>Data.Bifunctor</code> are always called <code>first</code> and <code>second</code>, but I see no reason to preserve such abstract naming in C#. In Haskell, these functions are part of the <code>Bifunctor</code> type class; the abstract names serve an actual purpose. This isn't the case in C#, so there's no reason to retain the abstract names. You might as well use names that communicate intent, which is what I've tried to do here. </p> <p> If you want to map only the internal nodes, you can implement a <code>SelectNode</code> method based on <code>SelectBoth</code>: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N1</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;SelectNode&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">N1</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;source, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">N1</span>&gt;&nbsp;selector) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;source.SelectBoth(selector,&nbsp;l&nbsp;=&gt;&nbsp;l); }</pre> </p> <p> This simply uses the <code>l =&gt; l</code> lambda expression as an ad-hoc <em>identity</em> function, while passing <code>selector</code> as the <code>selectNode</code> argument to the <code>SelectBoth</code> method. </p> <p> You can use this to map the above <code>tree</code> to a tree made entirely of numbers: </p> <p> <pre>&gt; <span style="color:blue;">var</span>&nbsp;tree&nbsp;=&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(<span style="color:#a31515;">&quot;foo&quot;</span>,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">int</span>&gt;(42),&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">int</span>&gt;(1337)); &gt; tree.SelectNode(s =&gt; s.Length) RoseNode&lt;int, int&gt;(3, IRoseTree&lt;int, int&gt;[2] { 42, 1337 })</pre> </p> <p> Such a tree is, incidentally, isomorphic to a <a href="/2018/08/06/a-tree-functor">'normal' tree</a>. It might be a good exercise, if you need one, to demonstrate the isormorphism by writing functions that convert a <code>Tree&lt;T&gt;</code> into an <code>IRoseTree&lt;T, T&gt;</code>, and vice versa. </p> <h3 id="baa9136b506241e39e13639e43679b31"> Mapping leaves <a href="#baa9136b506241e39e13639e43679b31" title="permalink">#</a> </h3> <p> Similar to <code>SelectNode</code>, you can also trivially implement <code>SelectLeaf</code>: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L1</span>&gt;&nbsp;SelectLeaf&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>,&nbsp;<span style="color:#2b91af;">L1</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;source, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">L</span>,&nbsp;<span style="color:#2b91af;">L1</span>&gt;&nbsp;selector) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;source.SelectBoth(n&nbsp;=&gt;&nbsp;n,&nbsp;selector); }</pre> </p> <p> This is another one-liner calling <code>SelectBoth</code>, with the difference that the identity function <code>n =&gt; n</code> is passed as the first argument, instead of as the last. This ensures that only <code>RoseLeaf</code> values are mapped: </p> <p> <pre>&gt; <span style="color:blue;">var</span>&nbsp;tree&nbsp;=&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(<span style="color:#a31515;">&quot;foo&quot;</span>,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">int</span>&gt;(42),&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">int</span>&gt;(1337)); &gt; tree.SelectLeaf(i =&gt; i % 2 == 0) RoseNode&lt;string, bool&gt;("foo", IRoseTree&lt;string, bool&gt;[2] { true, false })</pre> </p> <p> In the above <em>C# Interactive</em> session, the leaves are mapped to Boolean values, indicating whether they're even or odd. </p> <h3 id="afddb846bd244f4aa8f658fb5716b392"> Identity laws <a href="#afddb846bd244f4aa8f658fb5716b392" title="permalink">#</a> </h3> <p> Rose trees obey all the bifunctor laws. While it's formal work to prove that this is the case, you can get an intuition for it via examples. Often, I use a property-based testing library like <a href="https://fscheck.github.io/FsCheck">FsCheck</a> or <a href="https://github.com/hedgehogqa/fsharp-hedgehog">Hedgehog</a> to demonstrate (not prove) that laws hold, but in this article, I'll keep it simple and only cover each law with a parametrised test. </p> <p> <pre><span style="color:blue;">private</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">T</span>&nbsp;Id&lt;<span style="color:#2b91af;">T</span>&gt;(<span style="color:#2b91af;">T</span>&nbsp;x)&nbsp;=&gt;&nbsp;x; <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:blue;">object</span>[]&gt;&nbsp;BifunctorLawsData { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">get</span> &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">yield</span>&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;&quot;</span>)&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">yield</span>&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;foo&quot;</span>)&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">yield</span>&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(42)&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">yield</span>&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(42,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;bar&quot;</span>))&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">yield</span>&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>[]&nbsp;{&nbsp;exampleTree&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;} } [<span style="color:#2b91af;">Theory</span>,&nbsp;<span style="color:#2b91af;">MemberData</span>(<span style="color:blue;">nameof</span>(BifunctorLawsData))] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;SelectNodeObeysFirstFunctorLaw(<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;t) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.Equal(t,&nbsp;t.SelectNode(Id)); }</pre> </p> <p> This test uses <a href="https://xunit.github.io">xUnit.net</a>'s <code>[Theory]</code> feature to supply a small set of example input values. The input values are defined by the <code>BifunctorLawsData</code> property, since I'll reuse the same values for all the bifunctor law demonstration tests. The <code>exampleTree</code> object is the tree shown in <a href="/2019/07/29/church-encoded-rose-tree">Church-encoded rose tree</a>. </p> <p> The tests also use the identity function implemented as a <code>private</code> function called <code>Id</code>, since C# doesn't come equipped with such a function in the Base Class Library. </p> <p> For all the <code>IRoseTree&lt;int, string&gt;</code> objects <code>t</code>, the test simply verifies that the original tree <code>t</code> is equal to the tree projected over the first axis with the <code>Id</code> function. </p> <p> Likewise, the first functor law applies when translating over the second dimension: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>,&nbsp;<span style="color:#2b91af;">MemberData</span>(<span style="color:blue;">nameof</span>(BifunctorLawsData))] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;SelectLeafObeysFirstFunctorLaw(<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;t) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.Equal(t,&nbsp;t.SelectLeaf(Id)); }</pre> </p> <p> This is the same test as the previous test, with the only exception that it calls <code>SelectLeaf</code> instead of <code>SelectNode</code>. </p> <p> Both <code>SelectNode</code> and <code>SelectLeaf</code> are implemented by <code>SelectBoth</code>, so the real test is whether this method obeys the identity law: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>,&nbsp;<span style="color:#2b91af;">MemberData</span>(<span style="color:blue;">nameof</span>(BifunctorLawsData))] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;SelectBothObeysIdentityLaw(<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;t) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.Equal(t,&nbsp;t.SelectBoth(Id,&nbsp;Id)); }</pre> </p> <p> Projecting over both dimensions with the identity function does, indeed, return an object equal to the input object. </p> <h3 id="bfaa0b763e5346c488f4bd9576ab894c"> Consistency law <a href="#bfaa0b763e5346c488f4bd9576ab894c" title="permalink">#</a> </h3> <p> In general, it shouldn't matter whether you map with <code>SelectBoth</code> or a combination of <code>SelectNode</code> and <code>SelectLeaf</code>: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>,&nbsp;<span style="color:#2b91af;">MemberData</span>(<span style="color:blue;">nameof</span>(BifunctorLawsData))] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;ConsistencyLawHolds(<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;t) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">DateTime</span>&nbsp;f(<span style="color:blue;">int</span>&nbsp;i)&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTime</span>(i); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">bool</span>&nbsp;g(<span style="color:blue;">string</span>&nbsp;s)&nbsp;=&gt;&nbsp;<span style="color:blue;">string</span>.IsNullOrWhiteSpace(s); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.Equal(t.SelectBoth(f,&nbsp;g),&nbsp;t.SelectLeaf(g).SelectNode(f)); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.Equal( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;t.SelectNode(f).SelectLeaf(g), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;t.SelectLeaf(g).SelectNode(f)); }</pre> </p> <p> This example creates two local functions <code>f</code> and <code>g</code>. The first function, <code>f</code>, creates a new <code>DateTime</code> object from an integer, using one of the <code>DateTime</code> constructor overloads. The second function, <code>g</code>, just delegates to <code>string.IsNullOrWhiteSpace</code>, although I want to stress that this is just an example. The law should hold for any two (<a href="https://en.wikipedia.org/wiki/Pure_function">pure</a>) functions. </p> <p> The test then verifies that you get the same result from calling <code>SelectBoth</code> as when you call <code>SelectNode</code> followed by <code>SelectLeaf</code>, or the other way around. </p> <h3 id="dd3046c49d564991bb47924b6e8e65fb"> Composition laws <a href="#dd3046c49d564991bb47924b6e8e65fb" title="permalink">#</a> </h3> <p> The composition laws insist that you can compose functions, or translations, and that again, the choice to do one or the other doesn't matter. Along each of the axes, it's just the second functor law applied. This parametrised test demonstrates that the law holds for <code>SelectNode</code>: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>,&nbsp;<span style="color:#2b91af;">MemberData</span>(<span style="color:blue;">nameof</span>(BifunctorLawsData))] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;SecondFunctorLawHoldsForSelectNode(<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;t) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">char</span>&nbsp;f(<span style="color:blue;">bool</span>&nbsp;b)&nbsp;=&gt;&nbsp;b&nbsp;?&nbsp;<span style="color:#a31515;">&#39;T&#39;</span>&nbsp;:&nbsp;<span style="color:#a31515;">&#39;F&#39;</span>; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">bool</span>&nbsp;g(<span style="color:blue;">int</span>&nbsp;i)&nbsp;=&gt;&nbsp;i&nbsp;%&nbsp;2&nbsp;==&nbsp;0; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.Equal( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;t.SelectNode(x&nbsp;=&gt;&nbsp;f(g(x))), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;t.SelectNode(g).SelectNode(f)); }</pre> </p> <p> Here, <code>f</code> is a local function that returns the the character <code>'T'</code> for <code>true</code>, and <code>'F'</code> for <code>false</code>; <code>g</code> is the <em>even</em> function. The second functor law states that mapping <code>f(g(x))</code> in a single step is equivalent to first mapping over <code>g</code> and then map the result of that using <code>f</code>. </p> <p> The same law applies if you fix the first dimension and translate over the second: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>,&nbsp;<span style="color:#2b91af;">MemberData</span>(<span style="color:blue;">nameof</span>(BifunctorLawsData))] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;SecondFunctorLawHoldsForSelectLeaf(<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;t) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">bool</span>&nbsp;f(<span style="color:blue;">int</span>&nbsp;x)&nbsp;=&gt;&nbsp;x&nbsp;%&nbsp;2&nbsp;==&nbsp;0; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">int</span>&nbsp;g(<span style="color:blue;">string</span>&nbsp;s)&nbsp;=&gt;&nbsp;s.Length; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.Equal( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;t.SelectLeaf(x&nbsp;=&gt;&nbsp;f(g(x))), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;t.SelectLeaf(g).SelectLeaf(f)); }</pre> </p> <p> Here, <code>f</code> is the <em>even</em> function, whereas <code>g</code> is a local function that returns the length of a string. Again, the test demonstrates that the output is the same whether you map over an intermediary step, or whether you map using only a single step. </p> <p> This generalises to the composition law for <code>SelectBoth</code>: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>,&nbsp;<span style="color:#2b91af;">MemberData</span>(<span style="color:blue;">nameof</span>(BifunctorLawsData))] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;SelectBothCompositionLawHolds(<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;t) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">char</span>&nbsp;f(<span style="color:blue;">bool</span>&nbsp;b)&nbsp;=&gt;&nbsp;b&nbsp;?&nbsp;<span style="color:#a31515;">&#39;T&#39;</span>&nbsp;:&nbsp;<span style="color:#a31515;">&#39;F&#39;</span>; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">bool</span>&nbsp;g(<span style="color:blue;">int</span>&nbsp;x)&nbsp;=&gt;&nbsp;x&nbsp;%&nbsp;2&nbsp;==&nbsp;0; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">bool</span>&nbsp;h(<span style="color:blue;">int</span>&nbsp;x)&nbsp;=&gt;&nbsp;x&nbsp;%&nbsp;2&nbsp;==&nbsp;0; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">int</span>&nbsp;i(<span style="color:blue;">string</span>&nbsp;s)&nbsp;=&gt;&nbsp;s.Length; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.Equal( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;t.SelectBoth(x&nbsp;=&gt;&nbsp;f(g(x)),&nbsp;y&nbsp;=&gt;&nbsp;h(i(y))), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;t.SelectBoth(g,&nbsp;i).SelectBoth(f,&nbsp;h)); }</pre> </p> <p> Again, whether you translate in one or two steps shouldn't affect the outcome. </p> <p> As all of these tests demonstrate, the bifunctor laws hold for rose trees. The tests only showcase five examples, but I hope it gives you an intuition how any rose tree is a bifunctor. After all, the <code>SelectNode</code>, <code>SelectLeaf</code>, and <code>SelectBoth</code> methods are all generic, and they behave the same for all generic type arguments. </p> <h3 id="a1a5dea3d85d4ed1a3ee3fb0a4dca820"> Summary <a href="#a1a5dea3d85d4ed1a3ee3fb0a4dca820" title="permalink">#</a> </h3> <p> Rose trees are bifunctors. You can translate the node and leaf dimension of a rose tree independently of each other, and the bifunctor laws hold for any pure translation, no matter how you compose the projections. </p> <p> As always, there can be performance differences between the various compositions, but the outputs will be the same regardless of composition. </p> <p> A functor, and by extension, a bifunctor, is a structure-preserving map. This means that any projection preserves the structure of the underlying container. For rose trees this means that the shape of the tree remains the same. The number of leaves remain the same, as does the number of internal nodes. </p> <p> <strong>Next:</strong> <a href="/2018/01/08/software-design-isomorphisms">Software design isomorphisms</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Rose tree catamorphism https://blog.ploeh.dk/2019/08/05/rose-tree-catamorphism 2019-08-05T08:30:00+00:00 Mark Seemann <div id="post"> <p> <em>The catamorphism for a tree with different types of nodes and leaves is made up from two functions.</em> </p> <p> This article is part of an <a href="/2019/04/29/catamorphisms">article series about catamorphisms</a>. A catamorphism is a <a href="/2017/10/04/from-design-patterns-to-category-theory">universal abstraction</a> that describes how to digest a data structure into a potentially more compact value. </p> <p> This article presents the catamorphism for a <a href="https://en.wikipedia.org/wiki/Rose_tree">rose tree</a>, as well as how to identify it. The beginning of this article presents the catamorphism in C#, with examples. The rest of the article describes how to deduce the catamorphism. This part of the article presents my work in <a href="https://www.haskell.org">Haskell</a>. Readers not comfortable with Haskell can just read the first part, and consider the rest of the article as an optional appendix. </p> <p> A rose tree is a general-purpose data structure where each node in a tree has an associated value. Each node can have an arbitrary number of branches, including none. The distinguishing feature from a rose tree and just any <a href="https://en.wikipedia.org/wiki/Tree_(data_structure)">tree</a> is that internal nodes can hold values of a different type than leaf values. </p> <p> <img src="/content/binary/rose-tree-example.png" alt="A rose tree example diagram, with internal nodes containing integers, and leafs containing strings."> </p> <p> The diagram shows an example of a tree of internal integers and leaf strings. All internal nodes contain integer values, and all leaves contain strings. Each node can have an arbitrary number of branches. </p> <h3 id="078386d5f3924a63add86ff199fd88d0"> C# catamorphism <a href="#078386d5f3924a63add86ff199fd88d0" title="permalink">#</a> </h3> <p> As a C# representation of a rose tree, I'll use the <a href="/2019/07/29/church-encoded-rose-tree">Church-encoded rose tree I've previously described</a>. The catamorphism is this extension method: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">TResult</span>&nbsp;Cata&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;tree, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">TResult</span>&gt;,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;node, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">L</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;leaf) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;tree.Match( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;node:&nbsp;(n,&nbsp;branches)&nbsp;=&gt;&nbsp;node(n,&nbsp;branches.Select(t&nbsp;=&gt;&nbsp;t.Cata(node,&nbsp;leaf))), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;leaf:&nbsp;leaf); }</pre> </p> <p> Like most of the other catamorphisms shown in this article series, this one consists of two functions. One that handles the <em>leaf</em> case, and one that handles the partially reduced <em>node</em> case. Compare it with the <a href="/2019/06/10/tree-catamorphism">tree catamorphism</a>: notice that the rose tree catamorphism's <code>node</code> function is identical to the the tree catamorphism. The <code>leaf</code> function, however, is new. </p> <p> In previous articles, you've seen other examples of catamorphisms for <a href="/2018/05/22/church-encoding">Church-encoded</a> types. The most common pattern has been that the Church encoding (the <code>Match</code> method) was also the catamorphism, with the <a href="/2019/05/13/peano-catamorphism">Peano catamorphism</a> being the only exception so far. When it comes to the Peano catamorphism, however, I'm not entirely confident that the difference between Church encoding and catamorphism is real, or whether it's just an artefact of the way I originally designed the Church encoding. </p> <p> When it comes to the present rose tree, however, notice that the catamorphisms is distinctly different from the Church encoding. That's the reason I called the method <code>Cata</code> instead of <code>Match</code>. </p> <p> The method simply delegates the <code>leaf</code> handler to <code>Match</code>, while it adds behaviour to the <code>node</code> case. It works the same way as for the 'normal' tree catamorphism. </p> <h3 id="87e2c79711c24c63a5ed82fbe4f7b581"> Examples <a href="#87e2c79711c24c63a5ed82fbe4f7b581" title="permalink">#</a> </h3> <p> You can use <code>Cata</code> to implement most other behaviour you'd like <code>IRoseTree&lt;N, L&gt;</code> to have. In a future article, you'll see how to <a href="/2019/08/12/rose-tree-bifunctor">turn the rose tree into a bifunctor</a> and <a href="/2019/08/19/a-rose-tree-functor">functor</a>, so here, we'll look at some other, more ad hoc, examples. As is also the case for the 'normal' tree, you can calculate the sum of all nodes, if you can associate a number with each node. </p> <p> Consider the example tree in the above diagram. You can create it as an <code>IRoseTree&lt;int, string&gt;</code> object like this: </p> <p> <pre><span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;exampleTree&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(42, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(1337, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;foo&quot;</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;bar&quot;</span>)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(2112, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(90125, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;baz&quot;</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;qux&quot;</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;quux&quot;</span>)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;quuz&quot;</span>)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;corge&quot;</span>));</pre> </p> <p> If you want to calculate a sum for a tree like that, you can use the integers for the internal nodes, and perhaps the length of the strings of the leaves. That hardly makes much sense, but is technically possible: </p> <p> <pre>&gt; exampleTree.Cata((x,&nbsp;xs)&nbsp;=&gt;&nbsp;x&nbsp;+&nbsp;xs.Sum(),&nbsp;x&nbsp;=&gt;&nbsp;x.Length) 93641</pre> </p> <p> Perhaps slightly more useful is to count the number of leaves: </p> <p> <pre>&gt; exampleTree.Cata((_,&nbsp;xs)&nbsp;=&gt;&nbsp;xs.Sum(),&nbsp;_&nbsp;=&gt;&nbsp;1) 7</pre> </p> <p> A leaf node has, by definition, exactly one leaf node, so the <code>leaf</code> lambda expression always returns <code>1</code>. In the <code>node</code> case, <code>xs</code> contains the partially summed leaf node count, so just <code>Sum</code> those together while ignoring the value of the internal node. </p> <p> You can also measure the maximum depth of the tree: </p> <p> <pre>&gt; exampleTree.Cata((_,&nbsp;xs)&nbsp;=&gt;&nbsp;1&nbsp;+&nbsp;xs.Max(),&nbsp;_&nbsp;=&gt;&nbsp;0) 3</pre> </p> <p> Consistent with the example for 'normal' trees, you can arbitrarily decide that the depth of a leaf node is <code>0</code>, so again, the <code>leaf</code> lambda expression just returns a constant value. The <code>node</code> lambda expression takes the <code>Max</code> of the partially reduced <code>xs</code> and adds <code>1</code>, since an internal node represents another level of depth in a tree. </p> <h3 id="9e673c50edc14c1790a9e89a67d069d1"> Rose tree F-Algebra <a href="#9e673c50edc14c1790a9e89a67d069d1" title="permalink">#</a> </h3> <p> As in the <a href="/2019/06/10/tree-catamorphism">previous article</a>, I'll use <code>Fix</code> and <code>cata</code> as explained in <a href="https://bartoszmilewski.com">Bartosz Milewski</a>'s excellent <a href="https://bartoszmilewski.com/2017/02/28/f-algebras/">article on F-Algebras</a>. </p> <p> As always, start with the underlying endofunctor: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;RoseTreeF&nbsp;a&nbsp;b&nbsp;c&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;NodeF&nbsp;{&nbsp;nodeValue&nbsp;::&nbsp;a,&nbsp;nodes&nbsp;::&nbsp;ListFix&nbsp;c&nbsp;} &nbsp;&nbsp;|&nbsp;LeafF&nbsp;{&nbsp;leafValue&nbsp;::&nbsp;b&nbsp;} &nbsp;&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;(<span style="color:blue;">RoseTreeF</span>&nbsp;a&nbsp;b)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;f&nbsp;(NodeF&nbsp;x&nbsp;ns)&nbsp;=&nbsp;NodeF&nbsp;x&nbsp;$&nbsp;<span style="color:blue;">fmap</span>&nbsp;f&nbsp;ns &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;&nbsp;&nbsp;&nbsp;(LeafF&nbsp;x)&nbsp;=&nbsp;LeafF&nbsp;x</pre> </p> <p> Instead of using Haskell's standard list (<code>[]</code>) for the nodes, I've used <code>ListFix</code> from <a href="/2019/05/27/list-catamorphism">the article on list catamorphism</a>. This should, hopefully, demonstrate how you can build on already established definitions derived from first principles. </p> <p> As usual, I've called the 'data' types <code>a</code> and <code>b</code>, and the carrier type <code>c</code> (for <em>carrier</em>). The <code>Functor</code> instance as usual translates the carrier type; the <code>fmap</code> function has the type <code>(c -&gt; c1) -&gt; RoseTreeF a b c -&gt; RoseTreeF a b c1</code>. </p> <p> As was the case when deducing the recent catamorphisms, Haskell isn't too happy about defining instances for a type like <code>Fix (RoseTreeF a b)</code>. To address that problem, you can introduce a <code>newtype</code> wrapper: </p> <p> <pre><span style="color:blue;">newtype</span>&nbsp;RoseTreeFix&nbsp;a&nbsp;b&nbsp;= &nbsp;&nbsp;RoseTreeFix&nbsp;{&nbsp;unRoseTreeFix&nbsp;::&nbsp;Fix&nbsp;(RoseTreeF&nbsp;a&nbsp;b)&nbsp;}&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> You can define <code>Bifunctor</code>, <code>Bifoldable</code>, <code>Bitraversable</code>, etc. instances for this type without resorting to any funky GHC extensions. Keep in mind that ultimately, the purpose of all this code is just to figure out what the catamorphism looks like. This code isn't intended for actual use. </p> <p> A pair of helper functions make it easier to define <code>RoseTreeFix</code> values: </p> <p> <pre><span style="color:#2b91af;">roseLeafF</span>&nbsp;::&nbsp;b&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">RoseTreeFix</span>&nbsp;a&nbsp;b roseLeafF&nbsp;=&nbsp;RoseTreeFix&nbsp;.&nbsp;Fix&nbsp;.&nbsp;LeafF <span style="color:#2b91af;">roseNodeF</span>&nbsp;::&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;(<span style="color:blue;">RoseTreeFix</span>&nbsp;a&nbsp;b)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">RoseTreeFix</span>&nbsp;a&nbsp;b roseNodeF&nbsp;x&nbsp;=&nbsp;RoseTreeFix&nbsp;.&nbsp;Fix&nbsp;.&nbsp;NodeF&nbsp;x&nbsp;.&nbsp;<span style="color:blue;">fmap</span>&nbsp;unRoseTreeFix</pre> </p> <p> <code>roseLeafF</code> creates a leaf node: </p> <p> <pre>Prelude Fix List RoseTree&gt; roseLeafF "ploeh" RoseTreeFix {unRoseTreeFix = Fix (LeafF "ploeh")}</pre> </p> <p> <code>roseNodeF</code> is a helper function to create internal nodes: </p> <p> <pre>Prelude Fix List RoseTree&gt; roseNodeF 6 (consF (roseLeafF 0) nilF) RoseTreeFix {unRoseTreeFix = Fix (NodeF 6 (ListFix (Fix (ConsF (Fix (LeafF 0)) (Fix NilF)))))}</pre> </p> <p> Even with helper functions, construction of <code>RoseTreeFix</code> values is cumbersome, but keep in mind that the code shown here isn't meant to be used in practice. The goal is only to deduce catamorphisms from more basic universal abstractions, and you now have all you need to do that. </p> <h3 id="0bfc3f600a9e43eea1026f1a4a3b7604"> Haskell catamorphism <a href="#0bfc3f600a9e43eea1026f1a4a3b7604" title="permalink">#</a> </h3> <p> At this point, you have two out of three elements of an F-Algebra. You have an endofunctor (<code>RoseTreeF a b</code>), and an object <code>c</code>, but you still need to find a morphism <code>RoseTreeF a b c -&gt; c</code>. Notice that the algebra you have to find is the function that reduces the functor to its <em>carrier type</em> <code>c</code>, not any of the 'data types' <code>a</code> or <code>b</code>. This takes some time to get used to, but that's how catamorphisms work. This doesn't mean, however, that you get to ignore <code>a</code> or <code>b</code>, as you'll see. </p> <p> As in the previous articles, start by writing a function that will become the catamorphism, based on <code>cata</code>: </p> <p> <pre>roseTreeF&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unRoseTreeFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;(NodeF&nbsp;x&nbsp;ns)&nbsp;=&nbsp;<span style="color:blue;">undefined</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;(LeafF&nbsp;x)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> While this compiles, with its <code>undefined</code> implementations, it obviously doesn't do anything useful. I find, however, that it helps me think. How can you return a value of the type <code>c</code> from the <code>LeafF</code> case? You could pass a function argument to the <code>roseTreeF</code> function and use it with <code>x</code>: </p> <p> <pre>roseTreeF&nbsp;fl&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unRoseTreeFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;(NodeF&nbsp;x&nbsp;ns)&nbsp;=&nbsp;<span style="color:blue;">undefined</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;(LeafF&nbsp;x)&nbsp;=&nbsp;fl&nbsp;x</pre> </p> <p> While you could, technically, pass an argument of the type <code>c</code> to <code>roseTreeF</code> and then return that value from the <code>LeafF</code> case, that would mean that you would ignore the <code>x</code> value. This would be incorrect, so instead, make the argument a function and call it with <code>x</code>. Likewise, you can deal with the <code>NodeF</code> case in the same way: </p> <p> <pre><span style="color:#2b91af;">roseTreeF</span>&nbsp;::&nbsp;(a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;(b&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">RoseTreeFix</span>&nbsp;a&nbsp;b&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c roseTreeF&nbsp;fn&nbsp;fl&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unRoseTreeFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;(NodeF&nbsp;x&nbsp;ns)&nbsp;=&nbsp;fn&nbsp;x&nbsp;ns &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;(LeafF&nbsp;x)&nbsp;=&nbsp;fl&nbsp;x</pre> </p> <p> This works. Since <code>cata</code> has the type <code>Functor f =&gt; (f a -&gt; a) -&gt; Fix f -&gt; a</code>, that means that <code>alg</code> has the type <code>f a -&gt; a</code>. In the case of <code>RoseTreeF</code>, the compiler infers that the <code>alg</code> function has the type <code>RoseTreeF a b c -&gt; c</code>, which is just what you need! </p> <p> You can now see what the carrier type <code>c</code> is for. It's the type that the algebra extracts, and thus the type that the catamorphism returns. </p> <p> This, then, is the catamorphism for a rose tree. As has been the most common pattern so far, it's a pair, made from two functions. It's still not the only possible catamorphism, since you could trivially flip the arguments to <code>roseTreeF</code>, or the arguments to <code>fn</code>. </p> <p> I've chosen the representation shown here because it's similar to the catamorphism I've shown for a 'normal' tree, just with the added function for leaves. </p> <h3 id="256fd0a09c4a4651b6c27b5626b0fb33"> Basis <a href="#256fd0a09c4a4651b6c27b5626b0fb33" title="permalink">#</a> </h3> <p> You can implement most other useful functionality with <code>roseTreeF</code>. Here's the <code>Bifunctor</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Bifunctor</span>&nbsp;<span style="color:blue;">RoseTreeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;bimap&nbsp;f&nbsp;s&nbsp;=&nbsp;roseTreeF&nbsp;(roseNodeF&nbsp;.&nbsp;f)&nbsp;(roseLeafF&nbsp;.&nbsp;s)</pre> </p> <p> Notice how naturally the catamorphism implements <code>bimap</code>. </p> <p> From that instance, the <code>Functor</code> instance trivially follows: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;(<span style="color:blue;">RoseTreeFix</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;=&nbsp;second</pre> </p> <p> You could probably also add <code>Applicative</code> and <code>Monad</code> instances, but I find those hard to grasp, so I'm going to skip them in favour of <code>Bifoldable</code>: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Bifoldable</span>&nbsp;<span style="color:blue;">RoseTreeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;bifoldMap&nbsp;f&nbsp;=&nbsp;roseTreeF&nbsp;(\x&nbsp;xs&nbsp;-&gt;&nbsp;f&nbsp;x&nbsp;&lt;&gt;&nbsp;fold&nbsp;xs)</pre> </p> <p> The <code>Bifoldable</code> instance enables you to trivially implement the <code>Foldable</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Foldable</span>&nbsp;(<span style="color:blue;">RoseTreeFix</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;foldMap&nbsp;=&nbsp;bifoldMap&nbsp;mempty</pre> </p> <p> You may find the presence of <code>mempty</code> puzzling, since <code>bifoldMap</code> takes two functions as arguments. Is <code>mempty</code> a function? </p> <p> Yes, <code>mempty</code> can be a function. Here, it is. There's a <code>Monoid</code> instance for any function <code>a -&gt; m</code>, where <code>m</code> is a <code>Monoid</code> instance, and <code>mempty</code> is the identity for that monoid. That's the instance in use here. </p> <p> Just as <code>RoseTreeFix</code> is <code>Bifoldable</code>, it's also <code>Bitraversable</code>: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Bitraversable</span>&nbsp;<span style="color:blue;">RoseTreeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;bitraverse&nbsp;f&nbsp;s&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;roseTreeF&nbsp;(\x&nbsp;xs&nbsp;-&gt;&nbsp;roseNodeF&nbsp;&lt;$&gt;&nbsp;f&nbsp;x&nbsp;&lt;*&gt;&nbsp;sequenceA&nbsp;xs)&nbsp;(<span style="color:blue;">fmap</span>&nbsp;roseLeafF&nbsp;.&nbsp;s)</pre> </p> <p> You can comfortably implement the <code>Traversable</code> instance based on the <code>Bitraversable</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Traversable</span>&nbsp;(<span style="color:blue;">RoseTreeFix</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;sequenceA&nbsp;=&nbsp;bisequenceA&nbsp;.&nbsp;first&nbsp;pure</pre> </p> <p> That rose trees are <code>Traversable</code> turns out to be useful, as a future article will show. </p> <h3 id="c02950d3b4954435b384b1f7520d24d4"> Relationships <a href="#c02950d3b4954435b384b1f7520d24d4" title="permalink">#</a> </h3> <p> As was the case for 'normal' trees, the catamorphism for rose trees is more powerful than the <em>fold</em>. There are operations that you can express with the <code>Foldable</code> instance, but other operations that you can't. Consider the tree shown in the diagram at the beginning of the article. This is also the tree that the above C# examples use. In Haskell, using <code>RoseTreeFix</code>, you can define that tree like this: </p> <p> <pre>exampleTree&nbsp;= &nbsp;&nbsp;roseNodeF&nbsp;42&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;roseNodeF&nbsp;1337&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;(roseLeafF&nbsp;<span style="color:#a31515;">&quot;foo&quot;</span>)&nbsp;$ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;(roseLeafF&nbsp;<span style="color:#a31515;">&quot;bar&quot;</span>)&nbsp;nilF))&nbsp;$ &nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;roseNodeF&nbsp;2112&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;roseNodeF&nbsp;90125&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;(roseLeafF&nbsp;<span style="color:#a31515;">&quot;baz&quot;</span>)&nbsp;$ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;(roseLeafF&nbsp;<span style="color:#a31515;">&quot;qux&quot;</span>)&nbsp;$ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;(roseLeafF&nbsp;<span style="color:#a31515;">&quot;quux&quot;</span>)&nbsp;nilF))&nbsp;$ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;(roseLeafF&nbsp;<span style="color:#a31515;">&quot;quuz&quot;</span>)&nbsp;nilF))&nbsp;$ &nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;roseLeafF&nbsp;<span style="color:#a31515;">&quot;corge&quot;</span>) &nbsp;&nbsp;&nbsp;&nbsp;nilF)</pre> </p> <p> You can trivially calculate the sum of string lengths of all leaves, using only the <code>Foldable</code> instance: </p> <p> <pre>Prelude RoseTree&gt; sum $ length &lt;$&gt; exampleTree 25</pre> </p> <p> You can also fairly easily calculate a sum of all nodes, using the length of the strings as in the above C# example, but that requires the <code>Bifoldable</code> instance: </p> <p> <pre>Prelude Data.Bifoldable Data.Semigroup RoseTree&gt; bifoldMap Sum (Sum . length) exampleTree Sum {getSum = 93641}</pre> </p> <p> Fortunately, we get the same result as above. </p> <p> Counting leaves, or measuring the depth of a tree, on the other hand, is impossible with the <code>Foldable</code> instance, but interestingly, it turns out that counting leaves is possible with the <code>Bifoldable</code> instance: </p> <p> <pre><span style="color:#2b91af;">countLeaves</span>&nbsp;::&nbsp;(<span style="color:blue;">Bifoldable</span>&nbsp;p,&nbsp;<span style="color:blue;">Num</span>&nbsp;n)&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;p&nbsp;a&nbsp;b&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;n countLeaves&nbsp;=&nbsp;getSum&nbsp;.&nbsp;bifoldMap&nbsp;(<span style="color:blue;">const</span>&nbsp;$&nbsp;Sum&nbsp;0)&nbsp;(<span style="color:blue;">const</span>&nbsp;$&nbsp;Sum&nbsp;1)</pre> </p> <p> This works well with the example tree: </p> <p> <pre>Prelude RoseTree&gt; countLeaves exampleTree 7</pre> </p> <p> Notice, however, that <code>countLeaves</code> works for any <code>Bifoldable</code> instance. Does that mean that you can 'count the leaves' of a tuple? Yes, it does: </p> <p> <pre>Prelude RoseTree&gt; countLeaves ("foo", "bar") 1 Prelude RoseTree&gt; countLeaves (1, 42) 1</pre> </p> <p> Or what about <code>EitherFix</code>: </p> <p> <pre>Prelude RoseTree Either&gt; countLeaves $ leftF "foo" 0 Prelude RoseTree Either&gt; countLeaves $ rightF "bar" 1</pre> </p> <p> Notice that 'counting the leaves' of tuples always returns <code>1</code>, while 'counting the leaves' of <code>Either</code> always returns <code>0</code> for <code>Left</code> values, and <code>1</code> for <code>Right</code> values. This is because <code>countLeaves</code> considers the left, or <em>first</em>, data type to represent internal nodes, and the right, or <em>second</em>, data type to indicate leaves. </p> <p> You can further follow that train of thought to realise that you can convert both tuples and <code>EitherFix</code> values to small rose trees: </p> <p> <pre><span style="color:#2b91af;">fromTuple</span>&nbsp;::&nbsp;(a,&nbsp;b)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">RoseTreeFix</span>&nbsp;a&nbsp;b fromTuple&nbsp;(x,&nbsp;y)&nbsp;=&nbsp;roseNodeF&nbsp;x&nbsp;(consF&nbsp;(roseLeafF&nbsp;y)&nbsp;nilF) <span style="color:#2b91af;">fromEitherFix</span>&nbsp;::&nbsp;<span style="color:blue;">EitherFix</span>&nbsp;a&nbsp;b&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">RoseTreeFix</span>&nbsp;a&nbsp;b fromEitherFix&nbsp;=&nbsp;eitherF&nbsp;(`roseNodeF`&nbsp;nilF)&nbsp;roseLeafF</pre> </p> <p> The <code>fromTuple</code> function creates a small rose tree with one internal node and one leaf. The label of the internal node is the first value of the tuple, and the label of the leaf is the second value. Here's an example: </p> <p> <pre>Prelude RoseTree&gt; fromTuple ("foo", 42) RoseTreeFix {unRoseTreeFix = Fix (NodeF "foo" (ListFix (Fix (ConsF (Fix (LeafF 42)) (Fix NilF)))))}</pre> </p> <p> The <code>fromEitherFix</code> function turns a <em>left</em> value into an internal node with no leaves, and a <em>right</em> value into a leaf. Here are some examples: </p> <p> <pre>Prelude RoseTree Either&gt; fromEitherFix $ leftF "foo" RoseTreeFix {unRoseTreeFix = Fix (NodeF "foo" (ListFix (Fix NilF)))} Prelude RoseTree Either&gt; fromEitherFix $ rightF 42 RoseTreeFix {unRoseTreeFix = Fix (LeafF 42)}</pre> </p> <p> While counting leaves can be implemented using <code>Bifoldable</code>, that's not the case for measuring the depths of trees (I think; leave a comment if you know of a way to do this with one of the instances shown here). You can, however, measure tree depth with the catamorphism: </p> <p> <pre><span style="color:#2b91af;">treeDepth</span>&nbsp;::&nbsp;<span style="color:blue;">RoseTreeFix</span>&nbsp;a&nbsp;b&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">Integer</span> treeDepth&nbsp;=&nbsp;roseTreeF&nbsp;(\_&nbsp;xs&nbsp;-&gt;&nbsp;1&nbsp;+&nbsp;<span style="color:blue;">maximum</span>&nbsp;xs)&nbsp;(<span style="color:blue;">const</span>&nbsp;0)</pre> </p> <p> The implementation is similar to the implementation for 'normal' trees. I've arbitrarily decided that leaves have a depth of zero, so the function that handles leaves always returns <code>0</code>. The function that handles internal nodes receives <code>xs</code> as a partially reduced list of depths below the node in question. Take the maximum of those and add <code>1</code>, since each internal node has a depth of one. </p> <p> <pre>Prelude RoseTree&gt; treeDepth exampleTree 3</pre> </p> <p> This, hopefully, illustrates that the catamorphism is more capable, and that the fold is just a (list-biased) specialisation. </p> <h3 id="4276c6f8fab248c0acc52a7f14462e41"> Summary <a href="#4276c6f8fab248c0acc52a7f14462e41" title="permalink">#</a> </h3> <p> The catamorphism for rose trees is a pair of functions. One function transforms internal nodes with their partially reduced branches, while the other function transforms leaves. </p> <p> For a realistic example of using a rose tree in a real program, see <a href="/2019/09/09/picture-archivist-in-haskell">Picture archivist in Haskell</a>. </p> <p> This article series has so far covered progressively more complex data structures. The first examples (<a href="/2019/05/06/boolean-catamorphism">Boolean catamorphism</a> and <a href="/2019/05/13/peano-catamorphism">Peano catamorphism</a>) were neither <a href="/2018/03/22/functors">functors</a>, <a href="/2018/10/01/applicative-functors">applicatives</a>, nor monads. All subsequent examples, on the other hand, are all of these, and more. The next example presents a functor that's neither applicative nor monad, yet still foldable. Obviously, what functionality it offers is still based on a catamorphism. </p> <p> <strong>Next:</strong> <a href="/2019/06/24/full-binary-tree-catamorphism">Full binary tree catamorphism</a>. </p> </div> <hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Church-encoded rose tree https://blog.ploeh.dk/2019/07/29/church-encoded-rose-tree 2019-07-29T13:14:00+00:00 Mark Seemann <div id="post"> <p> <em>A rose tree is a tree with leaf nodes of one type, and internal nodes of another.</em> </p> <p> This article is part of <a href="/2018/05/22/church-encoding">a series of articles about Church encoding</a>. In the previous articles, you've seen <a href="/2018/06/04/church-encoded-maybe">how to implement a Maybe container</a>, and <a href="/2018/06/11/church-encoded-either">how to implement an Either container</a>. Through these examples, you've learned how to model <a href="https://en.wikipedia.org/wiki/Tagged_union">sum types</a> without explicit language support. In this article, you'll see how to model a <a href="https://en.wikipedia.org/wiki/Rose_tree">rose tree</a>. </p> <p> A rose tree is a general-purpose data structure where each node in a tree has an associated value. Each node can have an arbitrary number of branches, including none. The distinguishing feature from a rose tree and just any <a href="https://en.wikipedia.org/wiki/Tree_(data_structure)">tree</a> is that internal nodes can hold values of a different type than leaf values. </p> <p> <img src="/content/binary/rose-tree-example.png" alt="A rose tree example diagram, with internal nodes containing integers, and leaves containing strings."> </p> <p> The diagram shows an example of a tree of internal integers and leaf strings. All internal nodes contain integer values, and all leaves contain strings. Each node can have an arbitrary number of branches. </p> <h3 id="5255946728c14810a5aaef3c1022d126"> Contract <a href="#5255946728c14810a5aaef3c1022d126" title="permalink">#</a> </h3> <p> In C#, you can represent the fundamental structure of a rose tree with a Church encoding, starting with an interface: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">interface</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt; { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">TResult</span>&nbsp;Match&lt;<span style="color:#2b91af;">TResult</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&gt;,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;node, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">L</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;leaf); }</pre> </p> <p> The structure of a rose tree includes two mutually exclusive cases: internal nodes and leaf nodes. Since there's two cases, the <code>Match</code> method takes two arguments, one for each case. </p> <p> The interface is generic, with two type arguments: <code>N</code> (for <em>Node</em>) and <code>L</code> (for <em>leaf</em>). Any consumer of an <code>IRoseTree&lt;N, L&gt;</code> object must supply two functions when calling the <code>Match</code> method: a function that turns a node into a <code>TResult</code> value, and a function that turns a leaf into a <code>TResult</code> value. </p> <p> Both cases must have a corresponding implementation. </p> <h3 id="89c4833c4e4d46cc8eef2d5eb546f61d"> Leaves <a href="#89c4833c4e4d46cc8eef2d5eb546f61d" title="permalink">#</a> </h3> <p> The <em>leaf</em> implementation is the simplest: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">sealed</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;:&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt; { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">L</span>&nbsp;value; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;RoseLeaf(<span style="color:#2b91af;">L</span>&nbsp;value) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>.value&nbsp;=&nbsp;value; &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">TResult</span>&nbsp;Match&lt;<span style="color:#2b91af;">TResult</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&gt;,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;node, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">L</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;leaf) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;leaf(value); &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:blue;">override</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;Equals(<span style="color:blue;">object</span>&nbsp;obj) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(!(obj&nbsp;<span style="color:blue;">is</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;other)) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">false</span>; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;Equals(value,&nbsp;other.value); &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:blue;">override</span>&nbsp;<span style="color:blue;">int</span>&nbsp;GetHashCode() &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;value.GetHashCode(); &nbsp;&nbsp;&nbsp;&nbsp;} }</pre> </p> <p> The <code>RoseLeaf</code> class is an <a href="https://en.wikipedia.org/wiki/Adapter_pattern">Adapter</a> over a value of the generic type <code>L</code>. As is always the case with Church encoding, it implements the <code>Match</code> method by unconditionally calling one of the arguments, in this case the <code>leaf</code> function, with its adapted <code>value</code>. </p> <p> While it doesn't have to do this, it also overrides <code>Equals</code> and <code>GetHashCode</code>. This is an immutable class, so it's a great candidate to be a <a href="https://martinfowler.com/bliki/ValueObject.html">Value Object</a>. Making it a Value Object makes it easier to compare expected and actual values in unit tests, among other benefits. </p> <h3 id="f211476563fe40379eac66ee887ed75b"> Nodes <a href="#f211476563fe40379eac66ee887ed75b" title="permalink">#</a> </h3> <p> The <em>node</em> implementation is slightly more complex: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">sealed</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">RoseNode</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;:&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt; { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">N</span>&nbsp;value; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&gt;&nbsp;branches; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;RoseNode(<span style="color:#2b91af;">N</span>&nbsp;value,&nbsp;<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&gt;&nbsp;branches) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>.value&nbsp;=&nbsp;value; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>.branches&nbsp;=&nbsp;branches; &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">TResult</span>&nbsp;Match&lt;<span style="color:#2b91af;">TResult</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&gt;,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;node, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">L</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;leaf) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;node(value,&nbsp;branches); &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:blue;">override</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;Equals(<span style="color:blue;">object</span>&nbsp;obj) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(!(obj&nbsp;<span style="color:blue;">is</span>&nbsp;<span style="color:#2b91af;">RoseNode</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;other)) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">false</span>; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;Equals(value,&nbsp;other.value) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&amp;&amp;&nbsp;<span style="color:#2b91af;">Enumerable</span>.SequenceEqual(branches,&nbsp;other.branches); &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:blue;">override</span>&nbsp;<span style="color:blue;">int</span>&nbsp;GetHashCode() &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;value.GetHashCode()&nbsp;^&nbsp;branches.GetHashCode(); &nbsp;&nbsp;&nbsp;&nbsp;} }</pre> </p> <p> A node contains both a value (of the type <code>N</code>) and a collection of sub-trees, or <code>branches</code>. The class implements the <code>Match</code> method by unconditionally calling the <code>node</code> function argument with its constituent values. </p> <p> Again, it overrides <code>Equals</code> and <code>GetHashCode</code> for the same reasons as <code>RoseLeaf</code>. This isn't required to implement Church encoding, but makes comparison and unit testing easier. </p> <h3 id="a5c04c7e127349ed9b759e6361af5ab3"> Usage <a href="#a5c04c7e127349ed9b759e6361af5ab3" title="permalink">#</a> </h3> <p> You can use the <code>RoseLeaf</code> and <code>RoseNode</code> constructors to create new trees, but it sometimes helps to have a static helper method to create values. It turns out that there's little value in a helper method for leaves, but for nodes, it's marginally useful: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;Node&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;(<span style="color:#2b91af;">N</span>&nbsp;value,&nbsp;<span style="color:blue;">params</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;[]&nbsp;branches) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseNode</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;(value,&nbsp;branches); }</pre> </p> <p> This enables you to create tree objects, like this: </p> <p> <pre><span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">int</span>&gt;&nbsp;tree&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(<span style="color:#a31515;">&quot;foo&quot;</span>,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">int</span>&gt;(42),&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">int</span>&gt;(1337));</pre> </p> <p> That's a single node with the label <code>"foo"</code> and two leaves with the values <code>42</code> and <code>1337</code>, respectively. You can create the tree shown in the above diagram like this: </p> <p> <pre><span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;exampleTree&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(42, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(1337, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;foo&quot;</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;bar&quot;</span>)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(2112, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(90125, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;baz&quot;</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;qux&quot;</span>), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;quux&quot;</span>)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;quuz&quot;</span>)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:blue;">int</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;corge&quot;</span>));</pre> </p> <p> You can add various extension methods to implement useful functionality. In later articles, you'll see some more compelling examples, so here, I'm only going to show a few basic examples. One of the simplest features you can add is a method that will tell you if an <code>IRoseTree&lt;N, L&gt;</code> object is a node or a leaf: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IChurchBoolean</span>&nbsp;IsLeaf&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;source) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;source.Match&lt;<span style="color:#2b91af;">IChurchBoolean</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;node:&nbsp;(_,&nbsp;__)&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ChurchFalse</span>(), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;leaf:&nbsp;_&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ChurchTrue</span>()); } <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IChurchBoolean</span>&nbsp;IsNode&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">N</span>,&nbsp;<span style="color:#2b91af;">L</span>&gt;&nbsp;source) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ChurchNot</span>(source.IsLeaf()); }</pre> </p> <p> Since this article is part of the overall article series on Church encoding, and the purpose of that article series is also to show how basic language features can be created from Church encodings, these two methods return <a href="/2018/05/24/church-encoded-boolean-values">Church-encoded Boolean values</a> instead of the built-in <code>bool</code> type. I'm sure you can imagine how you could change the type to <code>bool</code> if you'd like. </p> <p> You can use these methods like this: </p> <p> <pre>&gt; <span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">Guid</span>,&nbsp;<span style="color:blue;">double</span>&gt;&nbsp;tree&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:#2b91af;">Guid</span>,&nbsp;<span style="color:blue;">double</span>&gt;(-3.2); &gt; tree.IsLeaf() ChurchTrue { } &gt; tree.IsNode() ChurchNot(ChurchTrue) &gt; <span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:blue;">long</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;tree&nbsp;=&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node&lt;<span style="color:blue;">long</span>,&nbsp;<span style="color:blue;">string</span>&gt;(42); &gt; tree.IsLeaf() ChurchFalse { } &gt; tree.IsNode() ChurchNot(ChurchFalse)</pre> </p> <p> In a <a href="/2019/09/16/picture-archivist-in-f">future article, you'll see some more compelling examples</a>. </p> <h3 id="3be01779f059443799df57342e2510cb"> Terminology <a href="#3be01779f059443799df57342e2510cb" title="permalink">#</a> </h3> <p> It's not entirely clear what to call a tree like the one shown here. <a href="https://en.wikipedia.org/wiki/Rose_tree">The Wikipedia entry</a> doesn't state one way or the other whether internal node types ought to be distinguishable from leaf node types, but there are <a href="https://twitter.com/kbattocchi/status/1072538730911752192">indications that this could be the case</a>. At least, it seems that the <a href="https://mail.haskell.org/pipermail/haskell-cafe/2015-May/119633.html">term isn't well-defined</a>, so I took the liberty to retcon the name <em>rose tree</em> to the data structure shown here. </p> <p> In the paper that introduces the <em>rose tree</em> term, Meertens writes: <blockquote> <p> "We consider trees whose internal nodes may fork into an arbitrary (natural) number of sub-trees. (If such a node has zero descendants, we still consider it internal.) Each external node carries a data item. No further information is stored in the tree; in particular, internal nodes are unlabelled." </p> <footer><cite><em>First Steps towards the Theory of Rose Trees</em>, Lambert Meertens, 1988</cite></footer> </blockquote> While the concept is foreign in C#, you can trivially introduce a <a href="/2018/01/15/unit-isomorphisms">unit</a> data type: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">Unit</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">Unit</span>&nbsp;Instance&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Unit</span>(); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;Unit()&nbsp;{&nbsp;} }</pre> </p> <p> This enables you to create a rose tree according to Meertens' definition: </p> <p> <pre><span style="color:#2b91af;">IRoseTree</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:blue;">int</span>&gt;&nbsp;meertensTree&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(<span style="color:#2b91af;">Unit</span>.Instance, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(<span style="color:#2b91af;">Unit</span>.Instance, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(<span style="color:#2b91af;">Unit</span>.Instance, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:blue;">int</span>&gt;(2112)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:blue;">int</span>&gt;(42), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:blue;">int</span>&gt;(1337), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:blue;">int</span>&gt;(90125)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">RoseTree</span>.Node(<span style="color:#2b91af;">Unit</span>.Instance, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:blue;">int</span>&gt;(1984)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">RoseLeaf</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:blue;">int</span>&gt;(666));</pre> </p> <p> Visually, you could draw it like this: </p> <p> <img src="/content/binary/meertens-tree-example.png" alt="A Meertens rose tree example diagram, with leaves containing integers."> </p> <p> Thus, the tree structure shown here seems to be a generalisation of Meertens' original definition. </p> <p> I'm not a mathematician, so I may have misunderstood some things. If you have a better name than <em>rose tree</em> for the data structure shown here, please leave a comment. </p> <h3 id="331fa8452cdd435c86ce87b5d39d51c5"> Yeats <a href="#331fa8452cdd435c86ce87b5d39d51c5" title="permalink">#</a> </h3> <p> Now that we're on the topic of <em>rose tree</em> as a term, you may, as a bonus, enjoy a similarly-titled poem: <blockquote> <h4>THE ROSE TREE</h4> <p> "O words are lightly spoken"<br> Said Pearse to Connolly,<br> "Maybe a breath of politic words<br> Has withered our Rose Tree;<br> Or maybe but a wind that blows<br> Across the bitter sea." </p> <p> "It needs to be but watered,"<br> James Connolly replied,<br> "To make the green come out again<br> And spread on every side,<br> And shake the blossom from the bud<br> To be the garden's pride."<br> </p> <p> "But where can we draw water"<br> Said Pearse to Connolly,<br> "When all the wells are parched away?<br> O plain as plain can be<br> There's nothing but our own red blood<br> Can make a right Rose Tree." </p> <footer><cite><a href="https://en.wikipedia.org/wiki/W._B._Yeats">W. B. Yeats</a></cite></footer> </blockquote> As far as I can tell, though, Yeats' metaphor is dissimilar to Meertens'. </p> <h3 id="9906b9a8856248f38b4f03e40252b761"> Summary <a href="#9906b9a8856248f38b4f03e40252b761" title="permalink">#</a> </h3> <p> You may occasionally find use for a tree that distinguishes between internal and leaf nodes. You can model such a tree with a Church encoding, as shown in this article. </p> <p> <strong>Next: </strong> <a href="/2019/04/29/catamorphisms">Catamorphisms</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Chain of Responsibility as catamorphisms https://blog.ploeh.dk/2019/07/22/chain-of-responsibility-as-catamorphisms 2019-07-22T14:11:00+00:00 Mark Seemann <div id="post"> <p> <em>The Chain of Responsibility design pattern can be viewed as a list fold over the First monoid, followed by a Maybe fold.</em> </p> <p> This article is part of <a href="/2018/03/05/some-design-patterns-as-universal-abstractions">a series of articles about specific design patterns and their category theory counterparts</a>. In it, you'll see how the <a href="https://en.wikipedia.org/wiki/Chain-of-responsibility_pattern">Chain of Responsibility design pattern</a> is equivalent to a succession of <a href="/2019/04/29/catamorphisms">catamorphisms</a>. First, you apply the <a href="/2018/04/03/maybe-monoids">First Maybe monoid</a> over the <a href="/2019/05/27/list-catamorphism">list catamorphism</a>, and then you conclude the reduction with the <a href="/2019/05/20/maybe-catamorphism">Maybe catamorphism</a>. </p> <h3 id="46a6c41949db446d9387c8befbf3fdb1"> Pattern <a href="#46a6c41949db446d9387c8befbf3fdb1" title="permalink">#</a> </h3> <p> The Chain of Responsibility design pattern gives you a way to model cascading conditionals with an object structure. It's a chain (or linked list) of objects that all implement the same interface (or base class). Each object (apart from the the last) has a reference to the next object in the list. </p> <p> <img src="/content/binary/chain-of-responsibility-diagram.png" alt="General diagram of the Chain of Responsibility design pattern."> </p> <p> A client (some other code) calls a method on the first object in the list. If that object can handle the request, it does so, and the interaction ends there. If the method returns a value, the object returns the value. </p> <p> If the first object determines that it can't handle the method call, it calls the next object in the chain. It only knows the next object as the interface, so the only way it can delegate the call is by calling the same method as the first one. In the above diagram, <em>Imp1</em> can't handle the method call, so it calls the same method on <em>Imp2</em>, which also can't handle the request and delegates responsibility to <em>Imp3</em>. In the diagram, <em>Imp3</em> can handle the method call, so it does so and returns a result that propagates back up the chain. In that particular example, <em>Imp4</em> never gets involved. </p> <p> You'll see an example below. </p> <p> One of the advantages of the pattern is that you can rearrange the chain to change its behaviour. You can even do this at run time, if you'd like, since all objects implement the same interface. </p> <h3 id="08a67dafd71f4bdd9a2e2577b0e43f9a"> User icon example <a href="#08a67dafd71f4bdd9a2e2577b0e43f9a" title="permalink">#</a> </h3> <p> Consider an online system that maintains user profiles for users. A user is modelled with the <code>User</code> class: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;User(<span style="color:blue;">int</span>&nbsp;id,&nbsp;<span style="color:blue;">string</span>&nbsp;name,&nbsp;<span style="color:blue;">string</span>&nbsp;email,&nbsp;<span style="color:blue;">bool</span>&nbsp;useGravatar,&nbsp;<span style="color:blue;">bool</span>&nbsp;useIdenticon)</pre> </p> <p> While I only show the signature of the class' constructor, it should be enough to give you an idea. If you need more details, the entire example code base is <a href="https://github.com/ploeh/UserProfile">available on GitHub</a>. </p> <p> Apart from an <code>id</code>, a <code>name</code> and <code>email</code> address, a user also has two flags. One flag tracks whether the user wishes to use his or her <a href="http://www.gravatar.com">Gravatar</a>, while another flag tracks if the user would like to use an <a href="https://en.wikipedia.org/wiki/Identicon">Identicon</a>. Obviously, both flags could be <code>true</code>, in which case the current business rule states that the Gravatar should take precedence. </p> <p> If none of the flags are set, users might still have a picture associated with their profile. This could be a picture that they've uploaded to the system, and is being tracked by a database. </p> <p> If no user icon can be found or generated, ultimately the system should use a fallback, default icon: </p> <p> <img src="/content/binary/default-user-icon.png" alt="Default user icon."> </p> <p> To summarise, the current rules are: <ol> <li>Use Gravatar if flag is set.</li> <li>Use Identicon if flag is set.</li> <li>Use uploaded picture if available.</li> <li>Use default icon.</li> </ol> The order of precedence could change in the future, new images sources could be added, or some of the present sources could be removed. Modelling this set of rules as a Chain of Responsibility makes it easy for you to reorder the rules, should you need to. </p> <p> To request an icon, a client can use the <code>IIconReader</code> interface: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">interface</span>&nbsp;<span style="color:#2b91af;">IIconReader</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Icon</span>&nbsp;ReadIcon(<span style="color:#2b91af;">User</span>&nbsp;user); }</pre> </p> <p> The <code>Icon</code> class is just a <a href="https://martinfowler.com/bliki/ValueObject.html">Value Object</a> wrapper around a URL. The idea is that such a URL can be used in an <code>img</code> tag to show the icon. Again, the full source code is available on GitHub if you'd like to investigate the details. </p> <p> The various rules for icon retrieval can be implemented using this interface. </p> <h3 id="b2a4cbfb576949c392ea0e0b3d440175"> Gravatar reader <a href="#b2a4cbfb576949c392ea0e0b3d440175" title="permalink">#</a> </h3> <p> Although you don't have to implement the classes in the order in which you are going to compose them, it seems natural to do so, starting with the Gravatar implementation. </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">GravatarReader</span>&nbsp;:&nbsp;<span style="color:#2b91af;">IIconReader</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">IIconReader</span>&nbsp;next; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;GravatarReader(<span style="color:#2b91af;">IIconReader</span>&nbsp;next) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>.next&nbsp;=&nbsp;next; &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Icon</span>&nbsp;ReadIcon(<span style="color:#2b91af;">User</span>&nbsp;user) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(user.UseGravatar) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Icon</span>(<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Gravatar</span>(user.Email).Url); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;next.ReadIcon(user); &nbsp;&nbsp;&nbsp;&nbsp;} }</pre> </p> <p> The <code>GravatarReader</code> class both implements the <code>IIconReader</code> interface, but also decorates another object of the same polymorphic type. If <code>user.UseGravatar</code> is <code>true</code>, it generates the appropriate Gravatar URL based on the user's <code>Email</code> address; otherwise, it delegates the work to the <code>next</code> object in the Chain of Responsibility. </p> <p> The <code>Gravatar</code> class contains the implementation details to generate the Gravatar <code>Url</code>. Again, please refer to the GitHub repository if you're interested in the details. </p> <h3 id="222ae025b264455695f1dbbd74cad17b"> Identicon reader <a href="#222ae025b264455695f1dbbd74cad17b" title="permalink">#</a> </h3> <p> When you compose the chain, according to the above business logic, the next type of icon you should attempt to generate is an Identicon. It's natural to implement the Identicon reader next, then: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">IdenticonReader</span>&nbsp;:&nbsp;<span style="color:#2b91af;">IIconReader</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">IIconReader</span>&nbsp;next; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;IdenticonReader(<span style="color:#2b91af;">IIconReader</span>&nbsp;next) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>.next&nbsp;=&nbsp;next; &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Icon</span>&nbsp;ReadIcon(<span style="color:#2b91af;">User</span>&nbsp;user) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(user.UseIdenticon) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Icon</span>(<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Uri</span>(baseUrl,&nbsp;HashUser(user))); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;next.ReadIcon(user); &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:green;">//&nbsp;Implementation&nbsp;details&nbsp;go&nbsp;here...</span> }</pre> </p> <p> Again, I'm omitting implementation details in order to focus on the Chain of Responsibility design pattern. If <code>user.UseIdenticon</code> is <code>true</code>, the <code>IdenticonReader</code> generates the appropriate Identicon and returns the URL for it; otherwise, it delegates the work to the <code>next</code> object in the chain. </p> <h3 id="e9f2904333b940c1a9a90522d19a41f3"> Database icon reader <a href="#e9f2904333b940c1a9a90522d19a41f3" title="permalink">#</a> </h3> <p> The <code>DBIconReader</code> class attempts to find an icon ID in a database. If it succeeds, it creates a URL corresponding to that ID. The assumption is that that resource exists; either it's a file on disk, or it's an image resource generated on the spot based on binary data stored in the database. </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">DBIconReader</span>&nbsp;:&nbsp;<span style="color:#2b91af;">IIconReader</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">IUserRepository</span>&nbsp;repository; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">IIconReader</span>&nbsp;next; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;DBIconReader(<span style="color:#2b91af;">IUserRepository</span>&nbsp;repository,&nbsp;<span style="color:#2b91af;">IIconReader</span>&nbsp;next) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>.repository&nbsp;=&nbsp;repository; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>.next&nbsp;=&nbsp;next; &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Icon</span>&nbsp;ReadIcon(<span style="color:#2b91af;">User</span>&nbsp;user) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(!repository.TryReadIconId(user.Id,&nbsp;<span style="color:blue;">out</span>&nbsp;<span style="color:blue;">string</span>&nbsp;iconId)) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;next.ReadIcon(user); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;parameters&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Dictionary</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">string</span>&gt; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{&nbsp;<span style="color:#a31515;">&quot;iconId&quot;</span>,&nbsp;iconId&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Icon</span>(urlTemplate.BindByName(baseUrl,&nbsp;parameters)); &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">Uri</span>&nbsp;baseUrl&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Uri</span>(<span style="color:#a31515;">&quot;https://example.com&quot;</span>); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">UriTemplate</span>&nbsp;urlTemplate&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">UriTemplate</span>(<span style="color:#a31515;">&quot;users/{iconId}/icon&quot;</span>); }</pre> </p> <p> This class demonstrates some variations in the way you can implement the Chain of Responsibility design pattern. The above <code>GravatarReader</code> and <code>IdenticonReader</code> classes both follow the same implementation pattern of checking a condition, and then performing work if the condition is <code>true</code>. The delegation to the next object in the chain happens, in those two classes, outside of the <code>if</code> statement. </p> <p> The <code>DBIconReader</code> class, on the other hand, reverses the structure of the code. It uses a <a href="https://refactoring.com/catalog/replaceNestedConditionalWithGuardClauses.html">Guard Clause</a> to detect whether to exit early, which is done by delegating work to the <code>next</code> object in the chain. </p> <p> If <code>TryReadIconId</code> returns <code>true</code>, however, the <code>ReadIcon</code> method proceeds to create the appropriate icon URL. </p> <p> Another variation on the Chain of Responsibility design pattern demonstrated by the <code>DBIconReader</code> class is that it takes a second dependency, apart from <code>next</code>. The <code>repository</code> is the usual misapplication of the Repository design pattern that everyone think they use correctly. Here, it's used in the common sense to provide access to a database. The main point, though, is that you can add as many other dependencies to a link in the chain as you'd like. All links, apart from the last, however, must have a reference to the <code>next</code> link in the chain. </p> <h3 id="cee40120578b4732892e6fd72329d5de"> Default icon reader <a href="#cee40120578b4732892e6fd72329d5de" title="permalink">#</a> </h3> <p> Like linked lists, a Chain of Responsibility has to ultimately terminate. You can use the following <code>DefaultIconReader</code> for that. </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">DefaultIconReader</span>&nbsp;:&nbsp;<span style="color:#2b91af;">IIconReader</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Icon</span>&nbsp;ReadIcon(<span style="color:#2b91af;">User</span>&nbsp;user) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:#2b91af;">Icon</span>.Default; &nbsp;&nbsp;&nbsp;&nbsp;} }</pre> </p> <p> This class unconditionally returns the <code>Default</code> icon. Notice that it doesn't have any <code>next</code> object it delegates to. This terminates the chain. If no previous implementation of the <code>IIconReader</code> has returned an <code>Icon</code> for the <code>user</code>, this one does. </p> <h3 id="8eb05bed2d98488a91c09bab52b00a53"> Chain composition <a href="#8eb05bed2d98488a91c09bab52b00a53" title="permalink">#</a> </h3> <p> With four implementations of <code>IIconReader</code>, you can now compose the Chain of Responsibility: </p> <p> <pre><span style="color:#2b91af;">IIconReader</span>&nbsp;reader&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">GravatarReader</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">IdenticonReader</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DBIconReader</span>(repo, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DefaultIconReader</span>())));</pre> </p> <p> The first link in the chain is a <code>GravatarReader</code> object that contains an <code>IdenticonReader</code> object as its <code>next</code> link, and so on. Referring back to the source code of <code>GravatarReader</code>, notice that its <code>next</code> dependency is declared as an <code>IIconReader</code>. Since the <code>IdenticonReader</code> class implements that interface, you can compose the chain like this, but if you later decide to change the order of the objects, you can do so simply by changing the composition. You could remove objects altogether, or add new classes, and you could even do this at run time, if required. </p> <p> The <code>DBIconReader</code> class requires an extra <code>IUserRepository</code> dependency, here simply an existing object called <code>repo</code>. </p> <p> The <code>DefaultIconReader</code> takes no other dependencies, so this effectively terminates the chain. If you try to pass another <code>IIconReader</code> to its constructor, the code doesn't compile. </p> <h3 id="fc1551665bb940b8ba5e75be81c0629a"> Haskell proof of concept <a href="#fc1551665bb940b8ba5e75be81c0629a" title="permalink">#</a> </h3> <p> When evaluating whether a design is <a href="/2018/11/19/functional-architecture-a-definition">a functional architecture</a>, I often port the relevant parts to <a href="https://www.haskell.org">Haskell</a>. You can do the same with the above example, and put it in a form where it's clearer that the Chain of Responsibility pattern is equivalent to two well-known catamorphisms. </p> <p> Readers not comfortable with Haskell can skip the next few sections. The object-oriented example continues below. </p> <p> <code>User</code> and <code>Icon</code> types are defined by types equivalent to above. There's no explicit interface, however. Creation of Gravatars and Identicons are both pure functions with the type <code>User -&gt; Maybe Icon</code>. Here's the Gravatar function, but the Identicon function looks similar: </p> <p> <pre><span style="color:#2b91af;">gravatarUrl</span>&nbsp;::&nbsp;<span style="color:#2b91af;">String</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">String</span> gravatarUrl&nbsp;email&nbsp;= &nbsp;&nbsp;<span style="color:#a31515;">&quot;https://www.gravatar.com/avatar/&quot;</span>&nbsp;++&nbsp;<span style="color:blue;">show</span>&nbsp;(hashString&nbsp;email&nbsp;::&nbsp;MD5Digest) <span style="color:#2b91af;">getGravatar</span>&nbsp;::&nbsp;<span style="color:blue;">User</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&nbsp;<span style="color:blue;">Icon</span> getGravatar&nbsp;u&nbsp;= &nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;useGravatar&nbsp;u &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">then</span>&nbsp;Just&nbsp;$&nbsp;Icon&nbsp;$&nbsp;gravatarUrl&nbsp;$&nbsp;userEmail&nbsp;u &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">else</span>&nbsp;Nothing</pre> </p> <p> Reading an icon ID from a database, however, is an impure operation, so the function to do this has the type <code>User -&gt; IO (Maybe Icon)</code>. </p> <h3 id="11adf8bd104d41fab9e6bcaef249210c"> Lazy I/O in Haskell <a href="#11adf8bd104d41fab9e6bcaef249210c" title="permalink">#</a> </h3> <p> Notice that the database icon-querying function has the return type <code>IO (Maybe Icon)</code>. In the introduction you read that the Chain of Responsibility design pattern is a sequence of catamorphisms - the first one over a list of <code>First</code> values. While <code>First</code> is, in itself, a <code>Semigroup</code> instance, it gives rise to a <code>Monoid</code> instance when combined with <code>Maybe</code>. Thus, to showcase the abstractions being used, you could create a list of <code>Maybe (First Icon)</code> values. This forms a <code>Monoid</code>, so is easy to fold. </p> <p> The problem with that, however, is that <code>IO</code> is strict under evaluation, so while it works, <a href="https://stackoverflow.com/q/47120384/126014">it's no longer lazy</a>. You can combine <code>IO (Maybe (First Icon))</code> values, but it leads to too much I/O activity. </p> <p> You can <a href="https://stackoverflow.com/q/47120384/126014">solve this problem with a newtype wrapper</a>: </p> <p> <pre><span style="color:blue;">newtype</span>&nbsp;FirstIO&nbsp;a&nbsp;=&nbsp;FirstIO&nbsp;(MaybeT&nbsp;IO&nbsp;a)&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Functor</span>,&nbsp;<span style="color:#2b91af;">Applicative</span>,&nbsp;<span style="color:#2b91af;">Monad</span>,&nbsp;<span style="color:#2b91af;">Alternative</span>) <span style="color:#2b91af;">firstIO</span>&nbsp;::&nbsp;<span style="color:#2b91af;">IO</span>&nbsp;(<span style="color:#2b91af;">Maybe</span>&nbsp;a)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">FirstIO</span>&nbsp;a firstIO&nbsp;=&nbsp;FirstIO&nbsp;.&nbsp;MaybeT <span style="color:#2b91af;">getFirstIO</span>&nbsp;::&nbsp;<span style="color:blue;">FirstIO</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">IO</span>&nbsp;(<span style="color:#2b91af;">Maybe</span>&nbsp;a) getFirstIO&nbsp;(FirstIO&nbsp;(MaybeT&nbsp;x))&nbsp;=&nbsp;x <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Semigroup</span>&nbsp;(<span style="color:blue;">FirstIO</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:#2b91af;">(&lt;&gt;)</span>&nbsp;=&nbsp;<span style="color:#2b91af;">(&lt;|&gt;)</span> <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Monoid</span>&nbsp;(<span style="color:blue;">FirstIO</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;mempty&nbsp;=&nbsp;empty</pre> </p> <p> This uses the <code>GeneralizedNewtypeDeriving</code> GHC extension to automatically make <code>FirstIO</code> <code>Functor</code>, <code>Applicative</code>, <code>Monad</code>, and <code>Alternative</code>. It also uses the <code>Alternative</code> instance to implement <code>Semigroup</code> and <code>Monoid</code>. You may recall from <a href="http://hackage.haskell.org/package/base/docs/Control-Applicative.html">the documentation</a> that <code>Alternative</code> is already a "monoid on applicative functors." </p> <h3 id="995f9ea8f8344aea93b2ffd0b3aad71f"> Alignment <a href="#995f9ea8f8344aea93b2ffd0b3aad71f" title="permalink">#</a> </h3> <p> You now have three functions with different types: two pure functions with the type <code>User -&gt; Maybe Icon</code> and one impure database-bound function with the type <code>User -&gt; IO (Maybe Icon)</code>. In order to have a common abstraction, you should align them so that all types match. At first glance, <code>User -&gt; IO (Maybe (First Icon))</code> seems like a type that fits all implementations, but that causes too much I/O to take place, so instead, use <code>User -&gt; FirstIO Icon</code>. Here's how to lift the pure <code>getGravatar</code> function: </p> <p> <pre><span style="color:#2b91af;">getGravatarIO</span>&nbsp;::&nbsp;<span style="color:blue;">User</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">FirstIO</span>&nbsp;<span style="color:blue;">Icon</span> getGravatarIO&nbsp;=&nbsp;firstIO&nbsp;.&nbsp;<span style="color:blue;">return</span>&nbsp;.&nbsp;getGravatar</pre> </p> <p> You can lift the other functions in similar fashion, to produce <code>getGravatarIO</code>, <code>getIdenticonIO</code>, and <code>getDBIconIO</code>, all with the mutual type <code>User -&gt; FirstIO Icon</code>. </p> <h3 id="f601a51f3006430398232e05b6595da0"> Haskell composition <a href="#f601a51f3006430398232e05b6595da0" title="permalink">#</a> </h3> <p> The goal of the Haskell proof of concept is to compose a function that can provide an <code>Icon</code> for any <code>User</code> - just like the above C# composition that uses Chain of Responsibility. There's, however, no way around impurity, because one of the steps involve a database, so the aim is a composition with the type <code>User -&gt; IO Icon</code>. </p> <p> While a more compact composition is possible, I'll show it in a way that makes the catamorphisms explicit: </p> <p> <pre><span style="color:#2b91af;">getIcon</span>&nbsp;::&nbsp;<span style="color:blue;">User</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">IO</span>&nbsp;<span style="color:blue;">Icon</span> getIcon&nbsp;u&nbsp;=&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;lazyIcons&nbsp;=&nbsp;<span style="color:blue;">fmap</span>&nbsp;(\f&nbsp;-&gt;&nbsp;f&nbsp;u)&nbsp;[getGravatarIO,&nbsp;getIdenticonIO,&nbsp;getDBIconIO] &nbsp;&nbsp;m&nbsp;&lt;-&nbsp;getFirstIO&nbsp;$&nbsp;fold&nbsp;lazyIcons &nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;$&nbsp;fromMaybe&nbsp;defaultIcon&nbsp;m</pre> </p> <p> The <code>getIcon</code> function starts with a list of all three functions. For each of them, it calls the function with the <code>User</code> value <code>u</code>. This may seem inefficient and redundant, because all three function calls may not be required, but since the return values are <code>FirstIO</code> values, all three function calls are lazily evaluated - even under <code>IO</code>. The result, <code>lazyIcons</code>, is a <code>[FirstIO Icon]</code> value; i.e. a lazily evaluated list of lazily evaluated values. </p> <p> This first step is just to put the potential values in a form that's recognisable. You can now <code>fold</code> the <code>lazyIcons</code> to a single <code>FirstIO Icon</code> value, and then use <code>getFirstIO</code> to unwrap it. Due to <code>do</code> notation, <code>m</code> is a <code>Maybe Icon</code> value. </p> <p> This is the first catamorphism. Granted, the generalisation that <code>fold</code> offers is not really required, since <code>lazyIcons</code> is a list; <code>mconcat</code> would have worked just as well. I did, however, choose to use <code>fold</code> (from <code>Data.Foldable</code>) to emphasise the point. While the <code>fold</code> function itself isn't the catamorphism for lists, we know that <a href="/2019/05/27/list-catamorphism">it's derived from the list catamorphism</a>. </p> <p> The final step is to utilise the Maybe catamorphism to reduce the <code>Maybe Icon</code> value to an <code>Icon</code> value. Again, the <code>getIcon</code> function doesn't use the Maybe catamorphism directly, but rather the derived <code>fromMaybe</code> function. The <a href="/2019/05/20/maybe-catamorphism">Maybe catamorphism</a> is the <code>maybe</code> function, but you can trivially implement <code>fromMaybe</code> with <code>maybe</code>. </p> <p> For <a href="https://en.wikipedia.org/wiki/Code_golf">golfers</a>, it's certainly possible to write this function in a more compact manner. Here's a <a href="https://en.wikipedia.org/wiki/Tacit_programming">point-free</a> version: </p> <p> <pre><span style="color:#2b91af;">getIcon</span>&nbsp;::&nbsp;<span style="color:blue;">User</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">IO</span>&nbsp;<span style="color:blue;">Icon</span> getIcon&nbsp;= &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;(fromMaybe&nbsp;defaultIcon)&nbsp;.&nbsp;getFirstIO&nbsp;.&nbsp;fold&nbsp;[getGravatarIO,&nbsp;getIdenticonIO,&nbsp;getDBIconIO]</pre> </p> <p> This alternative version utilises that <code>a -&gt; m</code> is a <code>Monoid</code> instance when <code>m</code> is a <code>Monoid</code> instance. That's the reason that you can <code>fold</code> a list of functions. The more explicit version above doesn't do that, but the behaviour is the same in both cases. </p> <p> That's all the Haskell code we need to discern the universal abstractions involved in the Chain of Responsibility design pattern. We can now return to the C# code example. </p> <h3 id="492ff50788784d7dbf6560ed08ed6bf7"> Chains as lists <a href="#492ff50788784d7dbf6560ed08ed6bf7" title="permalink">#</a> </h3> <p> The Chain of Responsibility design pattern is often illustrated like above, in a staircase-like diagram. There's, however, no inherent requirement to do so. You could also flatten the diagram: </p> <p> <img src="/content/binary/chain-of-responsibility-as-a-linked-list.png" alt="Chain of Responsibility illustrated as a linked list."> </p> <p> This looks a lot like a linked list. </p> <p> The difference is, however, that the terminator of a linked list is usually empty. Here, however, you have two types of objects. All objects apart from the rightmost object represent a <em>potential</em>. Each object may, or may not, handle the method call and produce an outcome; if an object can't handle the method call, it'll delegate to the next object in the chain. </p> <p> The rightmost object, however, is different. This object can't delegate any further, but <em>must</em> handle the method call. In the icon reader example, this is the <code>DefaultIconReader</code> class. </p> <p> Once you start to see most of the list as a list of potential values, you may realise that you'll be able to collapse into it a single potential value. This is possible because <a href="/2018/04/03/maybe-monoids">a list of values where you pick the first non-empty value forms a monoid</a>. This is sometimes called the <em>First</em> <a href="/2017/10/06/monoids">monoid</a>. </p> <p> In other words, you can reduce, or fold, all of the list, except the rightmost value, to a single potential value: </p> <p> <img src="/content/binary/chain-of-responsibility-as-a-linked-list-single-fold.png" alt="Chain of Responsibility illustrated as a linked list, with all but the rightmost objects folded to one."> </p> <p> When you do that, however, you're left with a single potential value. The result of folding most of the list is that you get the leftmost non-empty value in the list. There's no guarantee, however, that that value is non-empty. If all the values in the list are empty, the result is also empty. This means that you somehow need to combine a potential value with a value that's guaranteed to be present: the terminator. </p> <p> You can do that wither another fold: </p> <p> <img src="/content/binary/chain-of-responsibility-as-a-linked-list-double-fold.png" alt="Chain of Responsibility illustrated as a linked list, with two consecutive folds."> </p> <p> This second fold isn't a list fold, but rather a Maybe fold. </p> <h3 id="7632b9ff458d417fa49b1c65f7b198ed"> Maybe <a href="#7632b9ff458d417fa49b1c65f7b198ed" title="permalink">#</a> </h3> <p> The <em>First</em> monoid is a monoid over <a href="/2018/03/26/the-maybe-functor">Maybe</a>, so add a <code>Maybe</code> class to the code base. In Haskell, the catamorphism for Maybe is called <code>maybe</code>, but that's not a good method name in object-oriented design. Another option is some variation of <em>fold</em>, but in C#, this functionality tends to be called <code>Aggregate</code>, at least for <code>IEnumerable&lt;T&gt;</code>, so I'll reuse that terminology: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">TResult</span>&nbsp;Aggregate&lt;<span style="color:#2b91af;">TResult</span>&gt;(<span style="color:#2b91af;">TResult</span>&nbsp;@default,&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;func) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(func&nbsp;==&nbsp;<span style="color:blue;">null</span>) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">throw</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ArgumentNullException</span>(<span style="color:blue;">nameof</span>(func)); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;hasItem&nbsp;?&nbsp;func(item)&nbsp;:&nbsp;@default; }</pre> </p> <p> You can implement another, more list-like <code>Aggregate</code> overload from this one, but for this article, you don't need it. </p> <h3 id="8b60d0c605d14cffbfa5e237cf26b7b2"> From TryReadIconId to Maybe <a href="#8b60d0c605d14cffbfa5e237cf26b7b2" title="permalink">#</a> </h3> <p> In the above code examples, <code>DBIconReader</code> depends on <code>IUserRepository</code>, which defined this method: </p> <p> <pre><span style="color:blue;">bool</span>&nbsp;TryReadIconId(<span style="color:blue;">int</span>&nbsp;userId,&nbsp;<span style="color:blue;">out</span>&nbsp;<span style="color:blue;">string</span>&nbsp;iconId);</pre> </p> <p> From <a href="/2019/07/15/tester-doer-isomorphisms">Tester-Doer isomorphisms</a> we know, however, that such a design is isomorphic to returning a Maybe value, and since that's more composable, do that: </p> <p> <pre><span style="color:#2b91af;">Maybe</span>&lt;<span style="color:blue;">string</span>&gt;&nbsp;ReadIconId(<span style="color:blue;">int</span>&nbsp;userId);</pre> </p> <p> This requires you to refactor the <code>DBIconReader</code> implementation of the <code>ReadIcon</code> method: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Icon</span>&nbsp;ReadIcon(<span style="color:#2b91af;">User</span>&nbsp;user) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:blue;">string</span>&gt;&nbsp;mid&nbsp;=&nbsp;repository.ReadIconId(user.Id); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;&nbsp;lazyResult&nbsp;=&nbsp;mid.Aggregate( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;@default:&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;(()&nbsp;=&gt;&nbsp;next.ReadIcon(user)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;func:&nbsp;id&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;(()&nbsp;=&gt;&nbsp;CreateIcon(id))); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;lazyResult.Value; }</pre> </p> <p> A few things are worth a mention. Notice that the above <code>Aggregate</code> method (the Maybe catamorphism) requires you to supply a <code>@default</code> value (to be used if the Maybe object is empty). In the Chain of Responsibility design pattern, however, the fallback value is produced by calling the <code>next</code> object in the chain. If you do this unconditionally, however, you perform too much work. You're only supposed to call <code>next</code> if the current object can't handle the method call. </p> <p> The solution is to aggregate the <code>mid</code> object to a <code>Lazy&lt;Icon&gt;</code> and then return its <code>Value</code>. The <code>@default</code> value is now a lazy computation that calls <code>next</code> only if its <code>Value</code> is read. When <code>mid</code> is populated, on the other hand, the lazy computation calls the private <code>CreateIcon</code> method when <code>Value</code> is accessed. The private <code>CreateIcon</code> method contains the same logic as before the refactoring. </p> <p> This change of <code>DBIconReader</code> isn't strictly necessary in order to change the overall Chain of Responsibility to a pair of catamorphisms, but serves, I think, as a nice introduction to the use of the Maybe catamorphism. </p> <h3 id="ec329c8a0b70432d81d6f69e7084c13f"> Optional icon readers <a href="#ec329c8a0b70432d81d6f69e7084c13f" title="permalink">#</a> </h3> <p> Previously, the <code>IIconReader</code> interface <em>required</em> each implementation to return an <code>Icon</code> object: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">interface</span>&nbsp;<span style="color:#2b91af;">IIconReader</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Icon</span>&nbsp;ReadIcon(<span style="color:#2b91af;">User</span>&nbsp;user); }</pre> </p> <p> When you have an object like <code>GravatarReader</code> that may or may not return an <code>Icon</code>, this requirement leads toward the Chain of Responsibility design pattern. You can, however, shift the responsibility of what to do next by changing the interface: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">interface</span>&nbsp;<span style="color:#2b91af;">IIconReader</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;&nbsp;ReadIcon(<span style="color:#2b91af;">User</span>&nbsp;user); }</pre> </p> <p> An implementation like <code>GravatarReader</code> becomes simpler: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">GravatarReader</span>&nbsp;:&nbsp;<span style="color:#2b91af;">IIconReader</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;&nbsp;ReadIcon(<span style="color:#2b91af;">User</span>&nbsp;user) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(user.UseGravatar) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;(<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Icon</span>(<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Gravatar</span>(user.Email).Url)); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;(); &nbsp;&nbsp;&nbsp;&nbsp;} }</pre> </p> <p> No longer do you have to pass in a <code>next</code> dependency. Instead, you just return an empty <code>Maybe&lt;Icon&gt;</code> if you can't handle the method call. The same change applies to the <code>IdenticonReader</code> class. </p> <p> Since <a href="/2018/03/26/the-maybe-functor">Maybe is a functor</a>, and the <code>DBIconReader</code> already works on a <code>Maybe&lt;string&gt;</code> value, its implementation is greatly simplified: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;&nbsp;ReadIcon(<span style="color:#2b91af;">User</span>&nbsp;user) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;repository.ReadIconId(user.Id).Select(CreateIcon); }</pre> </p> <p> Since <code>ReadIconId</code> returns a <code>Maybe&lt;string&gt;</code>, you can simply use <code>Select</code> to transform the icon ID to an <code>Icon</code> object if the Maybe is populated. </p> <h3 id="94cac3b9e52e48c2a1768fd24c72e4bd"> Coalescing Composite <a href="#94cac3b9e52e48c2a1768fd24c72e4bd" title="permalink">#</a> </h3> <p> As an intermediate step, you can compose the various readers using a <a href="/2018/04/09/coalescing-composite-as-a-monoid">Coalescing Composite</a>: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">CompositeIconReader</span>&nbsp;:&nbsp;<span style="color:#2b91af;">IIconReader</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">IIconReader</span>[]&nbsp;iconReaders; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;CompositeIconReader(<span style="color:blue;">params</span>&nbsp;<span style="color:#2b91af;">IIconReader</span>[]&nbsp;iconReaders) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>.iconReaders&nbsp;=&nbsp;iconReaders; &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;&nbsp;ReadIcon(<span style="color:#2b91af;">User</span>&nbsp;user) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">foreach</span>&nbsp;(<span style="color:blue;">var</span>&nbsp;iconReader&nbsp;<span style="color:blue;">in</span>&nbsp;iconReaders) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;mIcon&nbsp;=&nbsp;iconReader.ReadIcon(user); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(IsPopulated(mIcon)) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;mIcon; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;(); &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;IsPopulated&lt;<span style="color:#2b91af;">T</span>&gt;(<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">T</span>&gt;&nbsp;m) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;m.Aggregate(<span style="color:blue;">false</span>,&nbsp;_&nbsp;=&gt;&nbsp;<span style="color:blue;">true</span>); &nbsp;&nbsp;&nbsp;&nbsp;} }</pre> </p> <p> I prefer a more explicit design over this one, so this is just an intermediate step. This <code>IIconReader</code> implementation composes an array of other <code>IIconReader</code> objects and queries each in order to return the first populated Maybe value it finds. If it doesn't find any populated value, it returns an empty Maybe object. </p> <p> You can now compose your <code>IIconReader</code> objects into a <a href="https://en.wikipedia.org/wiki/Composite_pattern">Composite</a>: </p> <p> <pre><span style="color:#2b91af;">IIconReader</span>&nbsp;reader&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">CompositeIconReader</span>( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">GravatarReader</span>(), &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">IdenticonReader</span>(), &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DBIconReader</span>(repo));</pre> </p> <p> While this gives you a single object on which you can call <code>ReadIcon</code>, the return value of that method is still a <code>Maybe&lt;Icon&gt;</code> object. You still need to reduce the <code>Maybe&lt;Icon&gt;</code> object to an <code>Icon</code> object. You can do this with a Maybe helper method: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">T</span>&nbsp;GetValueOrDefault(<span style="color:#2b91af;">T</span>&nbsp;@default) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;Aggregate(@default,&nbsp;x&nbsp;=&gt;&nbsp;x); }</pre> </p> <p> Given a <code>User</code> object named <code>user</code>, you can now use the composition and the <code>GetValueOrDefault</code> method to get an <code>Icon</code> object: </p> <p> <pre><span style="color:#2b91af;">Icon</span>&nbsp;icon&nbsp;=&nbsp;reader.ReadIcon(user).GetValueOrDefault(<span style="color:#2b91af;">Icon</span>.Default);</pre> </p> <p> First you use the composed <code>reader</code> to produce a <code>Maybe&lt;Icon&gt;</code> object, and then you use the <code>GetValueOrDefault</code> method to reduce the <code>Maybe&lt;Icon&gt;</code> object to an <code>Icon</code> object. </p> <p> The latter of these two steps, <code>GetValueOrDefault</code>, is already based on the Maybe catamorphism, but the first step is still too implicit to clearly show the nature of what's actually going on. The next step is to refactor the Coalescing Composite to a list of monoidal values. </p> <h3 id="c75ce57c2b4f4315a93eaa91b653a370"> First <a href="#c75ce57c2b4f4315a93eaa91b653a370" title="permalink">#</a> </h3> <p> While not strictly necessary, you can introduce a <code>First&lt;T&gt;</code> wrapper: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">sealed</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">T</span>&gt; { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;First(<span style="color:#2b91af;">T</span>&nbsp;item) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(item&nbsp;==&nbsp;<span style="color:blue;">null</span>) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">throw</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ArgumentNullException</span>(<span style="color:blue;">nameof</span>(item)); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Item&nbsp;=&nbsp;item; &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">T</span>&nbsp;Item&nbsp;{&nbsp;<span style="color:blue;">get</span>;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:blue;">override</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;Equals(<span style="color:blue;">object</span>&nbsp;obj) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(!(obj&nbsp;<span style="color:blue;">is</span>&nbsp;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">T</span>&gt;&nbsp;other)) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">false</span>; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;Equals(Item,&nbsp;other.Item); &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:blue;">override</span>&nbsp;<span style="color:blue;">int</span>&nbsp;GetHashCode() &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;Item.GetHashCode(); &nbsp;&nbsp;&nbsp;&nbsp;} }</pre> </p> <p> In this particular example, the <code>First&lt;T&gt;</code> class adds no new capabilities, so it's technically redundant. You could add to it methods to combine two <code>First&lt;T&gt;</code> objects into one (since <em>First</em> forms a <a href="/2017/11/27/semigroups">semigroup</a>), and perhaps a method or two to <a href="/2017/12/11/semigroups-accumulate">accumulate multiple values</a>, but in this article, none of those are required. </p> <p> While the class as shown above doesn't add any behaviour, I like that it signals intent, so I'll use it in that role. </p> <h3 id="c3feb40d90fc4d389fa0b3812abaa62c"> Lazy I/O in C# <a href="#c3feb40d90fc4d389fa0b3812abaa62c" title="permalink">#</a> </h3> <p> Like in the above Haskell code, you'll need to be able to combine two <code>First&lt;T&gt;</code> objects in a lazy fashion, in such a way that if the first object is populated, the I/O associated with producing the second value never happens. In Haskell I addressed that concern with a <code>newtype</code> that, among other abstractions, is a monoid. You can do the same in C# with an extension method: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">T</span>&gt;&gt;&gt;&nbsp;FindFirst&lt;<span style="color:#2b91af;">T</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">T</span>&gt;&gt;&gt;&nbsp;m, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">T</span>&gt;&gt;&gt;&nbsp;other) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(m.Value.IsPopulated()) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;m; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;other; } <span style="color:blue;">private</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;IsPopulated&lt;<span style="color:#2b91af;">T</span>&gt;(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">T</span>&gt;&nbsp;m) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;m.Aggregate(<span style="color:blue;">false</span>,&nbsp;_&nbsp;=&gt;&nbsp;<span style="color:blue;">true</span>); }</pre> </p> <p> The <code>FindFirst</code> method returns the first (leftmost) non-empty object of two options. It's a lazy version of the <em>First</em> monoid, and <a href="/2019/04/15/lazy-monoids">that's still a monoid</a>. It's truly lazy because it never accesses the <code>Value</code> property on <code>other</code>. While it has to force evaluation of the first lazy computation, <code>m</code>, it doesn't have to evaluate <code>other</code>. Thus, whenever <code>m</code> is populated, <code>other</code> can remain non-evaluated. </p> <p> Since <a href="/2017/11/20/monoids-accumulate">monoids accumulate</a>, you can also write an extension method to implement that functionality: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">T</span>&gt;&gt;&gt;&nbsp;FindFirst&lt;<span style="color:#2b91af;">T</span>&gt;(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">T</span>&gt;&gt;&gt;&gt;&nbsp;source) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;identity&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">T</span>&gt;&gt;&gt;(()&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">T</span>&gt;&gt;()); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;source.Aggregate(identity,&nbsp;(acc,&nbsp;x)&nbsp;=&gt;&nbsp;acc.FindFirst(x)); }</pre> </p> <p> This overload just uses the earlier <code>FindFirst</code> extension method to fold an arbitrary number of lazy <code>First&lt;T&gt;</code> objects into one. Notice that <code>Aggregate</code> is the C# name for the list catamorphisms. </p> <p> You can now compose the desired functionality using the basic building blocks of monoids, <a href="/2018/03/22/functors">functors</a>, and catamorphisms. </p> <h3 id="0fe80a69c74c463dacb8af0f86898518"> Composition from universal abstractions <a href="#0fe80a69c74c463dacb8af0f86898518" title="permalink">#</a> </h3> <p> The goal is still a function that takes a <code>User</code> object as input and produces an <code>Icon</code> object as output. While you could compose that functionality directly in-line where you need it, I think it may be helpful to package the composition in a <a href="https://en.wikipedia.org/wiki/Facade_pattern">Facade</a> object. </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">class</span>&nbsp;<span style="color:#2b91af;">IconReaderFacade</span> { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">IReadOnlyCollection</span>&lt;<span style="color:#2b91af;">IIconReader</span>&gt;&nbsp;readers; &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;IconReaderFacade(<span style="color:#2b91af;">IUserRepository</span>&nbsp;repository) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;readers&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">IIconReader</span>[] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">GravatarReader</span>(), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">IdenticonReader</span>(), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DBIconReader</span>(repository) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}; &nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Icon</span>&nbsp;ReadIcon(<span style="color:#2b91af;">User</span>&nbsp;user) &nbsp;&nbsp;&nbsp;&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;&gt;&gt;&gt;&nbsp;lazyIcons&nbsp;=&nbsp;readers &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;.Select(r&nbsp;=&gt; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;&gt;&gt;(()&nbsp;=&gt; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;r.ReadIcon(user).Select(i&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;(i)))); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">First</span>&lt;<span style="color:#2b91af;">Icon</span>&gt;&gt;&gt;&nbsp;m&nbsp;=&nbsp;lazyIcons.FindFirst(); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;m.Value.Aggregate(<span style="color:#2b91af;">Icon</span>.Default,&nbsp;fi&nbsp;=&gt;&nbsp;fi.Item); &nbsp;&nbsp;&nbsp;&nbsp;} }</pre> </p> <p> When you initialise an <code>IconReaderFacade</code> object, it creates an array of the desired <code>readers</code>. Whenever <code>ReadIcon</code> is invoked, it first transforms all those readers to a sequence of potential icons. All the values in the sequence are lazily evaluated, so in this step, nothing actually happens, even though it looks as though all readers' <code>ReadIcon</code> method gets called. The <code>Select</code> method is a structure-preserving map, so all readers are still potential producers of <code>Icon</code> objects. </p> <p> You now have an <code>IEnumerable&lt;Lazy&lt;Maybe&lt;First&lt;Icon&gt;&gt;&gt;&gt;</code>, which must be a good candidate for the prize for the <em>most nested generic .NET type of 2019</em>. It fits, though, the input type for the above <code>FindFirst</code> overload, so you can call that. The result is a single potential value <code>m</code>. That's the list catamorphism applied. </p> <p> Finally, you force evaluation of the lazy computation and apply the Maybe catamorphism (<code>Aggregate</code>). The <code>@default</code> value is <code>Icon.Default</code>, which gets returned if <code>m</code> turns out to be empty. When <code>m</code> is populated, you pull the <code>Item</code> out of the <code>First</code> object. In either case, you now have an <code>Icon</code> object to return. </p> <p> This composition has exactly the same behaviour as the initial Chain of Responsibility implementation, but is now composed from universal abstractions. </p> <h3 id="23819ca370344b94875ddbf5bde5aef3"> Summary <a href="#23819ca370344b94875ddbf5bde5aef3" title="permalink">#</a> </h3> <p> The Chain of Responsibility design pattern describes a flexible way to implement conditional logic. Instead of relying on keywords like <code>if</code> or <code>switch</code>, you can compose the conditional logic from polymorphic objects. This gives you several advantages. One is that you get better separations of concerns, which will tend to make it easier to refactor the code. Another is that it's possible to change the behaviour at run time, by moving the objects around. </p> <p> You can achieve a similar design, with equivalent advantages, by composing polymorphically similar functions in a list, map the functions to a list of potential values, and then use the list catamorphism to reduce many potential values to one. Finally, you apply the Maybe catamorphism to produce a value, even if the potential value is empty. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Tester-Doer isomorphisms https://blog.ploeh.dk/2019/07/15/tester-doer-isomorphisms 2019-07-15T07:35:00+00:00 Mark Seemann <div id="post"> <p> <em>The Tester-Doer pattern is equivalent to the Try-Parse idiom; both are equivalent to Maybe.</em> </p> <p> This article is part of <a href="/2018/01/08/software-design-isomorphisms">a series of articles about software design isomorphisms</a>. An isomorphism is when a bi-directional lossless translation exists between two representations. Such translations exist between the <em>Tester-Doer</em> pattern and the <em>Try-Parse</em> idiom. Both can also be translated into operations that return <a href="/2018/03/26/the-maybe-functor">Maybe</a>. </p> <p> <img src="/content/binary/tester-doer-try-parse-maybe-isomorphism.png" alt="Isomorphisms between Tester-Doer, Try-Parse, and Maybe."> </p> <p> Given an implementation that uses one of those three idioms or abstractions, you can translate your design into one of the other options. This doesn't imply that each is of equal value. When it comes to composability, Maybe is superior to the two other alternatives, and Tester-Doer isn't thread-safe. </p> <h3 id="e95c8f5d7a6445139b58445d30498493"> Tester-Doer <a href="#e95c8f5d7a6445139b58445d30498493" title="permalink">#</a> </h3> <p> The first time I explicitly encountered the Tester-Doer pattern was in the <a href="https://amzn.to/2zXCCfH">Framework Design Guidelines</a>, which is from where I've taken the name. The pattern is, however, older. The idea that you can query an object about whether a given operation would be possible, and then you only perform it if the answer is affirmative, is almost a leitmotif in <a href="http://amzn.to/1claOin">Object-Oriented Software Construction</a>. Bertrand Meyer often uses linked lists and stacks as examples, but I'll instead use the example that Krzysztof Cwalina and Brad Abrams use: </p> <p> <pre><span style="color:#2b91af;">ICollection</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;numbers&nbsp;=&nbsp;<span style="color:green;">//&nbsp;...</span> <span style="color:blue;">if</span>&nbsp;(!numbers.IsReadOnly) &nbsp;&nbsp;&nbsp;&nbsp;numbers.Add(1);</pre> </p> <p> The idea with the Tester-Doer pattern is that you test whether an intended operation is legal, and only perform it if the answer is affirmative. In the example, you only add to the <code>numbers</code> collection if <code>IsReadOnly</code> is <code>false</code>. Here, <code>IsReadOnly</code> is the <em>Tester</em>, and <code>Add</code> is the <em>Doer</em>. </p> <p> As Jeffrey Richter points out in the book, this is a dangerous pattern: <blockquote> "The potential problem occurs when you have multiple threads accessing the object at the same time. For example, one thread could execute the test method, which reports that all is OK, and before the doer method executes, another thread could change the object, causing the doer to fail." </blockquote> In other words, the pattern isn't thread-safe. While multi-threaded programming was always supported in .NET, this was less of a concern when the guidelines were first published (2006) than it is today. The guidelines were in internal use in Microsoft years before they were published, and there wasn't many multi-core processors in use back then. </p> <p> Another problem with the Tester-Doer pattern is with discoverability. If you're looking for a way to add an element to a collection, you'd usually consider your search over once you find the <code>Add</code> method. Even if you wonder <em>Is this operation safe? Can I always add an element to a collection?</em> you <em>might</em> consider looking for a <code>CanAdd</code> method, but not an <code>IsReadOnly</code> property. Most people don't even ask the question in the first place, though. </p> <h3 id="08bc9f42d8f048119f952aa9c2d94b34"> From Tester-Doer to Try-Parse <a href="#08bc9f42d8f048119f952aa9c2d94b34" title="permalink">#</a> </h3> <p> You could refactor such a Tester-Doer API to a single method, which is both thread-safe and discoverable. One option is a variation of the Try-Parse idiom (discussed in detail below). Using it could look like this: </p> <p> <pre><span style="color:#2b91af;">ICollection</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;numbers&nbsp;=&nbsp;<span style="color:green;">//&nbsp;...</span> <span style="color:blue;">bool</span>&nbsp;wasAdded&nbsp;=&nbsp;numbers.TryAdd(1);</pre> </p> <p> In this special case, you may not need the <code>wasAdded</code> variable, because the original <code>Add</code> operation never returned a value. If, on the other hand, you do care whether or not the element was added to the collection, you'd have to figure out what to do in the case where the return value is <code>true</code> and <code>false</code>, respectively. </p> <p> Compared to the more idiomatic example of the Try-Parse idiom below, you may have noticed that the <code>TryAdd</code> method shown here takes no <code>out</code> parameter. This is because the original <code>Add</code> method returns <code>void</code>; there's nothing to return. From <a href="/2018/01/15/unit-isomorphisms">unit isomorphisms</a>, however, we know that <em>unit</em> is isomorphic to <code>void</code>, so we could, more explicitly, have defined a <code>TryAdd</code> method with this signature: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;TryAdd(<span style="color:#2b91af;">T</span>&nbsp;item,&nbsp;<span style="color:blue;">out</span>&nbsp;<span style="color:#2b91af;">Unit</span>&nbsp;unit)</pre> </p> <p> There's no point in doing this, however, apart from demonstrating that the isomorphism holds. </p> <h3 id="e246bcfabcab42e8b76e2b3e314174c4"> From Tester-Doer to Maybe <a href="#e246bcfabcab42e8b76e2b3e314174c4" title="permalink">#</a> </h3> <p> You can also refactor the add-to-collection example to return a Maybe value, although in this degenerate case, it makes little sense. If you automate the refactoring process, you'd arrive at an API like this: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">Unit</span>&gt;&nbsp;TryAdd(<span style="color:#2b91af;">T</span>&nbsp;item)</pre> </p> <p> Using it would look like this: </p> <p> <pre><span style="color:#2b91af;">ICollection</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;numbers&nbsp;=&nbsp;<span style="color:green;">//&nbsp;...</span> <span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">Unit</span>&gt;&nbsp;m&nbsp;=&nbsp;numbers.TryAdd(1);</pre> </p> <p> The contract is consistent with what Maybe implies: You'd get an empty <code>Maybe&lt;Unit&gt;</code> object if the <em>add</em> operation 'failed', and a populated <code>Maybe&lt;Unit&gt;</code> object if the <em>add</em> operation succeeded. Even in the populated case, though, the value contained in the Maybe object would be <em>unit</em>, which carries no further information than its existence. </p> <p> To be clear, this isn't close to a proper functional design because all the interesting action happens as a side effect. Does the design have to be functional? No, it clearly isn't in this case, but Maybe is a concept that originated in functional programming, so you could be misled to believe that I'm trying to pass this particular design off as functional. It's not. </p> <p> A functional version of this API could look like this: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">ICollection</span>&lt;<span style="color:#2b91af;">T</span>&gt;&gt;&nbsp;TryAdd(<span style="color:#2b91af;">T</span>&nbsp;item)</pre> </p> <p> An implementation wouldn't mutate the object itself, but rather return a new collection with the added item, in case that was possible. This is, however, always possible, because you can always concatenate <code>item</code> to the front of the collection. In other words, this particular line of inquiry is increasingly veering into the territory of the absurd. This isn't, however, a counter-example of my proposition that the isomorphism exists; it's just a result of the initial example being degenerate. </p> <h3 id="9817f0d35d99428f93c38cab9fabc9ad"> Try-Parse <a href="#9817f0d35d99428f93c38cab9fabc9ad" title="permalink">#</a> </h3> <p> Another idiom described in the Framework Design Guidelines is the Try-Parse idiom. This seems to be a coding idiom more specific to the .NET framework, which is the reason I call it an <em>idiom</em> instead of a <em>pattern</em>. (Perhaps it is, after all, a pattern... I'm sure many of my readers are better informed about how problems like these are solved in other languages, and can enlighten me.) </p> <p> A better name might be <em>Try-Do</em>, since the idiom doesn't have to be constrained to parsing. The example that Cwalina and Abrams supply, however, relates to parsing a <code>string</code> into a <code>DateTime</code> value. Such an API is <a href="https://docs.microsoft.com/en-us/dotnet/api/system.datetime.tryparse">already available in the base class library</a>. Using it looks like this: </p> <p> <pre><span style="color:blue;">bool</span>&nbsp;couldParse&nbsp;=&nbsp;<span style="color:#2b91af;">DateTime</span>.TryParse(candidate,&nbsp;<span style="color:blue;">out</span>&nbsp;<span style="color:#2b91af;">DateTime</span>&nbsp;dateTime);</pre> </p> <p> Since <code>DateTime</code> is a <a href="https://docs.microsoft.com/en-us/dotnet/csharp/language-reference/keywords/value-types">value type</a>, the <code>out</code> parameter will never be <code>null</code>, even if parsing fails. You can, however, examine the return value <code>couldParse</code> to determine whether the <code>candidate</code> could be parsed. </p> <p> In the running commentary in the book, Jeffrey Richter likes this much better: <blockquote> "I like this guideline a lot. It solves the race-condition problem and the performance problem." </blockquote> I agree that it's better than Tester-Doer, but that doesn't mean that you can't refactor such a design to that pattern. </p> <h3 id="166ef01b6b64481a85fe64a6e9e07dc6"> From Try-Parse to Tester-Doer <a href="#166ef01b6b64481a85fe64a6e9e07dc6" title="permalink">#</a> </h3> <p> While I see no compelling reason to design parsing attempts with the Tester-Doer pattern, it's possible. You could create an API that enables interaction like this: </p> <p> <pre><span style="color:#2b91af;">DateTime</span>&nbsp;dateTime&nbsp;=&nbsp;<span style="color:blue;">default</span>(<span style="color:#2b91af;">DateTime</span>); <span style="color:blue;">bool</span>&nbsp;canParse&nbsp;=&nbsp;<span style="color:#2b91af;">DateTimeEnvy</span>.CanParse(candidate); <span style="color:blue;">if</span>&nbsp;(canParse) &nbsp;&nbsp;&nbsp;&nbsp;dateTime&nbsp;=&nbsp;<span style="color:#2b91af;">DateTime</span>.Parse(candidate);</pre> </p> <p> You'd need to add a new <code>CanParse</code> method with this signature: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;CanParse(<span style="color:blue;">string</span>&nbsp;candidate)</pre> </p> <p> In this particular example, you don't have to add a <code>Parse</code> method, because it already exists in the base class library, but in other examples, you'd have to add such a method as well. </p> <p> This example doesn't suffer from issues with thread safety, since strings are immutable, but in general, that problem is always a concern with the Tester-Doer <a href="/2019/01/21/some-thoughts-on-anti-patterns">anti-pattern</a>. Discoverability still suffers in this example. </p> <h3 id="ffd6284cfc8f4f528d1a3b80849fbf8c"> From Try-Parse to Maybe <a href="#ffd6284cfc8f4f528d1a3b80849fbf8c" title="permalink">#</a> </h3> <p> While the Try-Parse idiom is thread-safe, it isn't composable. Every time you run into an API modelled over this template, you have to stop what you're doing and check the return value. Did the operation succeed? Was should the code do if it didn't? </p> <p> <em>Maybe</em>, on the other hand, is composable, so is a much better way to model problems such as parsing. Typically, methods or functions that return Maybe values are still prefixed with <em>Try</em>, but there's no longer any <code>out</code> parameter. A Maybe-based <code>TryParse</code> function could look like this: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">DateTime</span>&gt;&nbsp;TryParse(<span style="color:blue;">string</span>&nbsp;candidate)</pre> </p> <p> You could use it like this: </p> <p> <pre><span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">DateTime</span>&gt;&nbsp;m&nbsp;=&nbsp;<span style="color:#2b91af;">DateTimeEnvy</span>.TryParse(candidate);</pre> </p> <p> If the <code>candidate</code> was successfully parsed, you get a populated <code>Maybe&lt;DateTime&gt;</code>; if the string was invalid, you get an empty <code>Maybe&lt;DateTime&gt;</code>. </p> <p> A Maybe object composes much better with other computations. Contrary to the Try-Parse idiom, you don't have to stop and examine a Boolean return value. You don't even have to deal with empty cases at the point where you parse. Instead, you can defer the decision about what to do in case of failure until a later time, where it may be more obvious what to do in that case. </p> <h3 id="4f27ce3476114a5f9b0f80fd415e5370"> Maybe <a href="#4f27ce3476114a5f9b0f80fd415e5370" title="permalink">#</a> </h3> <p> In my <a href="https://blog.ploeh.dk/encapsulation-and-solid">Encapsulation and SOLID</a> Pluralsight course, you get a walk-through of all three options for dealing with an operation that could potentially fail. Like in this article, the course starts with Tester-Doer, progresses over Try-Parse, and arrives at a Maybe-based implementation. In that course, the example involves reading a (previously stored) message from a text file. The final API looks like this: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&lt;<span style="color:blue;">string</span>&gt;&nbsp;Read(<span style="color:blue;">int</span>&nbsp;id)</pre> </p> <p> The protocol implied by such a signature is that you supply an ID, and if a message with that ID exists on disc, you receive a populated <code>Maybe&lt;string&gt;</code>; otherwise, an empty object. This is not only composable, but also thread-safe. For anyone who understands the <a href="/2017/10/04/from-design-patterns-to-category-theory">universal abstraction</a> of Maybe, it's clear that this is an operation that could fail. Ultimately, client code will have to deal with empty Maybe values, but this doesn't have to happen immediately. Such a decision can be deferred until a proper context exists for that purpose. </p> <h3 id="d35fbacb32bb4ef6afc843813ba901f1"> From Maybe to Tester-Doer <a href="#d35fbacb32bb4ef6afc843813ba901f1" title="permalink">#</a> </h3> <p> Since Tester-Doer is the least useful of the patterns discussed in this article, it makes little sense to refactor a Maybe-based API to a Tester-Doer implementation. Nonetheless, it's still possible. The API could look like this: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;Exists(<span style="color:blue;">int</span>&nbsp;id) <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">string</span>&nbsp;Read(<span style="color:blue;">int</span>&nbsp;id)</pre> </p> <p> Not only is this design not thread-safe, but it's another example of poor discoverability. While the doer is called <code>Read</code>, the tester isn't called <code>CanRead</code>, but rather <code>Exists</code>. If the class has other members, these could be listed interleaved between <code>Exists</code> and <code>Read</code>. It wouldn't be obvious that these two members were designed to be used together. </p> <p> Again, the intended usage is code like this: </p> <p> <pre><span style="color:blue;">string</span>&nbsp;message; <span style="color:blue;">if</span>&nbsp;(fileStore.Exists(49)) &nbsp;&nbsp;&nbsp;&nbsp;message&nbsp;=&nbsp;fileStore.Read(49);</pre> </p> <p> This is still problematic, because you need to decide what to do in the <code>else</code> case as well, although you don't see that case here. </p> <p> The point is, still, that you <em>can</em> translate from one representation to another without loss of information; not that you should. </p> <h3 id="3bbc92082af143d29681b2ce0bb11ccb"> From Maybe to Try-Parse <a href="#3bbc92082af143d29681b2ce0bb11ccb" title="permalink">#</a> </h3> <p> Of the three representations discussed in this article, I firmly believe that a Maybe-based API is superior. Unfortunately, the .NET base class library doesn't (yet) come with a built-in Maybe object, so if you're developing an API as part of a reusable library, you have two options: <ul> <li>Export the library's <code>Maybe&lt;T&gt;</code> type together with the methods that return it.</li> <li>Use Try-Parse for interoperability reasons.</li> </ul> This is the only reason I can think of to use the Try-Parse idiom. For the <code>FileStore</code> example from my Pluralsight course, this would imply not a <code>TryParse</code> method, but a <code>TryRead</code> method: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">bool</span>&nbsp;TryRead(<span style="color:blue;">int</span>&nbsp;id,&nbsp;<span style="color:blue;">out</span>&nbsp;<span style="color:blue;">string</span>&nbsp;message)</pre> </p> <p> This would enable you to expose the method in a reusable library. Client code could interact with it like this: </p> <p> <pre><span style="color:blue;">string</span>&nbsp;message; <span style="color:blue;">if</span>&nbsp;(!fileStore.TryRead(50,&nbsp;<span style="color:blue;">out</span>&nbsp;message)) &nbsp;&nbsp;&nbsp;&nbsp;message&nbsp;=&nbsp;<span style="color:#a31515;">&quot;&quot;</span>;</pre> </p> <p> This has all the problems associated with the Try-Parse idiom already discussed in this article, but it does, at least, have a basic use case. </p> <h3 id="c04073bcc534481eaaf1ba43dd2a22a4"> Isomorphism with Either <a href="#c04073bcc534481eaaf1ba43dd2a22a4" title="permalink">#</a> </h3> <p> At this point, I hope that you find it reasonable to believe that the three representations, Tester-Doer, Try-Parse, and Maybe, are isomorphic. You can translate between any of these representations to any other of these without loss of information. This also means that you can translate back again. </p> <p> While I've only argued with a series of examples, it's my experience that these three representations are truly isomorphic. You can always translate any of these representations into another. Mostly, though, I translate into Maybe. If you disagree with my proposition, all you have to do is to provide a counter-example. </p> <p> There's a fourth isomorphism that's already well-known, and that's between Maybe and <a href="/2018/06/11/church-encoded-either">Either</a>. Specifically, <code>Maybe&lt;T&gt;</code> is isomorphic to <code>Either&lt;Unit, T&gt;</code>. In <a href="https://www.haskell.org">Haskell</a>, this is easily demonstrated with this set of functions: </p> <p> <pre><span style="color:#2b91af;">toMaybe</span>&nbsp;::&nbsp;<span style="color:#2b91af;">Either</span>&nbsp;()&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&nbsp;a toMaybe&nbsp;(Left&nbsp;<span style="color:blue;">()</span>)&nbsp;=&nbsp;Nothing toMaybe&nbsp;(Right&nbsp;x)&nbsp;=&nbsp;Just&nbsp;x <span style="color:#2b91af;">fromMaybe</span>&nbsp;::&nbsp;<span style="color:#2b91af;">Maybe</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">Either</span>&nbsp;()&nbsp;a fromMaybe&nbsp;Nothing&nbsp;=&nbsp;Left&nbsp;<span style="color:blue;">()</span> fromMaybe&nbsp;(Just&nbsp;x)&nbsp;=&nbsp;Right&nbsp;x</pre> </p> <p> Translated to C#, using the <a href="/2018/06/04/church-encoded-maybe">Church-encoded Maybe</a> together with the Church-encoded Either, these two functions could look like the following, starting with the conversion from Maybe to Either: </p> <p> <pre><span style="color:green;">//&nbsp;On&nbsp;Maybe:</span> <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IEither</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;&nbsp;ToEither&lt;<span style="color:#2b91af;">T</span>&gt;(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IMaybe</span>&lt;<span style="color:#2b91af;">T</span>&gt;&nbsp;source) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;source.Match&lt;<span style="color:#2b91af;">IEither</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;&gt;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;nothing:&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Left</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;(<span style="color:#2b91af;">Unit</span>.Value), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;just:&nbsp;x&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Right</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;(x)); }</pre> </p> <p> Likewise, the conversion from Either to Maybe: </p> <p> <pre><span style="color:green;">//&nbsp;On&nbsp;Either:</span> <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IMaybe</span>&lt;<span style="color:#2b91af;">T</span>&gt;&nbsp;ToMaybe&lt;<span style="color:#2b91af;">T</span>&gt;(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IEither</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;&nbsp;source) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;source.Match&lt;<span style="color:#2b91af;">IMaybe</span>&lt;<span style="color:#2b91af;">T</span>&gt;&gt;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;onLeft:&nbsp;_&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Nothing</span>&lt;<span style="color:#2b91af;">T</span>&gt;(), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;onRight:&nbsp;x&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Just</span>&lt;<span style="color:#2b91af;">T</span>&gt;(x)); }</pre> </p> <p> You can convert back and forth to your heart's content, as this parametrised <a href="https://xunit.github.io">xUnit.net</a> 2.3.1 test shows: </p> <p> <pre>[<span style="color:#2b91af;">Theory</span>] [<span style="color:#2b91af;">InlineData</span>(42)] [<span style="color:#2b91af;">InlineData</span>(1337)] [<span style="color:#2b91af;">InlineData</span>(2112)] [<span style="color:#2b91af;">InlineData</span>(90125)] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">void</span>&nbsp;IsomorphicWithPopulatedMaybe(<span style="color:blue;">int</span>&nbsp;i) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;expected&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Right</span>&lt;<span style="color:#2b91af;">Unit</span>,&nbsp;<span style="color:blue;">int</span>&gt;(i); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;actual&nbsp;=&nbsp;expected.ToMaybe().ToEither(); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.Equal(expected,&nbsp;actual); }</pre> </p> <p> I decided to exclude <code>IEither&lt;Unit, T&gt;</code> from the overall theme of this article in order to better contrast three alternatives that may not otherwise look equivalent. That <code>IEither&lt;Unit, T&gt;</code> is isomorphic to <code>IMaybe&lt;T&gt;</code> is a well-known result. Besides, I think that both of these two representations already inhabit the same conceptual space. Either and Maybe are both well-known in statically typed functional programming. </p> <h3 id="8e3e7b55ac1e49568712675713426e59"> Summary <a href="#8e3e7b55ac1e49568712675713426e59" title="permalink">#</a> </h3> <p> The Tester-Doer pattern is a decades-old design pattern that attempts to model how to perform operations that can potentially fail, without relying on exceptions for flow control. It predates mainstream multi-core processors by decades, which can explain why it even exists as a pattern in the first place. At the time people arrived at the pattern, thread-safety wasn't a big concern. </p> <p> The Try-Parse idiom is a thread-safe alternative to the Tester-Doer pattern. It combines the two <em>tester</em> and <em>doer</em> methods into a single method with an <code>out</code> parameter. While thread-safe, it's not composable. </p> <p> <em>Maybe</em> offers the best of both worlds. It's both thread-safe and composable. It's also as discoverable as any Try-Parse method. </p> <p> These three alternatives are all, however, isomorphic. This means that you can refactor any of the three designs into one of the other designs, without loss of information. It also means that you can implement <a href="https://en.wikipedia.org/wiki/Adapter_pattern">Adapters</a> between particular implementations, should you so desire. You see this frequently in <a href="https://fsharp.org">F#</a> code, where functions that return <code>'a option</code> adapt Try-Parse methods from the .NET base class library. </p> <p> While all three designs are equivalent in the sense that you can translate one into another, it doesn't imply that they're equally useful. <em>Maybe</em> is the superior design, and Tester-Doer clearly inferior. </p> <p> <strong>Next:</strong> <a href="/2018/05/22/church-encoding">Church encoding</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Payment types catamorphism https://blog.ploeh.dk/2019/07/08/payment-types-catamorphism 2019-07-08T06:08:00+00:00 Mark Seemann <div id="post"> <p> <em>You can find the catamorphism for a custom sum type. Here's an example.</em> </p> <p> This article is part of an <a href="/2019/04/29/catamorphisms">article series about catamorphisms</a>. A catamorphism is a <a href="/2017/10/04/from-design-patterns-to-category-theory">universal abstraction</a> that describes how to digest a data structure into a potentially more compact value. </p> <p> This article presents the catamorphism for a domain-specific <a href="https://en.wikipedia.org/wiki/Tagged_union">sum type</a>, as well as how to identify it. The beginning of this article presents the catamorphism in C#, with a few examples. The rest of the article describes how to deduce the catamorphism. This part of the article presents my work in <a href="https://www.haskell.org">Haskell</a>. Readers not comfortable with Haskell can just read the first part, and consider the rest of the article as an optional appendix. </p> <p> In all previous articles in the series, you've seen catamorphisms for well-known data structures: <a href="/2019/05/06/boolean-catamorphism">Boolean values</a>, <a href="/2019/05/13/peano-catamorphism">Peano numbers</a>, <a href="/2019/05/20/maybe-catamorphism">Maybe</a>, <a href="/2019/06/10/tree-catamorphism">trees</a>, and so on. These are all general-purpose data structures, so you might be left with the impression that catamorphisms are only related to such general types. That's not the case. The point of this article is to demonstrate that you can find the catamorphism for your own custom, domain-specific sum type as well. </p> <h3 id="2b6f7df594c0474589ae9805f1e1a1d0"> C# catamorphism <a href="#2b6f7df594c0474589ae9805f1e1a1d0" title="permalink">#</a> </h3> <p> The custom type we'll examine in this article is the <a href="/2018/06/18/church-encoded-payment-types">Church-encoded payment types</a> I've previously written about. It's just an example of a custom data type, but it serves the purpose of illustration because I've already shown it as a Church encoding in C#, <a href="/2018/06/25/visitor-as-a-sum-type">as a Visitor in C#</a>, and <a href="/2016/11/28/easy-domain-modelling-with-types">as a discriminated union in F#</a>. </p> <p> The catamorphism for the <code>IPaymentType</code> interface is the <code>Match</code> method: </p> <p> <pre><span style="color:#2b91af;">T</span>&nbsp;Match&lt;<span style="color:#2b91af;">T</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">PaymentService</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;&nbsp;individual, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">PaymentService</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;&nbsp;parent, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">ChildPaymentService</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;&nbsp;child);</pre> </p> <p> As has turned out to be a common trait, the catamorphism is identical to the Church encoding. </p> <p> I'm not going to show more than a few examples of using the <code>Match</code> method, because you can find other examples in the previous articles, </p> <p> <pre>&gt; <span style="color:#2b91af;">IPaymentType</span>&nbsp;p&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Individual</span>(<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">PaymentService</span>(<span style="color:#a31515;">&quot;Visa&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;Pay&quot;</span>)); &gt; p.Match(ps&nbsp;=&gt;&nbsp;ps.Name,&nbsp;ps&nbsp;=&gt;&nbsp;ps.Name,&nbsp;cps&nbsp;=&gt;&nbsp;cps.PaymentService.Name) "Visa" &gt; <span style="color:#2b91af;">IPaymentType</span>&nbsp;p&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Parent</span>(<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">PaymentService</span>(<span style="color:#a31515;">&quot;Visa&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;Pay&quot;</span>)); &gt; p.Match(ps&nbsp;=&gt;&nbsp;ps.Name,&nbsp;ps&nbsp;=&gt;&nbsp;ps.Name,&nbsp;cps&nbsp;=&gt;&nbsp;cps.PaymentService.Name) "Visa" &gt; <span style="color:#2b91af;">IPaymentType</span>&nbsp;p&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Child</span>(<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ChildPaymentService</span>(<span style="color:#a31515;">&quot;1234&quot;</span>,&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">PaymentService</span>(<span style="color:#a31515;">&quot;Visa&quot;</span>,&nbsp;<span style="color:#a31515;">&quot;Pay&quot;</span>))); &gt; p.Match(ps&nbsp;=&gt;&nbsp;ps.Name,&nbsp;ps&nbsp;=&gt;&nbsp;ps.Name,&nbsp;cps&nbsp;=&gt;&nbsp;cps.PaymentService.Name) "Visa"</pre> </p> <p> These three examples from a <em>C# Interactive</em> session demonstrate that no matter which payment method you use, you can use the same <code>Match</code> method call to extract the payment name from the <code>p</code> object. </p> <h3 id="f2334a900eef421cb24c6e48a96e411b"> Payment types F-Algebra <a href="#f2334a900eef421cb24c6e48a96e411b" title="permalink">#</a> </h3> <p> As in the <a href="/2019/06/24/full-binary-tree-catamorphism">previous article</a>, I'll use <code>Fix</code> and <code>cata</code> as explained in <a href="https://bartoszmilewski.com">Bartosz Milewski</a>'s excellent <a href="https://bartoszmilewski.com/2017/02/28/f-algebras/">article on F-Algebras</a>. </p> <p> First, you'll have to define the auxiliary types involved in this API: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;PaymentService&nbsp;=&nbsp;PaymentService&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;paymentServiceName&nbsp;::&nbsp;String &nbsp;&nbsp;,&nbsp;paymentServiceAction&nbsp;::&nbsp;String &nbsp;&nbsp;}&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>) <span style="color:blue;">data</span>&nbsp;ChildPaymentService&nbsp;=&nbsp;ChildPaymentService&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;originalTransactionKey&nbsp;::&nbsp;String &nbsp;&nbsp;,&nbsp;parentPaymentService&nbsp;::&nbsp;PaymentService &nbsp;&nbsp;}&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> While F-Algebras and fixed points are mostly used for recursive data structures, you can also define an F-Algebra for a non-recursive data structure. You already saw examples of that in the articles about <a href="/2019/05/06/boolean-catamorphism">Boolean catamorphism</a>, <a href="/2019/05/20/maybe-catamorphism">Maybe catamorphism</a>, and <a href="/2019/06/03/either-catamorphism">Either catamorphism</a>. While each of the three payment types have associated data, none of it is parametrically polymorphic, so a single type argument for the carrier type suffices: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;PaymentTypeF&nbsp;c&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;IndividualF&nbsp;PaymentService &nbsp;&nbsp;|&nbsp;ParentF&nbsp;PaymentService &nbsp;&nbsp;|&nbsp;ChildF&nbsp;ChildPaymentService &nbsp;&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;<span style="color:blue;">PaymentTypeF</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;(IndividualF&nbsp;ps)&nbsp;=&nbsp;IndividualF&nbsp;ps &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(ParentF&nbsp;ps)&nbsp;=&nbsp;ParentF&nbsp;ps &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(ChildF&nbsp;cps)&nbsp;=&nbsp;ChildF&nbsp;cps</pre> </p> <p> I chose to call the carrier type <code>c</code> (for <em>carrier</em>). As was also the case with <code>BoolF</code>, <code>MaybeF</code>, and <code>EitherF</code>, the <code>Functor</code> instance ignores the map function because the carrier type is missing from all three cases. Like the <code>Functor</code> instances for <code>BoolF</code>, <code>MaybeF</code>, and <code>EitherF</code>, it'd seem that nothing happens, but at the type level, this is still a translation from <code>PaymentTypeF c</code> to <code>PaymentTypeF c1</code>. Not much of a function, perhaps, but definitely an <em>endofunctor</em>. </p> <p> Some helper functions make it a little easier to create <code>Fix PaymentTypeF</code> values, but there's really not much to them: </p> <p> <pre><span style="color:#2b91af;">individualF</span>&nbsp;::&nbsp;<span style="color:blue;">PaymentService</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">PaymentTypeF</span> individualF&nbsp;=&nbsp;Fix&nbsp;.&nbsp;IndividualF <span style="color:#2b91af;">parentF</span>&nbsp;::&nbsp;<span style="color:blue;">PaymentService</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">PaymentTypeF</span> parentF&nbsp;=&nbsp;Fix&nbsp;.&nbsp;ParentF <span style="color:#2b91af;">childF</span>&nbsp;::&nbsp;<span style="color:blue;">ChildPaymentService</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">PaymentTypeF</span> childF&nbsp;=&nbsp;Fix&nbsp;.&nbsp;ChildF</pre> </p> <p> That's all you need to identify the catamorphism. </p> <h3 id="da3c2c0fee2747bebb1db38c15110bcb"> Haskell catamorphism <a href="#da3c2c0fee2747bebb1db38c15110bcb" title="permalink">#</a> </h3> <p> At this point, you have two out of three elements of an F-Algebra. You have an endofunctor (<code>PaymentTypeF</code>), and an object <code>c</code>, but you still need to find a morphism <code>PaymentTypeF c -&gt; c</code>. </p> <p> As in the previous articles, start by writing a function that will become the catamorphism, based on <code>cata</code>: </p> <p> <pre>paymentF&nbsp;=&nbsp;cata&nbsp;alg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;(IndividualF&nbsp;ps)&nbsp;=&nbsp;<span style="color:blue;">undefined</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(ParentF&nbsp;ps)&nbsp;=&nbsp;<span style="color:blue;">undefined</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(ChildF&nbsp;cps)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> While this compiles, with its <code>undefined</code> implementations, it obviously doesn't do anything useful. I find, however, that it helps me think. How can you return a value of the type <code>c</code> from the <code>IndividualF</code> case? You could pass an argument to the <code>paymentF</code> function, but you shouldn't ignore the data <code>ps</code> contained in the case, so it has to be a function: </p> <p> <pre>paymentF&nbsp;fi&nbsp;=&nbsp;cata&nbsp;alg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;(IndividualF&nbsp;ps)&nbsp;=&nbsp;fi&nbsp;ps &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(ParentF&nbsp;ps)&nbsp;=&nbsp;<span style="color:blue;">undefined</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(ChildF&nbsp;cps)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> I chose to call the argument <code>fi</code>, for <em>function, individual</em>. You can pass a similar argument to deal with the <code>ParentF</code> case: </p> <p> <pre>paymentF&nbsp;fi&nbsp;fp&nbsp;=&nbsp;cata&nbsp;alg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;(IndividualF&nbsp;ps)&nbsp;=&nbsp;fi&nbsp;ps &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(ParentF&nbsp;ps)&nbsp;=&nbsp;fp&nbsp;ps &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(ChildF&nbsp;cps)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> And of course with the remaining <code>ChildF</code> case as well: </p> <p> <pre><span style="color:#2b91af;">paymentF</span>&nbsp;::&nbsp;(<span style="color:blue;">PaymentService</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span style="color:blue;">PaymentService</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span style="color:blue;">ChildPaymentService</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">Fix&nbsp;PaymentTypeF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c paymentF&nbsp;fi&nbsp;fp&nbsp;fc&nbsp;=&nbsp;cata&nbsp;alg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;(IndividualF&nbsp;ps)&nbsp;=&nbsp;fi&nbsp;ps &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(ParentF&nbsp;ps)&nbsp;=&nbsp;fp&nbsp;ps &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(ChildF&nbsp;cps)&nbsp;=&nbsp;fc&nbsp;cps</pre> </p> <p> This works. Since <code>cata</code> has the type <code>Functor f =&gt; (f a -&gt; a) -&gt; Fix f -&gt; a</code>, that means that <code>alg</code> has the type <code>f a -&gt; a</code>. In the case of <code>PaymentTypeF</code>, the compiler infers that the <code>alg</code> function has the type <code>PaymentTypeF c -&gt; c</code>, which is just what you need! </p> <p> You can now see what the carrier type <code>c</code> is for. It's the type that the algebra extracts, and thus the type that the catamorphism returns. </p> <p> This, then, is the catamorphism for the payment types. Except for the <a href="/2019/06/10/tree-catamorphism">tree catamorphism</a>, all catamorphisms so far have been pairs, but this one is a triplet of functions. This is because the sum type has three cases instead of two. </p> <p> As you've seen repeatedly, this isn't the only possible catamorphism, since you can, for example, trivially reorder the arguments to <code>paymentF</code>. The version shown here is, however, equivalent to the above C# <code>Match</code> method. </p> <h3 id="e6248a9ea34148c79c2b03acc92de5f7"> Usage <a href="#e6248a9ea34148c79c2b03acc92de5f7" title="permalink">#</a> </h3> <p> You can use the catamorphism as a basis for other functionality. If, for example, you want to convert a <code>Fix PaymentTypeF</code> value to JSON, you can first define an <a href="http://hackage.haskell.org/package/aeson/docs/Data-Aeson.html">Aeson</a> record type for that purpose: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;PaymentJson&nbsp;=&nbsp;PaymentJson&nbsp;{ &nbsp;&nbsp;&nbsp;&nbsp;name&nbsp;::&nbsp;String &nbsp;&nbsp;,&nbsp;action&nbsp;::&nbsp;String &nbsp;&nbsp;,&nbsp;startRecurrent&nbsp;::&nbsp;Bool &nbsp;&nbsp;,&nbsp;transactionKey&nbsp;::&nbsp;Maybe&nbsp;String &nbsp;&nbsp;}&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Generic</span>) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">ToJSON</span>&nbsp;<span style="color:blue;">PaymentJson</span></pre> </p> <p> Subsequently, you can use <code>paymentF</code> to implement a conversion from <code>Fix PaymentTypeF</code> to <code>PaymentJson</code>, as in the previous articles: </p> <p> <pre><span style="color:#2b91af;">toJson</span>&nbsp;::&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">PaymentTypeF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">PaymentJson</span> toJson&nbsp;= &nbsp;&nbsp;paymentF &nbsp;&nbsp;&nbsp;&nbsp;(\(PaymentService&nbsp;n&nbsp;a)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;-&gt;&nbsp;PaymentJson&nbsp;n&nbsp;a&nbsp;False&nbsp;Nothing) &nbsp;&nbsp;&nbsp;&nbsp;(\(PaymentService&nbsp;n&nbsp;a)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;-&gt;&nbsp;PaymentJson&nbsp;n&nbsp;a&nbsp;True&nbsp;Nothing) &nbsp;&nbsp;&nbsp;&nbsp;(\(ChildPaymentService&nbsp;k&nbsp;(PaymentService&nbsp;n&nbsp;a))&nbsp;-&gt;&nbsp;PaymentJson&nbsp;n&nbsp;a&nbsp;False&nbsp;$&nbsp;Just&nbsp;k)</pre> </p> <p> Testing it in GHCi, it works as it's supposed to: </p> <p> <pre>Prelude Data.Aeson B Payment&gt; B.putStrLn $ encode $ toJson $ parentF $ PaymentService "Visa" "Pay" {"transactionKey":null,"startRecurrent":true,"action":"Pay","name":"Visa"}</pre> </p> <p> Clearly, it would have been easier to define the payment types shown here as a regular Haskell sum type and just use standard pattern matching, but the purpose of this article isn't to present useful code; the only purpose of the code here is to demonstrate how to identify the catamorphism for a custom domain-specific sum type. </p> <h3 id="153479fffaf647f6ad6f5fc6a63fe025"> Summary <a href="#153479fffaf647f6ad6f5fc6a63fe025" title="permalink">#</a> </h3> <p> Even custom, domain-specific sum types have catamorphisms. This article presented the catamorphism for a custom payment sum type. Because this particular sum type has three cases, the catamorphism is a triplet, instead of a pair, which has otherwise been the most common shape of catamorphisms in previous articles. </p> <p> <strong>Next:</strong> <a href="/2018/03/05/some-design-patterns-as-universal-abstractions">Some design patterns as universal abstractions</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Yes silver bullet https://blog.ploeh.dk/2019/07/01/yes-silver-bullet 2019-07-01T07:38:00+00:00 Mark Seemann <div id="post"> <p> <em>Since Fred Brooks published his essay, I believe that we, contrary to his prediction, have witnessed several silver bullets.</em> </p> <p> I've been rereading <a href="https://en.wikipedia.org/wiki/Fred_Brooks">Fred Brooks</a>'s 1986 essay <a href="https://en.wikipedia.org/wiki/No_Silver_Bullet">No Silver Bullet</a> because I've become increasingly concerned that people seem to draw the wrong conclusions from it. <a href="https://martinfowler.com/bliki/SemanticDiffusion.html">Semantic diffusion</a> seems to have set in. These days, when people state something along the lines that there's <em>no silver bullet in software development</em>, I often get the impression that they mean that there's no panacea. </p> <p> Indeed; I agree. There's no miracle cure that will magically make all problems in software development go away. That's not what the essay states, however. It is, fortunately, more subtle than that. </p> <h3 id="712292e6c9c34663801dd40b4f278d3d"> No silver bullet reread <a href="#712292e6c9c34663801dd40b4f278d3d" title="permalink">#</a> </h3> <p> It's a great essay. It's not my intent to dispute the central argument of the essay, but I think that Brooks made one particular assumption that I disagree with. That doesn't make me smarter in any way. He wrote the essay in 1986. I'm writing this in 2019, with the benefit of the experience of all the years in-between. Hindsight is 20-20, so anyone could make the observations that I do here. </p> <p> Before we get to that, though, a brief summary of the essence of the essay is in order. In short, the conclusion is this: <blockquote> <p> "There is no single development, in either technology or management technique, which by itself promises even one order-of-magnitude improvement within a decade in productivity, in reliability, in simplicity." </p> <footer><cite>Fred Brooks, <em>No Silver Bullet</em>, 1986</cite></footer> </blockquote> The beginning of the essay is a brilliant analysis of the reasons why software development is inherently difficult. If you read this together with Jack Reeves <em>What Is Software Design?</em> (available various places on the internet, or as an appendix in <a href="http://amzn.to/19W4JHk">APPP</a>), you'll probably agree that there's an inherent complexity to software development that no invention is likely to dispel. </p> <p> Ostensibly in the tradition of <a href="https://en.wikipedia.org/wiki/Aristotle">Aristotle</a>, Brooks distinguishes between <em>essential</em> and <em>accidental</em> complexity. This distinction is central to his argument, so it's worth discussing for a minute. </p> <p> Software development problems are complex, i.e. made up of many interacting sub-problems. Some of that complexity is <em>accidental</em>. This doesn't imply randomness or sloppiness, but only that the complexity isn't inherent to the problem; that it's only the result of our (human) failure to achieve perfection. </p> <p> If you imagine that you could whittle away all the accidental complexity, you'd ultimately reach a point where, in the words of Saint Exupéry, <em>there is nothing more to remove</em>. What's left is the <em>essential</em> complexity. </p> <p> Brooks' conjecture is that a typical software development project comes with both essential and accidental complexity. In his 1995 reflections <em>"No Silver Bullet" Refired</em> (available in <a href="http://bit.ly/mythical-man-month">The Mythical Man-Month</a>), he clarifies what he already implied in 1986: <blockquote> <p> "It is my opinion, and that is all, that the accidental or representational part of the work is now down to about half or less of the total." </p> <footer><cite>Fred Brooks, <em>"No Silver Bullet" Refired</em>, 1995</cite></footer> </blockquote> This I fundamentally disagree with, but more on that later. It makes sense to me to graphically represent the argument like this: </p> <p> <img src="/content/binary/essential-accidental-complexity-shells-brooks-scenario.png" alt="Some, but not much, accidental complexity as a shell around essential complexity."> </p> <p> The way that I think of Brooks' argument is that any software project contains some essential and some accidental complexity. For a given project, the size of the essential complexity is fixed. </p> <p> Brooks believes that less than half of the overall complexity is accidental: </p> <p> <img src="/content/binary/essential-accidental-complexity-pie-chart-brooks-scenario.png" alt="Essential and accidental complexity pie chart."> </p> <p> While a pie chart better illustrates the supposed ratio between the two types of complexity, I prefer to view Brooks' arguments as the first diagram, above. In that visualisation, the essential complexity is a core of fixed size, while accidental complexity is something you can work at removing. If you keep improving your process and technology, you may, conceptually, be able to remove (almost) all of it. </p> <p> <img src="/content/binary/essential-almost-no-accidental-complexity-shells.png" alt="Essential complexity with a very thin shell of accidental complexity."> </p> <p> Brooks' point, with which I agree, is that if the essential complexity is inherent, then you can't reduce the size of it. The only way to decrease the overall complexity is to reduce the accidental complexity. </p> <p> If you agree with the assessment that less than half of the overall complexity in modern software development is accidental, then it follows that no dramatic improvements are available. Even if you remove all accidental complexity, you've only reduced overall complexity by, say, forty percent. </p> <h3 id="d8e6f84d104b4ff6ad6b5473e46a4e30"> Accidental complexity abounds <a href="#d8e6f84d104b4ff6ad6b5473e46a4e30" title="permalink">#</a> </h3> <p> I find Brooks' arguments compelling. I do not, however, accept the premise that there's only little accidental complexity left. Instead of the above diagrams, I believe that the situation looks more like this (not to scale): </p> <p> <img src="/content/binary/accidental-complexity-with-tiny-core-of-essential-complexity.png" alt="Accidental complexity with a tiny core of essential complexity."> </p> <p> I think that most of the complexity in software development is accidental. I'm not sure about today, but I believe that I have compelling evidence that this was the case in 1986, so I don't see why it shouldn't still be the case. </p> <p> To be clear, this is all anecdotal, since I don't believe that software development is quantifiable. In the essay, Brooks explicitly talks about the <em>invisibility</em> of software. Software is pure <em>thought stuff;</em> you can't measure it. I discuss this in my <a href="https://cleancoders.com/episode/humane-code-real-episode-1/show">Humane Code video</a>, but I also recommend that you read <a href="http://bit.ly/leprechauns-of-software-engineering">The Leprechauns of Software Engineering</a> if you have any illusions that we, as an industry, have any reliable measurements of productivity. </p> <p> Brooks predicts that, within the decade (from 1986 to 1996), there would be no single development that would increase productivity with an order of magnitude, i.e. by a factor of at least ten. Ironically, when he wrote <em>"No Silver Bullet" Refired</em> in 1995, at least two such developments were already in motion. </p> <p> We can't blame Brooks for not identifying those developments, because in 1995, their impact was not yet apparent. Again, hindsight is 20-20. </p> <p> Neither of these two developments are purely technological, although technology plays a role. Notice, though, that Brooks' prediction included <em>technology or management technique</em>. It's in the interaction between technology and the humane that the orders-of-magnitude developments emerged. </p> <h3 id="1d23f6fb89884b6d9833ce09d68a3b0f"> World Wide Web <a href="#1d23f6fb89884b6d9833ce09d68a3b0f" title="permalink">#</a> </h3> <p> I have a dirty little secret. In the beginning of my programming career, I became quite the expert on a programming framework called <a href="https://en.wikipedia.org/wiki/Microsoft_Commerce_Server">Microsoft Commerce Server</a>. In fact, I co-authored a chapter of <a href="https://amzn.to/2CpE4rr">Professional Commerce Server 2000 Programming</a>, and in 2003 I received an <a href="https://mvp.microsoft.com">MVP</a> award as an acknowledgement of my work in the Commerce Server community (such as it were; it was mostly on <a href="https://en.wikipedia.org/wiki/Usenet">Usenet</a>). </p> <p> The Commerce Server framework was a black box. This was long before Microsoft embraced open source, and while there was a bit of official documentation, it was superficial; it was mostly of the <em>getting-started</em> kind. </p> <p> Over several years, I managed to figure out how the framework really worked, and thus, how one could extend it. This was a painstaking process. Since it was a black box, I couldn't just go and read the code to figure out how it worked. The framework was written in C++ and Visual Basic, so there wasn't even IL code to decompile. </p> <p> I had one window into the framework. It relied on SQL Server, and I could attach the profiler tool to spy on its interaction with the database. Painstakingly, over several years, I managed to wrest the framework's secrets from it. </p> <p> I wasted much time doing detective work like that. </p> <p> In general, programming in the late nineties and early two-thousands was less productive, not because the languages or tools were orders-of-magnitude worse than today, but because when you hit a snag, you were in trouble. </p> <p> These days, if you run into a problem beyond your abilities, you can ask for help on the World Wide Web. Usually, you'll find an existing answer on <a href="https://stackoverflow.com">Stack Overflow</a>, and you'll be able to proceed without too much delay. </p> <p> Compared to twenty years ago, I believe that the World Wide Web has increased my productivity more than ten-fold. While it also existed in 1995, there wasn't much content. It's not the technology itself that provides the productivity increase, but rather the synergy of technology and human knowledge. </p> <p> I think that Brooks vastly underestimated how much time one can waste when one is stuck. That's a sort of accidental complexity, although in the development process rather than in the technology itself. </p> <h3 id="a3b19483cd6a4c509d8c3a77fe324872"> Automated testing <a href="#a3b19483cd6a4c509d8c3a77fe324872" title="permalink">#</a> </h3> <p> In the late nineties, I was developing web sites (with Commerce Server). When I wanted to run my code to see if it worked, I'd launch the web site on my laptop, log in, click around and enter data until I was convinced that the functionality was working as it should. Most of the time, however, it wasn't, so I'd change a bit of the code, and go through the same process again. </p> <p> I think that's a common way to 'test' software; at least, it was back then. </p> <p> While you could get good at going through these motions quickly, verifying a single, or a handful of related functionalities, could easily take at least a couple of seconds, and usually more like half a minute. </p> <p> If you had dozens, or even hundreds, of different scenarios to address, you obviously wouldn't run through them all every time you changed the code. At the very best, you'd click your way through three of four usage scenarios that you thought were relevant to the change you'd made. Other functionality, earlier declared <em>done</em>, you just considered to be unaffected. </p> <p> Needless to say, regressions were regular occurrences. </p> <p> In 2003 I discovered test-driven development, and through that, automated testing. While you can't directly compare unit tests with whole usage scenarios, I think it's fair to compare something like automated integration tests or user-scenario tests (whatever you want to call them) with manually clicking through an application. </p> <p> Even an integration test, if written properly, can verify a scenario <em>at least</em> ten times faster than you can do it by hand. A more realistic estimate is probably hundred times faster, or more. </p> <p> Granted, you have to write the automated test as well, and I know that it's not always trivial. Still, once you have an automated test suite in place, you can run it all the time. </p> <p> I never ran through <em>all</em> usage scenarios when I manually 'tested' my software. With automated tests, I do. This saves me from most regressions. </p> <p> This improvement is, in my opinion, a no-brainer. It's easily a factor ten improvement. All the time wasted manually 'testing' the software, plus the time wasted fixing regressions, can be put to better use. </p> <p> At the time Brooks was writing his own retrospective (in 1995), Kent Beck was beginning to talk to other people about test-driven development. As is a common theme in this article, hindsight is 20-20. </p> <h3 id="c7ca9269cce04b3ab934c97bc8cf0328"> Honourable mentions <a href="#c7ca9269cce04b3ab934c97bc8cf0328" title="permalink">#</a> </h3> <p> There's been other improvements in software development since 1986. I considered including several other improvements as bona fide orders-of-magnitude improvements, but I think that's probably going too far. Each of the following developments have, however, offered significant improvements: <ul> <li> <strong>Git.</strong> It's surprising how much more productive Git can make you. While it's somewhat better than centralised source control systems at the functionality also available with those other systems, the productivity increase comes from all the new, unanticipated workflows it enables. Before I started using DVCS, I'd have lots of code that was commented out, so that I could experiment with various alternatives. With Git, I just create a new branch, or stash my changes, and experiment with abandon. While it's probably not a ten-fold increase in productivity, I believe it's the simplest technology change you can make to dramatically increase your productivity. </li> <li> <strong>Garbage collection.</strong> Since I've admitted that I worked with Microsoft Commerce Server, I've probably lost all credibility with my reader already, but let's see if I can win back a little. While Commerce Server programming involved <a href="https://en.wikipedia.org/wiki/VBScript">VBScript</a> programming, it also often involved <a href="https://en.wikipedia.org/wiki/Component_Object_Model">COM</a> programming, and I did quite a bit of that in C++. Having to make sure that you've cleaned up all memory after use is a bother. Garbage collection just makes this work go away. It's hardly a ten-fold improvement in productivity, but I do find it significant. </li> <li> <strong>Agile software development.</strong> The methodology of decreasing the feedback time between implementation and deployment has made me much more productive. I'm not interested in peddling any particular methodology like Scrum as much as just the general concept of getting rapid feedback. Particularly if you combine continuous delivery with Git, you have a powerful combination. Brooks already talked about incremental software development, and had some hopes attached to this as well. My personal experience can only agree with his sentiment. Again, probably not in itself a ten-fold increase in productivity, but enough that I wouldn't want to work on a project where rapid feedback and incremental development wasn't valued. </li> </ul> I'm probably forgetting lots of other improvements that have happened in the last decades. That's fine. The purpose of this article isn't to produce an exhaustive list, but rather to make the argument that significant improvements have been made since Brooks wrote his essay. I think it'd be folly, then, to believe that we've seen the last of such improvements. </p> <p> Personally, I'm inclined to believe another order-of-magnitude improvement is right at our feet. </p> <h3 id="bd2d47d8dac2401e936ca7902bc9109d"> Statically typed functional programming <a href="#bd2d47d8dac2401e936ca7902bc9109d" title="permalink">#</a> </h3> <p> This section is conjecture on my part. The improvements I've so far covered are already realised (at least for those who choose to take advantage of them). The improvement I'll cover here is more speculative. </p> <p> I believe that statically typed functional programming offers another order-of-magnitude improvement over existing software development. Twenty years ago, I believed that object-oriented programming was a good idea. I now believe that I was wrong about that, so it's possible that in another twenty years, I'll also believe that I was wrong about functional programming. Take the following for what it is. </p> <p> When I carefully reread <em>No Silver Bullet</em>, I got the distinct impression that Brooks considered low-level details of programming part of its essential complexity: <blockquote> <p> "Much of the complexity in a software construct is, however, not due to conformity to the external world but rather to the implementation itself - its data structures, its algorithms, its connectivity." </p> <footer><cite>Fred Brooks, <em>"No Silver Bullet" Refired</em>, 1995</cite></footer> </blockquote> It's unreasonable to blame anyone writing in 1986, or 1995 for that matter, to think that <code>for</code> loops, variables, program state, and such other programming stables were anything but essential parts of the complexity of developing software. </p> <p> Someone, unfortunately I forget who, once made the point that all mainstream programming languages are layers of abstractions of how a CPU works. Assembly language is basically just mnemonics on top of a CPU instruction set, then C can be thought of as an abstraction over assembly language, C++ as the next step in abstraction, Java and C# as sort of abstractions of C++, and so on. The origin of the design is the physical CPU. You could say that these languages are designed in a bottom-up fashion. </p> <p> <img src="/content/binary/imperative-bottom-up-functional-top-down.png" alt="Imperative languages depicted as designed bottom-up, and functional languages as designed top-down."> </p> <p> Some functional languages (perhaps most famously <a href="https://www.haskell.org">Haskell</a>, but also <a href="https://en.wikipedia.org/wiki/APL_(programming_language)">APL</a>, and, possibly, <a href="https://en.wikipedia.org/wiki/Lisp_(programming_language)">Lisp</a>) are designed in a much more top-down fashion. You start with mathematical abstractions like <a href="https://en.wikipedia.org/wiki/Category_theory">category theory</a> and then figure out how to crystallise the theory into a programming language, and then again, via more layers of abstractions, how to turn the abstract language into machine code. </p> <p> The more you learn about the <a href="https://en.wikipedia.org/wiki/Pure_function">pure</a> functional alternative to programming, the more you begin to see mutable program state, variables, <code>for</code> loops, and similar language constructs merely as artefacts of the underlying model. Brooks, I think, thought of these as part of the essential complexity of programming. I don't think that that's the case. You can get by just fine with other abstractions instead. </p> <p> Besides, Brooks writes, under the heading of <em>Complexity:</em> <blockquote> <p> "From the complexity comes the difficulty of enumerating, much less understanding, all the possible states of the program, and from that comes the unreliability. From the complexity of the functions comes the difficulty of invoking those functions, which makes programs hard to use." </p> <footer><cite>Fred Brooks, <em>No Silver Bullet</em>, 1986</cite></footer> </blockquote> When he writes <em>functions</em>, I don't think that he means functions in the Haskell sense. I think that he means <em>operations</em>, <em>procedures</em>, or <em>methods</em>. </p> <p> Indeed, when you look at a C# method signature like the following, it's hard to enumerate, understand, or remember, all that it does: </p> <p> <pre><span style="color:blue;">int</span>?&nbsp;TryAccept(<span style="color:#2b91af;">Reservation</span>&nbsp;reservation);</pre> </p> <p> If this is a high-level function, many things could happen when you call that method. It could change the state of a database. It could send an email. It could mutate a variable. Not only that, but the behaviour could depend on non-deterministic factors, such as the date, time of day, or just raw randomness. Finally, how should you handle the return value? What does it mean if the return value is <em>null</em>? What if it's not? Is <code>0</code> a valid value? Are negative numbers valid? Are they different from positive values? </p> <p> It is, indeed, difficult to enumerate all the possible states of such a function. </p> <p> Consider, instead, a Haskell function with a type like this: </p> <p> <pre><span style="color:#2b91af;">tryAccept</span>&nbsp;::&nbsp;<span style="color:#2b91af;">Int</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Reservation</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">MaybeT</span>&nbsp;<span style="color:blue;">ReservationsProgram</span>&nbsp;<span style="color:#2b91af;">Int</span></pre> </p> <p> What happens if you invoke this function? It returns a value. Does it send any emails? Does it mutate any state? No, it can't, because the static type informs us that this is a pure function. If any programmer, anywhere inside of the function, or the functions it calls, or functions they call, etc. tried to do something impure, it wouldn't have compiled. </p> <p> Can we enumerate the states of the program? Certainly. We just have to figure out what <code>ReservationsProgram</code> is. After following a few types, we find this statically typed enumeration: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;ReservationsInstruction&nbsp;next&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;IsReservationInFuture&nbsp;Reservation&nbsp;(Bool&nbsp;-&gt;&nbsp;next) &nbsp;&nbsp;|&nbsp;ReadReservations&nbsp;UTCTime&nbsp;([Reservation]&nbsp;-&gt;&nbsp;next) &nbsp;&nbsp;|&nbsp;Create&nbsp;Reservation&nbsp;(Int&nbsp;-&gt;&nbsp;next) &nbsp;&nbsp;<span style="color:blue;">deriving</span>&nbsp;Functor</pre> </p> <p> Essentially, there's three 'actions' that this type enables. The <code>tryAccept</code> function returns the <code>ReservationsProgram</code> inside of a <code>MaybeT</code> container, so there's a fourth option that something short-circuits along the way. </p> <p> You don't even have to keep track of this yourself. The compiler keeps you honest. Whenever you invoke the <code>tryAccept</code> function, the compiler will insist that you write code that can handle all possible outcomes. If you turn on the right compiler flags, the code is not going to compile if you don't. </p> <p> (Both code examples are taken from <a href="https://github.com/ploeh/dependency-injection-revisited">the same repository</a>.) </p> <p> Haskellers jokingly declare that <em>if Haskell code compiles, it works</em>. While humorous, there's a kernel of truth in that. An advanced type system can carry much information about the behaviour of a program. Some people, particularly programmers who come from a dynamically typed background, find Haskell's type system rigid. That's not an unreasonable criticism, but often, in dynamically typed languages, you have to write many automated tests to ensure that your program behaves as desired, and that it correctly handles various edge cases. A type system like Haskell's, on the other hand, embeds those rules in types instead of in tests. </p> <p> While you should still write automated tests for Haskell programs, fewer are needed. How many fewer? Compared to C-based languages, a factor ten isn't an unreasonable guess. </p> <p> After a few false starts, in 2014 I finally decided that <a href="https://fsharp.org">F#</a> would be my default choice of language on .NET. The reason for that decision was that I felt so much more productive in F# compared to C#. While F#'s type system doesn't embed information about pure versus impure functions, it does support <a href="https://en.wikipedia.org/wiki/Tagged_union">sum types</a>, which is what enables the sort of compile-time <em>enumeration</em> that Brooks discusses. </p> <p> F# is still my .NET language of choice, but I find that I mostly 'think in' Haskell these days. My conjecture is that a sufficiently advanced type system (like Haskell's) could easily represent another order-of-magnitude improvement over mainstream imperative languages. </p> <h3 id="a75ae35933314755b1a0cdb665262bc5"> Improvements for those who want them <a href="#a75ae35933314755b1a0cdb665262bc5" title="permalink">#</a> </h3> <p> The essay <em>No Silver Bullet</em> is a perspicacious work. I think more people should read at least the first part, where Brooks explains why software development is hard. I find that analysis brilliant, and I agree: software development presupposes essential complexity. It's inherently hard. </p> <p> There's no reason to make it harder than it has to be, though. </p> <p> More than once, I've discussed productivity improvements with people, only to be met with the dismissal that 'there's no silver bullet'. </p> <p> Granted, there's no magical solution that will solve all problems with software development, but that doesn't mean that improvements can't be had. </p> <p> Consider the improvements I've argued for here. Everyone now uses the World Wide Web and sites like Stack Overflow for research; that particular improvement is firmly embedded in all organisations. On the other hand, I still regularly talk to organisations that don't routinely use automated testing. </p> <p> People still use centralised version control (like TFS or SVN). If there was ever a low-hanging fruit, changing to Git is one. Git is <em>free</em>, and there's plenty of tools you can use to migrate your version history to it. There's also plenty of training and help to be had. Yes, it'll require a small investment to make the change, but the productivity increase is significant. <blockquote> <p> "The future is already here — it's just not very evenly distributed." </p> <footer><cite>William Gibson</cite></footer> </blockquote> So it is with technology improvements. Automated testing is available, but not ubiquitous. Git is free, but still organisations stick to suboptimal version control. Haskell and F# are mature languages, yet programmers still program in C# or Java. </p> <h3 id="864e39a22bc84129bfecaafe33dd1757"> Summary <a href="#864e39a22bc84129bfecaafe33dd1757" title="permalink">#</a> </h3> <p> The essay <em>No Silver Bullet</em> was written in 1986, but seems to me to be increasingly misunderstood. When people today talk about it at all, it's mostly as an excuse to stay where they are. "There's no silver bullets," they'll say. </p> <p> The essay, however, doesn't argue that no improvements can be had. It only argues that no more order-of-magnitude improvements can be had. </p> <p> In the present essay I argue that, since Brooks wrote <em>No Silver Bullet</em>, more than one such improvement happened. Once the World Wide Web truly began furnishing <em>information at your fingertips</em>, you could be more productive because you wouldn't be <em>stuck</em> for days or weeks. Automated testing reduces the work that manual testers used to perform, as well as limiting regressions. </p> <p> If you accept my argument, that order-of-magnitude improvements appeared after 1986, this implies that Brooks' premise was wrong. In that case, there's no reason to believe that we've seen the last significant improvement to software development. </p> <p> I think that more such improvements await us. I suggest that statically typed functional programming offers such an advance, but if history teaches us anything, it seems that breakthroughs tend to be unpredictable. </p> </div> <div id="comments"> <hr> <h2 id="comments-header"> Comments </h2> <div class="comment" id="7e7e932f5eea47f3bab328c58e9d164a"> <div class="comment-author"><a href="http://blog.strobaek.org">Karsten Strøbæk</a></div> <div class="comment-content"> <p> As always I enjoy reading your blog, even though I don't understand half of it most of the time. Or is that most of it half of the time? Allow me to put a few observations forward. </p> <p> First I should confess, that I have actually not read the whole of Brook's essay. When I initially tried I got about half way through; it sounds like I should make another go at it. That of course will not stop me from commenting on the above. </p> <p> Brook talks about complexity. To me designing and implementing a software system is not complex. Quantum physics is complex. Flying an airplane is difficult. Software development may be difficult depending on the task at hand (and unfortunately the qualifications of the team), but I would argue that it at most falls into the same category as flying an airplane. </p> <p> I would properly also state, that there are no silver bullets. But like you I feel that people understand it incorrectly and there is definetely no reason for making things harder than they are. I think the examples of technology that helps are excellent and exactly describe that things do move forward. </p> <p> That being said, it does not take away the creativity of the right decomposition, the responsibility for getting the use cases right, and especially the liability for getting it wrong. Sadly especially the last of overlooked. People should be reminded of where the phrase 'live under the bridge' comes from. </p> <p> To end my ramblins, I would also look a little into the future. As you know I am somewhat sceptial about machine learning and AI. However, looking at the recent break throughs and use cases in these areas, I would not be surprised of a future where software development is done by 'an AI' assemblying pre-defined 'entities' to create the software we need. Like an F16 cannot be flown without a computer, future software cannot be created by a human. </p> </div> <div class="comment-date">2019-07-04 18:29:00 UTC</div> </div> <div class="comment" id="756066e5cb0e42368ff9eeb9569fa47f"> <div class="comment-author"><a href="/">Mark Seemann</a></div> <div class="comment-content"> <p> Karsten, thank you for writing. I'm not inclined to agree that software development falls into the same category of complexity as flying a plane. It seems to me to be orders of magnitudes more complex. </p> <p> Just look at error rates. </p> <p> Would you ever board an air plane if flying had error rates similar to those observed in software development? Would you fly even if only one percent of all flights ended with plane crash? </p> <p> In reality, flying is extremely safe. Would you claim that software development is as safe, predictable, and manageable as flying? </p> <p> I see no evidence of that. </p> <p> Are pilots significantly more capable human beings than software developers, or does something else explain the discrepancy in failure rates? </p> </div> <div class="comment-date">2019-07-05 15:47 UTC</div> </div> <div class="comment" id="7e7e932f5eea47f3bab328c58e9d164b"> <div class="comment-author"><a href="http://blog.strobaek.org">Karsten Strøbæk</a></div> <div class="comment-content"> <p> Hi Mark. The fact that error rates are higher in software development is more a statement to the bad state our industry is in and has been for a milinium or more. </p> <p> Why do we except that we produce crappy systems or in your words software that is no safe, predictable, and manageble? The list of excuses is very long and the list of results is very short. We as an industry are simply doing it wrong, but most people prefers hand waving and marketing than simple and plausible heuristic. </p> <p> To use your analogy about planes I could ask if you would fly with a place that had (only) been unit tested? Properly not as it is never the unit that fails, but always the integration. Should be test all integrations then? Yes, why not? </p> <p> The used of planes or pilots (or whatever) may have been bad. My point was, the I do not see software development as complex. </p> </div> <div class="comment-date">2019-07-05 20:12 UTC</div> </div> <div class="comment" id="0df7412992fb499d915e6f4cdbb644a0"> <div class="comment-author"><a href="/">Mark Seemann</a></div> <div class="comment-content"> <p> Karsten, if we, as an industry, are doing it wrong, then why are we doing that? </p> <p> And what should we be doing instead? </p> </div> <div class="comment-date">2019-07-06 16:00 UTC</div> </div> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Full binary tree catamorphism https://blog.ploeh.dk/2019/06/24/full-binary-tree-catamorphism 2019-06-24T06:00:00+00:00 Mark Seemann <div id="post"> <p> <em>The catamorphism for a full binary tree is a pair of functions.</em> </p> <p> This article is part of an <a href="/2019/04/29/catamorphisms">article series about catamorphisms</a>. A catamorphism is a <a href="/2017/10/04/from-design-patterns-to-category-theory">universal abstraction</a> that describes how to digest a data structure into a potentially more compact value. </p> <p> This article presents the catamorphism for a full <a href="https://en.wikipedia.org/wiki/Binary_tree">binary tree</a>, as well as how to identify it. The beginning of this article presents the catamorphism in C#, with examples. The rest of the article describes how to deduce the catamorphism. This part of the article presents my work in <a href="https://www.haskell.org">Haskell</a>. Readers not comfortable with Haskell can just read the first part, and consider the rest of the article as an optional appendix. </p> <p> A <em>full binary tree</em> (also known as a <em>proper</em> or <em>plane</em> binary tree) is a tree in which each node has either two or no branches. </p> <p> <img src="/content/binary/full-binary-tree-example.png" alt="A full binary tree example diagram, with each node containing integers."> </p> <p> The diagram shows an example of a tree of integers. The left branch contains two children, of which the right branch again contains two sub-branches. The rest of the nodes are leaf-nodes with no sub-branches. </p> <h3 id="d6b9699fa3894a4383f9b2b2992a9e8f"> C# catamorphism <a href="#d6b9699fa3894a4383f9b2b2992a9e8f" title="permalink">#</a> </h3> <p> As a C# representation of a full binary tree, I'll start with the <code>IBinaryTree&lt;T&gt;</code> API from <a href="/2018/08/13/a-visitor-functor">A Visitor functor</a>. The catamorphism is the <code>Accept</code> method: </p> <p> <pre><span style="color:#2b91af;">TResult</span>&nbsp;Accept&lt;<span style="color:#2b91af;">TResult</span>&gt;(<span style="color:#2b91af;">IBinaryTreeVisitor</span>&lt;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;visitor);</pre> </p> <p> So far in this article series, you've mostly seen <a href="/2018/05/22/church-encoding">Church-encoded</a> catamorphisms, so a catamorphism represented as a <a href="https://en.wikipedia.org/wiki/Visitor_pattern">Visitor</a> may be too big of a cognitive leap. We know, however, from <a href="/2018/06/25/visitor-as-a-sum-type">Visitor as a sum type</a> that a Visitor representation is isomorphic to a Church encoding. Since these are isomorphic, it's possible to refactor <code>IBinaryTree&lt;T&gt;</code> to a Church encoding. The <a href="https://github.com/ploeh/ChurchEncoding">GitHub repository</a> contains a series of commits that demonstrates how that refactoring works. Once you're done, you arrive at this <code>Match</code> method, which is the refactored <code>Accept</code> method: </p> <p> <pre><span style="color:#2b91af;">TResult</span>&nbsp;Match&lt;<span style="color:#2b91af;">TResult</span>&gt;(<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">TResult</span>,&nbsp;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;node,&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;leaf);</pre> </p> <p> This method takes a pair of functions as arguments. The <code>node</code> function deals with an internal node in the tree (the blue nodes in the above diagram), whereas the <code>leaf</code> function deals with the leaf nodes (the green nodes in the diagram). </p> <p> The <code>leaf</code> function may be the easiest one to understand. A leaf node only contains a value of the type <code>T</code>, so the only operation the function has to support is translating the <code>T</code> value to a <code>TResult</code> value. This is also the premise of the <code>Leaf</code> class' implementation of the method: </p> <p> <pre><span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">T</span>&nbsp;item; <span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">TResult</span>&nbsp;Match&lt;<span style="color:#2b91af;">TResult</span>&gt;(<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">TResult</span>,&nbsp;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;node,&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;leaf) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;leaf(item); }</pre> </p> <p> The <code>node</code> function is more tricky. It takes three input arguments, of the types <code>TResult</code>, <code>T</code>, and <code>TResult</code>. The roles of these are respectively <em>left</em>, <em>item</em>, and <em>right</em>. This is a typical representation of a binary node. Since there's always a left and a right branch, you put the node's value in the middle. As was the case with the <a href="/2019/06/10/tree-catamorphism">tree catamorphism</a>, the catamorphism function receives the branches as already-translated values; that is, both the left and right branch have already been translated to <code>TResult</code> when <code>node</code> is called. While it looks like magic, as always it's just the result of recursion: </p> <p> <pre><span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">IBinaryTree</span>&lt;<span style="color:#2b91af;">T</span>&gt;&nbsp;left; <span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">T</span>&nbsp;item; <span style="color:blue;">private</span>&nbsp;<span style="color:blue;">readonly</span>&nbsp;<span style="color:#2b91af;">IBinaryTree</span>&lt;<span style="color:#2b91af;">T</span>&gt;&nbsp;right; <span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">TResult</span>&nbsp;Match&lt;<span style="color:#2b91af;">TResult</span>&gt;(<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">TResult</span>,&nbsp;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;node,&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;leaf) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;node(left.Match(node,&nbsp;leaf),&nbsp;item,&nbsp;right.Match(node,&nbsp;leaf)); }</pre> </p> <p> This is the <code>Node&lt;T&gt;</code> class implementation of the <code>Match</code> method. It calls <code>node</code> and returns whatever it returns, but notice that as the <code>left</code> and <code>right</code> arguments, if first, recursively, calls <code>left.Match</code> and <code>right.Match</code>. This is how it can call <code>node</code> with the translated branches, as well as with the basic <code>item</code>. </p> <p> The recursion stops and unwinds on <code>left</code> and <code>right</code> whenever one of those are <code>Leaf</code> instances. </p> <h3 id="c64210d585c94cb78653b96380cbf0e6"> Examples <a href="#c64210d585c94cb78653b96380cbf0e6" title="permalink">#</a> </h3> <p> You can use <code>Match</code> to implement most other behaviour you'd like <code>IBinaryTree&lt;T&gt;</code> to have. In <a href="/2018/08/13/a-visitor-functor">the original article on the full binary tree functor</a> you saw how to implement <code>Select</code> with a Visitor, but now that the API is Church-encoded, you can derive <code>Select</code> from <code>Match</code>: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IBinaryTree</span>&lt;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;Select&lt;<span style="color:#2b91af;">TResult</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IBinaryTree</span>&lt;<span style="color:#2b91af;">T</span>&gt;&nbsp;tree, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;selector) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(tree&nbsp;==&nbsp;<span style="color:blue;">null</span>) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">throw</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ArgumentNullException</span>(<span style="color:blue;">nameof</span>(tree)); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">if</span>&nbsp;(selector&nbsp;==&nbsp;<span style="color:blue;">null</span>) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">throw</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ArgumentNullException</span>(<span style="color:blue;">nameof</span>(selector)); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;tree.Match( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;node:&nbsp;(l,&nbsp;x,&nbsp;r)&nbsp;=&gt;&nbsp;Create(l,&nbsp;selector(x),&nbsp;r), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;leaf:&nbsp;x&nbsp;=&gt;&nbsp;Leaf(selector(x))); }</pre> </p> <p> In the <code>leaf</code> case, the <code>Select</code> method simply calls <code>selector</code> with the <code>x</code> value it receives, and puts the resulting <code>TResult</code> object into a new <code>Leaf</code> object. </p> <p> In the <code>node</code> case, the lambda expression receives three arguments: <code>l</code> and <code>r</code> are the <em>already-translated</em> left and right branches, so you only need to call <code>selector</code> on <code>x</code> and call the <code>Create</code> helper method to produce a new <code>Node</code> object. </p> <p> You can also implement more specialised functionality, like calculating the sum of nodes, measuring the depth of the tree, and similar functions. You saw equivalent examples in the <a href="/2019/06/10/tree-catamorphism">previous article</a>. </p> <p> For the examples in this article, I'll use the tree shown in the above diagram. Using static helper methods, you can write it like this: </p> <p> <pre><span style="color:blue;">var</span>&nbsp;tree&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">BinaryTree</span>.Create( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">BinaryTree</span>.Create( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">BinaryTree</span>.Leaf(42), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1337, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">BinaryTree</span>.Create( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">BinaryTree</span>.Leaf(2112), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;5040, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">BinaryTree</span>.Leaf(1984))), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">BinaryTree</span>.Leaf(90125));</pre> </p> <p> To calculate the sum of all nodes, you can write a function like this: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">int</span>&nbsp;Sum(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IBinaryTree</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;tree) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;tree.Match((l,&nbsp;x,&nbsp;r)&nbsp;=&gt;&nbsp;l&nbsp;+&nbsp;x&nbsp;+&nbsp;r,&nbsp;x&nbsp;=&gt;&nbsp;x); }</pre> </p> <p> The <code>leaf</code> function just returns the value of the node, while the <code>node</code> function adds the numbers together. It works for the above <code>tree</code>: </p> <p> <pre>&gt; tree.Sum() 100642</pre> </p> <p> To find the maximum value, you can write another extension method: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">int</span>&nbsp;Max(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IBinaryTree</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;tree) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;tree.Match((l,&nbsp;x,&nbsp;r)&nbsp;=&gt;&nbsp;<span style="color:#2b91af;">Math</span>.Max(<span style="color:#2b91af;">Math</span>.Max(l,&nbsp;r),&nbsp;x),&nbsp;x&nbsp;=&gt;&nbsp;x); }</pre> </p> <p> Again, the <code>leaf</code> function just returns the value of the node. The <code>node</code> function receives the value of the current node <code>x</code>, as well as the already-found maximum value of the left branch and the right branch; it then returns the maximum of these three values: </p> <p> <pre>&gt; tree.Max() 90125</pre> </p> <p> As was also the case for trees, both of these operations are part of the standard repertoire available via a data structure's <em>fold</em>. That's not the case for the next two functions, which can't be implemented using a fold, but which can be defined with the catamorphism. The first is a function to count the leaves of a tree: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">int</span>&nbsp;CountLeaves&lt;<span style="color:#2b91af;">T</span>&gt;(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IBinaryTree</span>&lt;<span style="color:#2b91af;">T</span>&gt;&nbsp;tree) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;tree.Match((l,&nbsp;_,&nbsp;r)&nbsp;=&gt;&nbsp;l&nbsp;+&nbsp;r,&nbsp;_&nbsp;=&gt;&nbsp;1); }</pre> </p> <p> Since the <code>leaf</code> function handles a leaf node, the number of leaf nodes in a leaf node is, by definition, <em>one</em>. Thus, that function can ignore the value of the node and always return <code>1</code>. The <code>node</code> function, on the other hand, receives the number of leaf nodes on the left-hand side (<code>l</code>), the value of the current node, and the number of leaf nodes on the right-hand side (<code>r</code>). Notice that since an internal node is never a leaf node, it doesn't count; instead, just add <code>l</code> and <code>r</code> together. Notice that, again, the value of the node itself is irrelevant. </p> <p> How many leaf nodes does the above tree have? </p> <p> <pre>&gt; tree.CountLeaves() 4</pre> </p> <p> You can also measure the maximum depth of a tree: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">int</span>&nbsp;MeasureDepth&lt;<span style="color:#2b91af;">T</span>&gt;(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IBinaryTree</span>&lt;<span style="color:#2b91af;">T</span>&gt;&nbsp;tree) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;tree.Match((l,&nbsp;_,&nbsp;r)&nbsp;=&gt;&nbsp;1&nbsp;+&nbsp;<span style="color:#2b91af;">Math</span>.Max(l,&nbsp;r),&nbsp;_&nbsp;=&gt;&nbsp;0); }</pre> </p> <p> Like in the previous article, I've arbitrarily decided that the depth of a leaf node is <em>zero</em>; therefore, the <code>leaf</code> function always returns <code>0</code>. The <code>node</code> function receives the depth of the left and right branches, and returns the maximum of those two values, plus one, since the current node adds one level of depth. </p> <p> <pre>&gt; tree.MeasureDepth() 3</pre> </p> <p> You may not have much need for working with full binary trees in your normal, day-to-day C# work, but I found it worthwhile to include this example for a couple of reasons. First, because the original of the API shows that a catamorphism may be hiding in a Visitor. Second, because binary trees are interesting, in that they're foldable <a href="/2018/03/22/functors">functors</a>, but not monads. </p> <p> Where does the catamorphism come from, though? How can you trust that the <code>Match</code> method is the catamorphism? </p> <h3 id="d015bcc9afe742408d7c8ba6c6edce2a"> Binary tree F-Algebra <a href="#d015bcc9afe742408d7c8ba6c6edce2a" title="permalink">#</a> </h3> <p> As in the <a href="/2019/06/10/tree-catamorphism">previous article</a>, I'll use <code>Fix</code> and <code>cata</code> as explained in <a href="https://bartoszmilewski.com">Bartosz Milewski</a>'s excellent <a href="https://bartoszmilewski.com/2017/02/28/f-algebras/">article on F-Algebras</a>. </p> <p> As always, start with the underlying endofunctor. You can think of this one as a specialisation of the rose tree from the previous article: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;FullBinaryTreeF&nbsp;a&nbsp;c&nbsp;=&nbsp;LeafF&nbsp;a&nbsp;|&nbsp;NodeF&nbsp;c&nbsp;a&nbsp;c&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;(<span style="color:blue;">FullBinaryTreeF</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(LeafF&nbsp;x)&nbsp;=&nbsp;LeafF&nbsp;x &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;f&nbsp;(NodeF&nbsp;l&nbsp;x&nbsp;r)&nbsp;=&nbsp;NodeF&nbsp;(f&nbsp;l)&nbsp;x&nbsp;(f&nbsp;r)</pre> </p> <p> As usual, I've called the 'data' type <code>a</code> and the carrier type <code>c</code> (for <em>carrier</em>). The <code>Functor</code> instance as usual translates the carrier type; the <code>fmap</code> function has the type <code>(c -&gt; c1) -&gt; FullBinaryTreeF a c -&gt; FullBinaryTreeF a c1</code>. </p> <p> As was the case when deducing the recent catamorphisms, Haskell isn't too happy about defining instances for a type like <code>Fix (FullBinaryTreeF a)</code>. To address that problem, you can introduce a <code>newtype</code> wrapper: </p> <p> <pre><span style="color:blue;">newtype</span>&nbsp;FullBinaryTreeFix&nbsp;a&nbsp;= &nbsp;&nbsp;FullBinaryTreeFix&nbsp;{&nbsp;unFullBinaryTreeFix&nbsp;::&nbsp;Fix&nbsp;(FullBinaryTreeF&nbsp;a)&nbsp;} &nbsp;&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> You can define <code>Functor</code>, <code>Foldable</code>, and <code>Traversable</code> instances (but not <code>Monad</code>) for this type without resorting to any funky GHC extensions. Keep in mind that ultimately, the purpose of all this code is just to figure out what the catamorphism looks like. This code isn't intended for actual use. </p> <p> A pair of helper functions make it easier to define <code>FullBinaryTreeFix</code> values: </p> <p> <pre><span style="color:#2b91af;">fbtLeafF</span>&nbsp;::&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">FullBinaryTreeFix</span>&nbsp;a fbtLeafF&nbsp;=&nbsp;FullBinaryTreeFix&nbsp;.&nbsp;Fix&nbsp;.&nbsp;LeafF <span style="color:#2b91af;">fbtNodeF</span>&nbsp;::&nbsp;<span style="color:blue;">FullBinaryTreeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">FullBinaryTreeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">FullBinaryTreeFix</span>&nbsp;a fbtNodeF&nbsp;(FullBinaryTreeFix&nbsp;l)&nbsp;x&nbsp;(FullBinaryTreeFix&nbsp;r)&nbsp;=&nbsp;FullBinaryTreeFix&nbsp;$&nbsp;Fix&nbsp;$&nbsp;NodeF&nbsp;l&nbsp;x&nbsp;r</pre> </p> <p> In order to distinguish these helper functions from the ones that create <code>TreeFix a</code> values, I prefixed them with <code>fbt</code> (for <em>Full Binary Tree</em>). <code>fbtLeafF</code> creates a leaf node: </p> <p> <pre>Prelude Fix FullBinaryTree&gt; fbtLeafF "fnaah" FullBinaryTreeFix {unFullBinaryTreeFix = Fix (LeafF "fnaah")}</pre> </p> <p> <code>fbtNodeF</code> is a helper function to create an internal node: </p> <p> <pre>Prelude Fix FullBinaryTree&gt; fbtNodeF (fbtLeafF 1337) 42 (fbtLeafF 2112) FullBinaryTreeFix {unFullBinaryTreeFix = Fix (NodeF (Fix (LeafF 1337)) 42 (Fix (LeafF 2112)))}</pre> </p> <p> The <code>FullBinaryTreeFix</code> type, or rather the underlying <code>FullBinaryTreeF a</code> functor, is all you need to identify the catamorphism. </p> <h3 id="ced0da7dc61943b0be872ec79b4e3651"> Haskell catamorphism <a href="#ced0da7dc61943b0be872ec79b4e3651" title="permalink">#</a> </h3> <p> At this point, you have two out of three elements of an F-Algebra. You have an endofunctor (<code>FullBinaryTreeF a</code>), and an object <code>c</code>, but you still need to find a morphism <code>FullBinaryTreeF a c -&gt; c</code>. Notice that the algebra you have to find is the function that reduces the functor to its <em>carrier type</em> <code>c</code>, not the 'data type' <code>a</code>. This takes some time to get used to, but that's how catamorphisms work. This doesn't mean, however, that you get to ignore <code>a</code>, as you'll see. </p> <p> As in the previous articles, start by writing a function that will become the catamorphism, based on <code>cata</code>: </p> <p> <pre>fullBinaryTreeF&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unFullBinaryTreeFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(LeafF&nbsp;x)&nbsp;=&nbsp;<span style="color:blue;">undefined</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(NodeF&nbsp;l&nbsp;x&nbsp;r)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> While this compiles, with its <code>undefined</code> implementation of <code>alg</code>, it obviously doesn't do anything useful. I find, however, that it helps me think. How can you return a value of the type <code>c</code> from <code>alg</code>? You could pass a function argument to the <code>fullBinaryTreeF</code> function and use it with <code>x</code>: </p> <p> <pre>fullBinaryTreeF&nbsp;fl&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unFullBinaryTreeFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(LeafF&nbsp;x)&nbsp;=&nbsp;fl&nbsp;x &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(NodeF&nbsp;l&nbsp;x&nbsp;r)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> I called the function <code>fl</code> for <em>function, leaf</em>, because we're also going to need a function for the <code>NodeF</code> case: </p> <p> <pre><span style="color:#2b91af;">fullBinaryTreeF</span>&nbsp;::&nbsp;(c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;(a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">FullBinaryTreeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c fullBinaryTreeF&nbsp;fn&nbsp;fl&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unFullBinaryTreeFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(LeafF&nbsp;x)&nbsp;=&nbsp;fl&nbsp;x &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(NodeF&nbsp;l&nbsp;x&nbsp;r)&nbsp;=&nbsp;fn&nbsp;l&nbsp;x&nbsp;r</pre> </p> <p> This works. Since <code>cata</code> has the type <code>Functor f =&gt; (f a -&gt; a) -&gt; Fix f -&gt; a</code>, that means that <code>alg</code> has the type <code>f a -&gt; a</code>. In the case of <code>FullBinaryTreeF</code>, the compiler infers that the <code>alg</code> function has the type <code>FullBinaryTreeF a c -&gt; c</code>, which is just what you need! </p> <p> You can now see what the carrier type <code>c</code> is for. It's the type that the algebra extracts, and thus the type that the catamorphism returns. </p> <p> This, then, is the catamorphism for a full binary tree. As always, it's not the only possible catamorphism, since you can easily reorder the arguments to both <code>fullBinaryTreeF</code>, <code>fn</code>, and <code>fl</code>. These would all be isomorphic, though. </p> <h3 id="3f87d49db58f4cd59dec76a97d31c0d2"> Basis <a href="#3f87d49db58f4cd59dec76a97d31c0d2" title="permalink">#</a> </h3> <p> You can implement most other useful functionality with <code>treeF</code>. Here's the <code>Functor</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;<span style="color:blue;">FullBinaryTreeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;f&nbsp;=&nbsp;fullBinaryTreeF&nbsp;(\l&nbsp;x&nbsp;r&nbsp;-&gt;&nbsp;fbtNodeF&nbsp;l&nbsp;(f&nbsp;x)&nbsp;r)&nbsp;(fbtLeafF&nbsp;.&nbsp;f)</pre> </p> <p> The <code>fl</code> function first invokes <code>f</code>, followed by <code>fbtLeafF</code>. The <code>fn</code> function uses the <code>fbtNodeF</code> helper function to create a new internal node. <code>l</code> and <code>r</code> are already-translated branches, so you just need to call <code>f</code> with the node value <code>x</code>. </p> <p> There's no <code>Monad</code> instance for binary trees, because you can't flatten a binary tree of binary trees. You can, on the other hand, define a <code>Foldable</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Foldable</span>&nbsp;<span style="color:blue;">FullBinaryTreeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;foldMap&nbsp;f&nbsp;=&nbsp;fullBinaryTreeF&nbsp;(\l&nbsp;x&nbsp;r&nbsp;-&gt;&nbsp;l&nbsp;&lt;&gt;&nbsp;f&nbsp;x&nbsp;&lt;&gt;&nbsp;r)&nbsp;f</pre> </p> <p> The <code>f</code> function passed to <code>foldMap</code> has the type <code>Monoid m =&gt; (a -&gt; m)</code>, so the <code>fl</code> function that handles leaf nodes simply calls <code>f</code> with the contents of the node. The <code>fn</code> function receives two branches already translated to <code>m</code>, so it just has to call <code>f</code> with <code>x</code> and combine all the <code>m</code> values using the <code>&lt;&gt;</code> operator. </p> <p> The <code>Traversable</code> instance follows right on the heels of <code>Foldable</code>: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Traversable</span>&nbsp;<span style="color:blue;">FullBinaryTreeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;sequenceA&nbsp;=&nbsp;fullBinaryTreeF&nbsp;(liftA3&nbsp;fbtNodeF)&nbsp;(<span style="color:blue;">fmap</span>&nbsp;fbtLeafF)</pre> </p> <p> There are operations on binary trees that you can implement with a fold, but some that you can't. Consider the tree shown in the diagram at the beginning of the article. This is also the tree that the above C# examples use. In Haskell, using <code>FullBinaryTreeFix</code>, you can define that tree like this: </p> <p> <pre>tree&nbsp;=&nbsp; &nbsp;&nbsp;fbtNodeF &nbsp;&nbsp;&nbsp;&nbsp;(fbtNodeF &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(fbtLeafF&nbsp;42) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1337 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(fbtNodeF &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(fbtLeafF&nbsp;2112) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;5040 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(fbtLeafF&nbsp;1984))) &nbsp;&nbsp;&nbsp;&nbsp;2 &nbsp;&nbsp;&nbsp;&nbsp;(fbtLeafF&nbsp;90125)</pre> </p> <p> Since <code>FullBinaryTreeFix</code> is <code>Foldable</code>, and that type class already comes with <code>sum</code> and <code>maximum</code> functions, no further work is required to repeat the first two of the above C# examples: </p> <p> <pre>Prelude Fix FullBinaryTree&gt; sum tree 100642 Prelude Fix FullBinaryTree&gt; maximum tree 90125</pre> </p> <p> Counting leaves, or measuring the depth of a tree, on the other hand, is impossible with the <code>Foldable</code> instance, but can be implemented using the catamorphism: </p> <p> <pre><span style="color:#2b91af;">countLeaves</span>&nbsp;::&nbsp;<span style="color:blue;">Num</span>&nbsp;n&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;<span style="color:blue;">FullBinaryTreeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;n countLeaves&nbsp;=&nbsp;fullBinaryTreeF&nbsp;(\l&nbsp;_&nbsp;r&nbsp;-&gt;&nbsp;l&nbsp;+&nbsp;r)&nbsp;(<span style="color:blue;">const</span>&nbsp;1) <span style="color:#2b91af;">treeDepth</span>&nbsp;::&nbsp;(<span style="color:blue;">Ord</span>&nbsp;n,&nbsp;<span style="color:blue;">Num</span>&nbsp;n)&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;<span style="color:blue;">FullBinaryTreeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;n treeDepth&nbsp;=&nbsp;fullBinaryTreeF&nbsp;(\l&nbsp;_&nbsp;r&nbsp;-&gt;&nbsp;1&nbsp;+&nbsp;<span style="color:blue;">max</span>&nbsp;l&nbsp;r)&nbsp;(<span style="color:blue;">const</span>&nbsp;0)</pre> </p> <p> The reasoning is the same as already explained in the above C# examples. The functions also produce the same results: </p> <p> <pre>Prelude Fix FullBinaryTree&gt; countLeaves tree 4 Prelude Fix FullBinaryTree&gt; treeDepth tree 3</pre> </p> <p> This, hopefully, illustrates that the catamorphism is more capable, and that the fold is just a (list-biased) specialisation. </p> <h3 id="81b3e77b6fbe4760bc8c74805e4edba8"> Summary <a href="#81b3e77b6fbe4760bc8c74805e4edba8" title="permalink">#</a> </h3> <p> The catamorphism for a full binary tree is a pair of functions. One function handles internal nodes, while the other function handles leaf nodes. </p> <p> I thought it was interesting to show this example for two reasons: First, the original example was a Visitor implementation, and I think it's worth realising that a Visitor's <code>Accept</code> method can also be viewed as a catamorphism. Second, a binary tree is an example of a data structure that has a fold, but isn't a monad. </p> <p> All articles in the article series have, so far, covered data structures well-known from computer science. The next example will, on the other hand, demonstrate that even completely ad-hoc domain-specific data structures have catamorphisms. </p> <p> <strong>Next:</strong> <a href="/2019/07/08/payment-types-catamorphism">Payment types catamorphism</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Composition Root location https://blog.ploeh.dk/2019/06/17/composition-root-location 2019-06-17T05:55:00+00:00 Mark Seemann <div id="post"> <p> <em>A Composition Root should be located near the point where user code first executes.</em> </p> <p> Prompted by a recent Internet discussion, my <a href="https://amzn.to/2TE8tJx">DIPPP</a> co-author <a href="https://blogs.cuttingedge.it/steven/">Steven van Deursen</a> wrote to me in order to help clarify the <a href="/2011/07/28/CompositionRoot">Composition Root</a> pattern. </p> <p> In the email, Steven ponders whether it's defensible to use an API that <a href="/2010/11/01/PatternRecognitionAbstractFactoryorServiceLocator">looks like a Service Locator</a> from within a unit test. He specifically calls out my article that describes the <a href="/2013/03/11/auto-mocking-container">Auto-mocking Container design pattern</a>. </p> <p> In that article, I show how to use Castle Windsor's <code>Resolve</code> method from within a unit test: </p> <p> <pre style="margin: 0px;">[<span style="color: #2b91af;">Fact</span>] <span style="color: blue;">public</span> <span style="color: blue;">void</span> SutIsController() { &nbsp;&nbsp;&nbsp; <span style="color: blue;">var</span> container = <span style="color: blue;">new</span> <span style="color: #2b91af;">WindsorContainer</span>().Install(<span style="color: blue;">new</span> <span style="color: #2b91af;">ShopFixture</span>()); &nbsp;&nbsp;&nbsp; <span style="color: blue;">var</span> sut = container.Resolve&lt;<span style="color: #2b91af;">BasketController</span>&gt;(); &nbsp;&nbsp;&nbsp; <span style="color: #2b91af;">Assert</span>.IsAssignableFrom&lt;<span style="color: #2b91af;">IHttpController</span>&gt;(sut); }</pre> </p> <p> Is the test using a <a href="/2010/02/03/ServiceLocatorisanAnti-Pattern">Service Locator</a>? If so, why is that okay? If not, why isn't it a Service Locator? </p> <p> This article argues that that this use of <code>Resolve</code> isn't a Service Locator. </p> <h3 id="e9a6c124fa1d4610ae57b3cba83254b0"> Entry points defined <a href="#e9a6c124fa1d4610ae57b3cba83254b0" title="permalink">#</a> </h3> <p> The <a href="/2011/07/28/CompositionRoot">original article about the Composition Root pattern</a> defines a Composition Root as the place where you compose your object graph(s). It repeatedly describes how this ought to happen in, or as close as possible to, the application's entry point. I believe that this definition is compatible with the pattern description given in <a href="https://amzn.to/2TE8tJx">our book</a>. </p> <p> I do realise, however, that we may never have explicitly defined what an <em>entry point</em> is. </p> <p> In order to do so, it may be helpful to establish a bit of terminology. In the following, I'll use the terms <em>user code</em> as opposed to <em>framework code</em>. </p> <p> Much of the code you write probably runs within some sort of framework. If you're writing a web application, you're probably using a web framework. If you're writing a message-based application, you might be using some message bus, or actor, framework. If you're writing an app for a mobile device, you're probably using some sort of framework for that, too. </p> <p> Even as a programmer, you're a <em>user</em> of frameworks. </p> <p> As I usually do, I'll use <a href="http://tomasp.net">Tomas Petricek</a>'s distinction between <a href="http://tomasp.net/blog/2015/library-frameworks">libraries and frameworks</a>. A library is a collection of APIs that you can call. A framework is a software system that calls your code. </p> <p> <img src="/content/binary/user-code-in-framework.png" alt="User code running in a framework."> </p> <p> The reality is often more complex, as illustrated by the figure. While a framework will call your code, you can also invoke APIs afforded by the framework. </p> <p> The point, however, is that <em>user code</em> is code that you write, while <em>framework code</em> is code that someone else wrote to develop the framework. The framework starts up first, and at some point in its lifetime, it calls your code. </p> <p class="text-center"> <strong>Definition:</strong> The <em>entry point</em> is the user code that the framework calls first. </p> <p> As an example, in ASP.NET Core, the (conventional) <code>Startup</code> class is the first user code that the framework calls. (If you follow Tomas Petricek's definition to the letter, ASP.NET Core isn't a framework, but a library, because you have to write a <code>Main</code> method and call <code>WebHost.CreateDefaultBuilder(args).UseStartup&lt;Startup&gt;().Build().Run()</code>. In reality, though, you're supposed to configure the application from your <code>Startup</code> class, making it the <em>de facto</em> entry point.) </p> <h3 id="61e3f212e0e244f18ac998f4b9fbb635"> Unit testing endpoints <a href="#61e3f212e0e244f18ac998f4b9fbb635" title="permalink">#</a> </h3> <p> Most .NET-based unit testing packages are frameworks. There's typically little explicit configuration. Instead, you just write a method and adorn it with an attribute: </p> <p> <pre>[<span style="color:#2b91af;">Fact</span>] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">async</span>&nbsp;<span style="color:#2b91af;">Task</span>&nbsp;ReservationSucceeds() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;repo&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">FakeReservationsRepository</span>(); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;sut&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ReservationsController</span>(10,&nbsp;repo); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;reservation&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;date:&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTimeOffset</span>(2018,&nbsp;8,&nbsp;13,&nbsp;16,&nbsp;53,&nbsp;0,&nbsp;<span style="color:#2b91af;">TimeSpan</span>.FromHours(2)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;email:&nbsp;<span style="color:#a31515;">&quot;mark@example.com&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;name:&nbsp;<span style="color:#a31515;">&quot;Mark&nbsp;Seemann&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;quantity:&nbsp;4); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;actual&nbsp;=&nbsp;<span style="color:blue;">await</span>&nbsp;sut.Post(reservation); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.True(repo.Contains(reservation.Accept())); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;expectedId&nbsp;=&nbsp;repo.GetId(reservation.Accept()); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;ok&nbsp;=&nbsp;<span style="color:#2b91af;">Assert</span>.IsAssignableFrom&lt;<span style="color:#2b91af;">OkActionResult</span>&gt;(actual); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.Equal(expectedId,&nbsp;ok.Value); } [<span style="color:#2b91af;">Fact</span>] <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">async</span>&nbsp;<span style="color:#2b91af;">Task</span>&nbsp;ReservationFails() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;repo&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">FakeReservationsRepository</span>(); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;sut&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ReservationsController</span>(10,&nbsp;repo); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;reservation&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Reservation</span>( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;date:&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">DateTimeOffset</span>(2018,&nbsp;8,&nbsp;13,&nbsp;16,&nbsp;53,&nbsp;0,&nbsp;<span style="color:#2b91af;">TimeSpan</span>.FromHours(2)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;email:&nbsp;<span style="color:#a31515;">&quot;mark@example.com&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;name:&nbsp;<span style="color:#a31515;">&quot;Mark&nbsp;Seemann&quot;</span>, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;quantity:&nbsp;11); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">var</span>&nbsp;actual&nbsp;=&nbsp;<span style="color:blue;">await</span>&nbsp;sut.Post(reservation); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.False(reservation.IsAccepted); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.False(repo.Contains(reservation)); &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Assert</span>.IsAssignableFrom&lt;<span style="color:#2b91af;">InternalServerErrorActionResult</span>&gt;(actual); }</pre> </p> <p> With <a href="https://xunit.net">xUnit.net</a>, the attribute is called <code>[Fact]</code>, but the principle is the same in <a href="https://nunit.org">NUnit</a> and MSTest, only that names are different. </p> <p> Where's the entry point? </p> <p> Each test is it's own entry point. The test is (typically) the first user code that the test runner executes. Furthermore, each test runs independently of any other. </p> <p> For the sake of argument, you could write each test case in a new application, and run all your test applications in parallel. It would be impractical, but it oughtn't change the way you organise the tests. Each test method is, conceptually, a mini-application. </p> <p> A test method is its own Composition Root; or, more generally, each test has its own Composition Root. In fact, xUnit.net has various extensibility points that enable you to hook into the framework before each test method executes. You can, for example, <a href="/2010/10/08/AutoDataTheorieswithAutoFixture">combine a <code>[Theory]</code> attribute with a custom <code>AutoDataAttribute</code></a>, or you can adorn your tests with a <code>BeforeAfterTestAttribute</code>. This doesn't change that the test runner will run each test case independently of all the other tests. Those pre-execution hooks play the same role as middleware in real applications. </p> <p> You can, therefore, consider the <a href="/2013/06/24/a-heuristic-for-formatting-code-according-to-the-aaa-pattern">Arrange phase</a> the Composition Root for each test. </p> <p> Thus, I don't consider the use of an Auto-mocking Container to be a Service Locator, since <a href="/2011/08/25/ServiceLocatorrolesvs.mechanics">its role is to resolve object graphs at the entry point instead of locating services from arbitrary locations in the code base</a>. </p> <h3 id="200be4483e4b4369abe5912b2a8213c3"> Summary <a href="#200be4483e4b4369abe5912b2a8213c3" title="permalink">#</a> </h3> <p> A Composition Root is located at, or near, the <em>entry point</em>. An entry point is where <em>user code</em> is first executed by a framework. Each unit test method constitutes a separate, independent entry point. Therefore, it's consistent with these definitions to use an Auto-mocking Container in a unit test. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Tree catamorphism https://blog.ploeh.dk/2019/06/10/tree-catamorphism 2019-06-10T09:10:00+00:00 Mark Seemann <div id="post"> <p> <em>The catamorphism for a tree is just a single function with a particular type.</em> </p> <p> This article is part of an <a href="/2019/04/29/catamorphisms">article series about catamorphisms</a>. A catamorphism is a <a href="/2017/10/04/from-design-patterns-to-category-theory">universal abstraction</a> that describes how to digest a data structure into a potentially more compact value. </p> <p> This article presents the catamorphism for a <a href="https://en.wikipedia.org/wiki/Tree_(data_structure)">tree</a>, as well as how to identify it. The beginning of this article presents the catamorphism in C#, with examples. The rest of the article describes how to deduce the catamorphism. This part of the article presents my work in <a href="https://www.haskell.org">Haskell</a>. Readers not comfortable with Haskell can just read the first part, and consider the rest of the article as an optional appendix. </p> <p> A tree is a general-purpose data structure where each node in a tree has an associated value. Each node can have an arbitrary number of branches, including none. </p> <p> <img src="/content/binary/tree-example.png" alt="A tree example diagram, with each node containing integers."> </p> <p> The diagram shows an example of a tree of integers. The left branch contains a sub-tree with only a single branch, whereas the right branch contains a sub-tree with three branches. Each of the leaf nodes are trees in their own right, but they all have zero branches. </p> <p> In this example, each branch at the 'same level' has the same depth, but this isn't required. </p> <h3 id="7d3f657d0c6b443f83eac89370e0c660"> C# catamorphism <a href="#7d3f657d0c6b443f83eac89370e0c660" title="permalink">#</a> </h3> <p> As a C# representation of a tree, I'll use the <code>Tree&lt;T&gt;</code> class from <a href="/2018/08/06/a-tree-functor">A Tree functor</a>. The catamorphism is this instance method on <code>Tree&lt;T&gt;</code>: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">TResult</span>&nbsp;Cata&lt;<span style="color:#2b91af;">TResult</span>&gt;(<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">IReadOnlyCollection</span>&lt;<span style="color:#2b91af;">TResult</span>&gt;,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;func) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;func(Item,&nbsp;children.Select(c&nbsp;=&gt;&nbsp;c.Cata(func)).ToArray()); }</pre> </p> <p> Contrary to previous articles, I didn't call this method <code>Match</code>, but simply <code>Cata</code> (for <em>catamorphism</em>). The reason is that those other methods are called <code>Match</code> for a particular reason. The data structures for which they are catamorphisms are all <a href="/2018/05/22/church-encoding">Church-encoded</a> <a href="https://en.wikipedia.org/wiki/Tagged_union">sum types</a>. For those types, the <code>Match</code> methods enable a syntax similar to pattern matching in <a href="https://fsharp.org">F#</a>. That's not the case for <code>Tree&lt;T&gt;</code>. It's not a sum type, and it isn't Church-encoded. </p> <p> The method takes a single function as an input argument. This is the first catamorphism in this article series that isn't made up of a pair of some sort. The <a href="/2019/05/06/boolean-catamorphism">Boolean catamorphism</a> is a pair of values, the <a href="/2019/05/20/maybe-catamorphism">Maybe catamorphism</a> is a pair made up of a value and a function, and the <a href="/2019/06/03/either-catamorphism">Either catamorphism</a> is a pair of functions. The tree catamorphism, in contrast, is just a single function. </p> <p> The first argument to the function is a value of the type <code>T</code>. This will be an <code>Item</code> value. The second argument to the function is a finite collection of <code>TResult</code> values. This may take a little time getting used to, but it's a collection of already reduced sub-trees. When you supply such a function to <code>Cata</code>, that function must return a single value of the type <code>TResult</code>. Thus, the function must be able to digest a finite collection of <code>TResult</code> values, as well as a <code>T</code> value, to a single <code>TResult</code> value. </p> <p> The <code>Cata</code> method accomplishes this by calling <code>func</code> with the current <code>Item</code>, as well as by recursively applying itself to each of the sub-trees. Eventually, <code>Cata</code> will recurse into leaf nodes, which means that <code>children</code> will be empty. When that happens, the lambda expression inside <code>children.Select</code> never runs, and recursion stops and unwinds. </p> <h3 id="167ba023ee654db39fb5eb448d35a8df"> Examples <a href="#167ba023ee654db39fb5eb448d35a8df" title="permalink">#</a> </h3> <p> You can use <code>Cata</code> to implement most other behaviour you'd like <code>Tree&lt;T&gt;</code> to have. In <a href="/2018/08/06/a-tree-functor">the original article on the Tree functor</a> you saw an ad-hoc implementation of <code>Select</code>, but instead, you can derive <code>Select</code> from <code>Cata</code>: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:#2b91af;">Tree</span>&lt;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;Select&lt;<span style="color:#2b91af;">TResult</span>&gt;(<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;selector) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;Cata&lt;<span style="color:#2b91af;">Tree</span>&lt;<span style="color:#2b91af;">TResult</span>&gt;&gt;((x,&nbsp;nodes)&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Tree</span>&lt;<span style="color:#2b91af;">TResult</span>&gt;(selector(x),&nbsp;nodes)); }</pre> </p> <p> The lambda expression receives <code>x</code>, an object of the type <code>T</code>, as well as <code>nodes</code>, which is a finite collection of already translated sub-trees. It simply translates <code>x</code> with <code>selector</code> and returns a <code>new Tree&lt;TResult&gt;</code> with the translated value and the already translated <code>nodes</code>. </p> <p> This works just as well as the ad-hoc implementation; it passes all the same tests as shown in the previous article. </p> <p> If you have a tree of numbers, you can add them all together: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">int</span>&nbsp;Sum(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">Tree</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;tree) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;tree.Cata&lt;<span style="color:blue;">int</span>&gt;((x,&nbsp;xs)&nbsp;=&gt;&nbsp;x&nbsp;+&nbsp;xs.Sum()); }</pre> </p> <p> This uses the built-in <a href="https://docs.microsoft.com/dotnet/api/system.linq.enumerable.sum">Sum method</a> for <code>IEnumerable&lt;T&gt;</code> to add all the partly calculated sub-trees together, and then adds the value of the current node. In this and remaining examples, I'll use the tree shown in the above diagram: </p> <p> <pre><span style="color:#2b91af;">Tree</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;tree&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Tree</span>.Create(42, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Tree</span>.Create(1337, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Tree</span>.Leaf(-3)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Tree</span>.Create(7, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Tree</span>.Leaf(-99), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Tree</span>.Leaf(100), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Tree</span>.Leaf(0)));</pre> </p> <p> You can now calculate the sum of all these nodes: </p> <p> <pre>&gt; tree.Sum() 1384</pre> </p> <p> Another option is to find the maximum value anywhere in a tree: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">int</span>&nbsp;Max(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">Tree</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;tree) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;tree.Cata&lt;<span style="color:blue;">int</span>&gt;((x,&nbsp;xs)&nbsp;=&gt;&nbsp;xs.Any()&nbsp;?&nbsp;<span style="color:#2b91af;">Math</span>.Max(x,&nbsp;xs.Max())&nbsp;:&nbsp;x); }</pre> </p> <p> This method again utilises one of the LINQ methods available via the .NET base class library: <a href="https://docs.microsoft.com/dotnet/api/system.linq.enumerable.max">Max</a>. It is, however, necessary to first check whether the partially reduced <code>xs</code> is empty or not, because the <code>Max</code> extension method on <code>IEnumerable&lt;int&gt;</code> doesn't know how to deal with an empty collection (it throws an exception). When <code>xs</code> is empty that implies a leaf node, in which case you can simply return <code>x</code>; otherwise, you'll first have to use the <code>Max</code> method on <code>xs</code> to find the maximum value there, and then use <code>Math.Max</code> to find the maximum of those two. (I'll here remind the attentive reader that finding the maximum number forms a <a href="/2017/11/27/semigroups">semigroup</a> and that <a href="/2017/12/11/semigroups-accumulate">semigroups accumulate</a> when collections are non-empty. It all fits together. Isn't maths lovely?) </p> <p> Using the same <code>tree</code> as before, you can see that this method, too, works as expected: </p> <p> <pre>&gt; tree.Max() 1337</pre> </p> <p> So far, these two extension methods are just specialised <em>folds</em>. In Haskell, <code>Foldable</code> is a specific type class, and <code>sum</code> and <code>max</code> are available for all instances. As promised in <a href="/2019/04/29/catamorphisms">the introduction to the series</a>, though, there are some functions on trees that you can't implement using a fold. One of these is to count all the leaf nodes. You can still derive that functionality from the catamorphism, though: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">int</span>&nbsp;CountLeaves() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;Cata&lt;<span style="color:blue;">int</span>&gt;((x,&nbsp;xs)&nbsp;=&gt;&nbsp;xs.Any()&nbsp;?&nbsp;xs.Sum()&nbsp;:&nbsp;1); }</pre> </p> <p> Like <code>Max</code>, the lambda expression used to implement <code>CountLeaves</code> uses <a href="https://docs.microsoft.com/dotnet/api/system.linq.enumerable.any">Any</a> to detect whether or not <code>xs</code> is empty, which is when <code>Any</code> is <code>false</code>. Empty <code>xs</code> indicates that you've found a leaf node, so return <code>1</code>. When <code>xs</code> isn't empty, it contains a collection of <code>1</code> values - one for each leaf node recursively found; add them together with <code>Sum</code>. </p> <p> This also works for the same <code>tree</code> as before: </p> <p> <pre>&gt; tree.CountLeaves() 4</pre> </p> <p> You can also measure the maximum depth of a tree: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">int</span>&nbsp;MeasureDepth() { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;Cata&lt;<span style="color:blue;">int</span>&gt;((x,&nbsp;xs)&nbsp;=&gt;&nbsp;xs.Any()&nbsp;?&nbsp;1&nbsp;+&nbsp;xs.Max()&nbsp;:&nbsp;0); }</pre> </p> <p> This implementation considers a leaf node to have no depth: </p> <p> <pre>&gt; <span style="color:#2b91af;">Tree</span>.Leaf(<span style="color:#a31515;">&quot;foo&quot;</span>).MeasureDepth() 0</pre> </p> <p> This is a discretionary definition; you could also argue that, by definition, a leaf node ought to have a depth of one. If you think so, you'll need to change the <code>0</code> to <code>1</code> in the above <code>MeasureDepth</code> implementation. </p> <p> Once more, you can use <code>Any</code> to detect leaf nodes. Whenever you find a leaf node, you return its depth, which, by definition, is <code>0</code>. Otherwise, you find the maximum depth already found among <code>xs</code>, and add <code>1</code>, because <code>xs</code> contains the maximum depths of all immediate sub-trees. </p> <p> Using the same <code>tree</code> again: </p> <p> <pre>&gt; tree.MeasureDepth() 2</pre> </p> <p> The above <code>tree</code> has the same depth for all sub-trees, so here's an example of a tilted tree: </p> <p> <pre>&gt; <span style="color:#2b91af;">Tree</span>.Create(3, . <span style="color:#2b91af;">Tree</span>.Create(1, . <span style="color:#2b91af;">Tree</span>.Leaf(0), . <span style="color:#2b91af;">Tree</span>.Leaf(0)), . <span style="color:#2b91af;">Tree</span>.Leaf(0), . <span style="color:#2b91af;">Tree</span>.Leaf(0), . <span style="color:#2b91af;">Tree</span>.Create(2, . <span style="color:#2b91af;">Tree</span>.Create(1, . <span style="color:#2b91af;">Tree</span>.Leaf(0)))) . .MeasureDepth() 3</pre> </p> <p> To make it easier to understand, I've labelled all the leaf nodes with <code>0</code>, because that's their depth. I've then labelled the other nodes with the maximum number 'under' them, plus one. That's the algorithm used. </p> <h3 id="82e5042d05534db2a67c8f7c37f78419"> Tree F-Algebra <a href="#82e5042d05534db2a67c8f7c37f78419" title="permalink">#</a> </h3> <p> As in the <a href="/2019/06/03/either-catamorphism">previous article</a>, I'll use <code>Fix</code> and <code>cata</code> as explained in <a href="https://bartoszmilewski.com">Bartosz Milewski</a>'s excellent <a href="https://bartoszmilewski.com/2017/02/28/f-algebras/">article on F-Algebras</a>. </p> <p> As always, start with the underlying endofunctor. I've taken some inspiration from <code>Tree a</code> from <code>Data.Tree</code>, but changed some names: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;TreeF&nbsp;a&nbsp;c&nbsp;=&nbsp;NodeF&nbsp;{&nbsp;nodeValue&nbsp;::&nbsp;a,&nbsp;nodes&nbsp;::&nbsp;ListFix&nbsp;c&nbsp;}&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;(<span style="color:blue;">TreeF</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;f&nbsp;(NodeF&nbsp;x&nbsp;ns)&nbsp;=&nbsp;NodeF&nbsp;x&nbsp;$&nbsp;<span style="color:blue;">fmap</span>&nbsp;f&nbsp;ns</pre> </p> <p> Instead of using Haskell's standard list (<code>[]</code>) for the sub-forest, I've used <code>ListFix</code> from <a href="/2019/05/27/list-catamorphism">the article on list catamorphism</a>. This should, hopefully, demonstrate how you can build on already established definitions derived from first principles. </p> <p> As usual, I've called the 'data' type <code>a</code> and the carrier type <code>c</code> (for <em>carrier</em>). The <code>Functor</code> instance as usual translates the carrier type; the <code>fmap</code> function has the type <code>(c -&gt; c1) -&gt; TreeF a c -&gt; TreeF a c1</code>. </p> <p> As was the case when deducing the recent catamorphisms, Haskell isn't too happy about defining instances for a type like <code>Fix (TreeF a)</code>. To address that problem, you can introduce a <code>newtype</code> wrapper: </p> <p> <pre><span style="color:blue;">newtype</span>&nbsp;TreeFix&nbsp;a&nbsp;=&nbsp;TreeFix&nbsp;{&nbsp;unTreeFix&nbsp;::&nbsp;Fix&nbsp;(TreeF&nbsp;a)&nbsp;}&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> You can define <code>Functor</code>, <code>Applicative</code>, <code>Monad</code>, etc. instances for this type without resorting to any funky GHC extensions. Keep in mind that ultimately, the purpose of all this code is just to figure out what the catamorphism looks like. This code isn't intended for actual use. </p> <p> A pair of helper functions make it easier to define <code>TreeFix</code> values: </p> <p> <pre><span style="color:#2b91af;">leafF</span>&nbsp;::&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;a leafF&nbsp;x&nbsp;=&nbsp;TreeFix&nbsp;$&nbsp;Fix&nbsp;$&nbsp;NodeF&nbsp;x&nbsp;nilF <span style="color:#2b91af;">nodeF</span>&nbsp;::&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;(<span style="color:blue;">TreeFix</span>&nbsp;a)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;a nodeF&nbsp;x&nbsp;=&nbsp;TreeFix&nbsp;.&nbsp;Fix&nbsp;.&nbsp;NodeF&nbsp;x&nbsp;.&nbsp;<span style="color:blue;">fmap</span>&nbsp;unTreeFix</pre> </p> <p> <code>leafF</code> creates a leaf node: </p> <p> <pre>Prelude Fix List Tree&gt; leafF "ploeh" TreeFix {unTreeFix = Fix (NodeF "ploeh" (ListFix (Fix NilF)))}</pre> </p> <p> <code>nodeF</code> is a helper function to create a non-leaf node: </p> <p> <pre>Prelude Fix List Tree&gt; nodeF 4 (consF (leafF 9) nilF) TreeFix {unTreeFix = Fix (NodeF 4 (ListFix (Fix (ConsF (Fix (NodeF 9 (ListFix (Fix NilF)))) (Fix NilF)))))}</pre> </p> <p> Even with helper functions, construction of <code>TreeFix</code> values is cumbersome, but keep in mind that the code shown here isn't meant to be used in practice. The goal is only to deduce catamorphisms from more basic universal abstractions, and you now have all you need to do that. </p> <h3 id="ca5669298d814809a3f0d4b0422b860f"> Haskell catamorphism <a href="#ca5669298d814809a3f0d4b0422b860f" title="permalink">#</a> </h3> <p> At this point, you have two out of three elements of an F-Algebra. You have an endofunctor (<code>TreeF a</code>), and an object <code>c</code>, but you still need to find a morphism <code>TreeF a c -&gt; c</code>. Notice that the algebra you have to find is the function that reduces the functor to its <em>carrier type</em> <code>c</code>, not the 'data type' <code>a</code>. This takes some time to get used to, but that's how catamorphisms work. This doesn't mean, however, that you get to ignore <code>a</code>, as you'll see. </p> <p> As in the previous articles, start by writing a function that will become the catamorphism, based on <code>cata</code>: </p> <p> <pre>treeF&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unTreeFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;(NodeF&nbsp;x&nbsp;ns)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> While this compiles, with its <code>undefined</code> implementation of <code>alg</code>, it obviously doesn't do anything useful. I find, however, that it helps me think. How can you return a value of the type <code>c</code> from <code>alg</code>? You could pass a function argument to the <code>treeF</code> function and use it with <code>x</code> and <code>ns</code>: </p> <p> <pre><span style="color:#2b91af;">treeF</span>&nbsp;::&nbsp;(a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c treeF&nbsp;f&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unTreeFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;(NodeF&nbsp;x&nbsp;ns)&nbsp;=&nbsp;f&nbsp;x&nbsp;ns</pre> </p> <p> This works. Since <code>cata</code> has the type <code>Functor f =&gt; (f a -&gt; a) -&gt; Fix f -&gt; a</code>, that means that <code>alg</code> has the type <code>f a -&gt; a</code>. In the case of <code>TreeF</code>, the compiler infers that the <code>alg</code> function has the type <code>TreeF a c -&gt; c</code>, which is just what you need! </p> <p> You can now see what the carrier type <code>c</code> is for. It's the type that the algebra extracts, and thus the type that the catamorphism returns. </p> <p> This, then, is the catamorphism for a tree. So far in this article series, all previous catamorphisms have been pairs, but this one is just a single function. It's still not the only possible catamorphism, since you could trivially flip the arguments to <code>f</code>. </p> <p> I've chosen the representation shown here because it's isomorphic to the <code>foldTree</code> function from Haskell's built-in <code>Data.Tree</code> module, which explicitly documents that the function "is also known as the catamorphism on trees." <code>foldTree</code> is defined using Haskell's standard list type (<code>[]</code>), so the type is simpler: <code>(a -&gt; [b] -&gt; b) -&gt; Tree a -&gt; b</code>. The two representations of trees, <code>TreeFix</code> and <code>Tree</code> are, however, isomorphic, so <code>foldTree</code> is equivalent to <code>treeF</code>. Notice how both of these functions are also equivalent to the above C# <code>Cata</code> method. </p> <h3 id="8647c7bd03aa4d4b8a01a8252058830f"> Basis <a href="#8647c7bd03aa4d4b8a01a8252058830f" title="permalink">#</a> </h3> <p> You can implement most other useful functionality with <code>treeF</code>. Here's the <code>Functor</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;f&nbsp;=&nbsp;treeF&nbsp;(nodeF&nbsp;.&nbsp;f)</pre> </p> <p> <code>nodeF . f</code> is just the <a href="https://en.wikipedia.org/wiki/Tacit_programming">point-free</a> version of <code>\x ns -&gt; nodeF (f x) ns</code>, which follows the exact same implementation logic as the above C# <code>Select</code> implementation. </p> <p> The <code>Applicative</code> instance is, I'm afraid, the most complex code you've seen so far in this article series: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Applicative</span>&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;pure&nbsp;=&nbsp;leafF &nbsp;&nbsp;ft&nbsp;&lt;*&gt;&nbsp;xt&nbsp;=&nbsp;treeF&nbsp;(\f&nbsp;ts&nbsp;-&gt;&nbsp;addNodes&nbsp;ts&nbsp;$&nbsp;f&nbsp;&lt;$&gt;&nbsp;xt)&nbsp;ft &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;addNodes&nbsp;ns&nbsp;(TreeFix&nbsp;(Fix&nbsp;(NodeF&nbsp;x&nbsp;xs)))&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;TreeFix&nbsp;(Fix&nbsp;(NodeF&nbsp;x&nbsp;(xs&nbsp;&lt;&gt;&nbsp;(unTreeFix&nbsp;&lt;$&gt;&nbsp;ns))))</pre> </p> <p> I'd be surprised if it's impossible to make this terser, but I thought that it was just complicated enough that I needed to make one of the steps explicit. The <code>addNodes</code> helper function has the type <code>ListFix (TreeFix a) -&gt; TreeFix a -&gt; TreeFix a</code>, and it adds a list of sub-trees to the top node of a tree. It looks worse than it is, but it really just peels off the wrappers (<code>TreeFix</code>, <code>Fix</code>, and <code>NodeF</code>) to access the data (<code>x</code> and <code>xs</code>) of the top node. It then concatenates <code>xs</code> with <code>ns</code>, and puts all the wrappers back on. </p> <p> I have to admit, though, that the <code>Applicative</code> and <code>Monad</code> instance in general are mind-binding to me. The <code>&lt;*&gt;</code> operation, particularly, takes a <em>tree of functions</em> and has to combine it with a <em>tree of values</em>. What does that even mean? How does one do that? </p> <p> Like the above, apparently. I took the <code>Applicative</code> behaviour from <code>Data.Tree</code> and made sure that my implementation is isomorphic. I even have a property to make 'sure' that's the case: </p> <p> <pre>testProperty&nbsp;<span style="color:#a31515;">&quot;Applicative&nbsp;behaves&nbsp;like&nbsp;Data.Tree&quot;</span>&nbsp;$&nbsp;<span style="color:blue;">do</span> &nbsp;&nbsp;<span style="color:#2b91af;">xt</span>&nbsp;::&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;<span style="color:#2b91af;">Integer</span>&nbsp;&lt;-&nbsp;fromTree&nbsp;&lt;$&gt;&nbsp;resize&nbsp;10&nbsp;arbitrary &nbsp;&nbsp;<span style="color:#2b91af;">ft</span>&nbsp;::&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;(<span style="color:#2b91af;">Integer</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">String</span>)&nbsp;&lt;-&nbsp;fromTree&nbsp;&lt;$&gt;&nbsp;resize&nbsp;5&nbsp;arbitrary &nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;actual&nbsp;=&nbsp;ft&nbsp;&lt;*&gt;&nbsp;xt &nbsp;&nbsp;<span style="color:blue;">let</span>&nbsp;expected&nbsp;=&nbsp;toTree&nbsp;ft&nbsp;&lt;*&gt;&nbsp;toTree&nbsp;xt &nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;$&nbsp;expected&nbsp;===&nbsp;toTree&nbsp;actual</pre> </p> <p> The <code>Monad</code> instance looks similar to the <code>Applicative</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Monad</span>&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;t&nbsp;&gt;&gt;=&nbsp;f&nbsp;=&nbsp;treeF&nbsp;(\x&nbsp;ns&nbsp;-&gt;&nbsp;addNodes&nbsp;ns&nbsp;$&nbsp;f&nbsp;x)&nbsp;t &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;addNodes&nbsp;ns&nbsp;(TreeFix&nbsp;(Fix&nbsp;(NodeF&nbsp;x&nbsp;xs)))&nbsp;= &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;TreeFix&nbsp;(Fix&nbsp;(NodeF&nbsp;x&nbsp;(xs&nbsp;&lt;&gt;&nbsp;(unTreeFix&nbsp;&lt;$&gt;&nbsp;ns))))</pre> </p> <p> The <code>addNodes</code> helper function is the same as for <code>&lt;*&gt;</code>, so you may wonder why I didn't extract that as a separate, reusable function. I decided, however, to apply the <a href="https://en.wikipedia.org/wiki/Rule_of_three_(computer_programming)">rule of three</a>, and since, ultimately, <code>addNodes</code> appear only twice, I left them as the implementation details they are. </p> <p> Fortunately, the <code>Foldable</code> instance is easier on the eyes: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Foldable</span>&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;foldMap&nbsp;f&nbsp;=&nbsp;treeF&nbsp;(\x&nbsp;xs&nbsp;-&gt;&nbsp;f&nbsp;x&nbsp;&lt;&gt;&nbsp;fold&nbsp;xs)</pre> </p> <p> Since <code>f</code> is a function of the type <code>a -&gt; m</code>, where <code>m</code> is a <code>Monoid</code> instance, you can use <code>fold</code> and <code>&lt;&gt;</code> to accumulate everything to a single <code>m</code> value. </p> <p> The <code>Traversable</code> instance is similarly terse: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Traversable</span>&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;sequenceA&nbsp;=&nbsp;treeF&nbsp;(\x&nbsp;ns&nbsp;-&gt;&nbsp;nodeF&nbsp;&lt;$&gt;&nbsp;x&nbsp;&lt;*&gt;&nbsp;sequenceA&nbsp;ns)</pre> </p> <p> Finally, you can implement conversions to and from the <code>Tree</code> type from <code>Data.Tree</code>, using <code>ana</code> as the dual of <code>cata</code>: </p> <p> <pre><span style="color:#2b91af;">toTree</span>&nbsp;::&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Tree</span>&nbsp;a toTree&nbsp;=&nbsp;treeF&nbsp;(\x&nbsp;ns&nbsp;-&gt;&nbsp;Node&nbsp;x&nbsp;$&nbsp;toList&nbsp;ns) <span style="color:#2b91af;">fromTree</span>&nbsp;::&nbsp;<span style="color:blue;">Tree</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;a fromTree&nbsp;=&nbsp;TreeFix&nbsp;.&nbsp;ana&nbsp;coalg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;coalg&nbsp;(Node&nbsp;x&nbsp;ns)&nbsp;=&nbsp;NodeF&nbsp;x&nbsp;(fromList&nbsp;ns)</pre> </p> <p> This demonstrates that <code>TreeFix</code> is isomorphic to <code>Tree</code>, which again establishes that <code>treeF</code> and <code>foldTree</code> are equivalent. </p> <h3 id="7180d37efb404a70b707f8c9b8639a35"> Relationships <a href="#7180d37efb404a70b707f8c9b8639a35" title="permalink">#</a> </h3> <p> In this series, you've seen various examples of catamorphisms of structures that have no folds, catamorphisms that coincide with folds, and catamorphisms that are more general than the fold. The introduction to the series included this diagram: </p> <p> <img src="/content/binary/catamorphism-and-fold-relations.png" alt="Catamorphisms and folds as sets, for various sum types."> </p> <p> The <a href="/2019/06/03/either-catamorphism">Either catamorphism</a> is another example of a catamorphism that is more general than the fold, but that one turned out to be identical to the <em>bifold</em>. That's not the case here, because <code>TreeFix</code> isn't a <code>Bifoldable</code> instance at all. </p> <p> There are operations on trees that you can implement with a fold, but some that you can't. Consider the tree in shown in the diagram at the beginning of the article. This is also the tree that the above C# examples use. In Haskell, using <code>TreeFix</code>, you can define that tree like this: </p> <p> <pre>tree&nbsp;= &nbsp;&nbsp;nodeF&nbsp;42 &nbsp;&nbsp;&nbsp;&nbsp;(consF&nbsp;(nodeF&nbsp;1337 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(consF&nbsp;(leafF&nbsp;(-3))&nbsp;nilF))&nbsp;$ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;(nodeF&nbsp;7 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(consF&nbsp;(leafF&nbsp;(-99))&nbsp;$ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;(leafF&nbsp;100)&nbsp;$ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;consF&nbsp;(leafF&nbsp;0)&nbsp;nilF)) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;nilF)</pre> </p> <p> Yes, that almost looks like some Lisp dialect... </p> <p> Since <code>TreeFix</code> is <code>Foldable</code>, and that type class already comes with <code>sum</code> and <code>maximum</code> functions, no further work is required to repeat the first two of the above C# examples: </p> <p> <pre>*Tree Fix List Tree&gt; sum tree 1384 *Tree Fix List Tree&gt; maximum tree 1337</pre> </p> <p> Counting leaves, or measuring the depth of a tree, on the other hand, is impossible with the <code>Foldable</code> instance, but can be implemented using the catamorphism: </p> <p> <pre><span style="color:#2b91af;">countLeaves</span>&nbsp;::&nbsp;<span style="color:blue;">Num</span>&nbsp;n&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;n countLeaves&nbsp;=&nbsp;treeF&nbsp;(\_&nbsp;xs&nbsp;-&gt;&nbsp;<span style="color:blue;">if</span>&nbsp;<span style="color:blue;">null</span>&nbsp;xs&nbsp;<span style="color:blue;">then</span>&nbsp;1&nbsp;<span style="color:blue;">else</span>&nbsp;<span style="color:blue;">sum</span>&nbsp;xs) <span style="color:#2b91af;">treeDepth</span>&nbsp;::&nbsp;(<span style="color:blue;">Ord</span>&nbsp;n,&nbsp;<span style="color:blue;">Num</span>&nbsp;n)&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;<span style="color:blue;">TreeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;n treeDepth&nbsp;=&nbsp;treeF&nbsp;(\_&nbsp;xs&nbsp;-&gt;&nbsp;<span style="color:blue;">if</span>&nbsp;<span style="color:blue;">null</span>&nbsp;xs&nbsp;<span style="color:blue;">then</span>&nbsp;0&nbsp;<span style="color:blue;">else</span>&nbsp;1&nbsp;+&nbsp;<span style="color:blue;">maximum</span>&nbsp;xs)</pre> </p> <p> The reasoning is the same as already explained in the above C# examples. The functions also produce the same results: </p> <p> <pre>*Tree Fix List Tree&gt; countLeaves tree 4 *Tree Fix List Tree&gt; treeDepth tree 2</pre> </p> <p> This, hopefully, illustrates that the catamorphism is more capable, and that the fold is just a (list-biased) specialisation. </p> <h3 id="66f69ba33cef4ed58d01f1f0bafef14a"> Summary <a href="#66f69ba33cef4ed58d01f1f0bafef14a" title="permalink">#</a> </h3> <p> The catamorphism for a tree is just a single function, which is recursively evaluated. It enables you to translate, traverse, and reduce trees in many interesting ways. </p> <p> You can use the catamorphism to implement a (list-biased) fold, including enumerating all nodes as a flat list, but there are operations (such as counting leaves) that you can implement with the catamorphism, but not with the fold. </p> <p> <strong>Next:</strong> <a href="/2019/08/05/rose-tree-catamorphism">Rose tree catamorphism</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Either catamorphism https://blog.ploeh.dk/2019/06/03/either-catamorphism 2019-06-03T06:05:00+00:00 Mark Seemann <div id="post"> <p> <em>The catamorphism for Either is a generalisation of its fold. The catamorphism enables operations not available via fold.</em> </p> <p> This article is part of an <a href="/2019/04/29/catamorphisms">article series about catamorphisms</a>. A catamorphism is a <a href="/2017/10/04/from-design-patterns-to-category-theory">universal abstraction</a> that describes how to digest a data structure into a potentially more compact value. </p> <p> This article presents the catamorphism for <a href="/2018/06/11/church-encoded-either">Either</a> (also known as <em>Result</em>), as well as how to identify it. The beginning of this article presents the catamorphism in C#, with examples. The rest of the article describes how to deduce the catamorphism. This part of the article presents my work in <a href="https://www.haskell.org">Haskell</a>. Readers not comfortable with Haskell can just read the first part, and consider the rest of the article as an optional appendix. </p> <p> <em>Either</em> is a <a href="https://bartoszmilewski.com/2014/01/14/functors-are-containers">data container</a> that models two mutually exclusive results. It's often used to return values that may be either correct (<em>right</em>), or incorrect (<em>left</em>). In statically typed functional programming with a preference for total functions, Either offers a saner, more reasonable way to model success and error results than throwing exceptions. </p> <h3 id="d8214c38aac04ee7b80b9352d57d3bd1"> C# catamorphism <a href="#d8214c38aac04ee7b80b9352d57d3bd1" title="permalink">#</a> </h3> <p> This article uses <a href="/2018/06/11/church-encoded-either">the Church encoding of Either</a>. The catamorphism for Either is the <code>Match</code> method: </p> <p> <pre><span style="color:#2b91af;">T</span>&nbsp;Match&lt;<span style="color:#2b91af;">T</span>&gt;(<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">L</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;&nbsp;onLeft,&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">R</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;&nbsp;onRight);</pre> </p> <p> Until this article, all previous catamorphisms have been a pair made from an initial value and a function. The Either catamorphism is a generalisation, since it's a pair of functions. One function handles the case where there's a <em>left</em> value, and the other function handles the case where there's a <em>right</em> value. Both functions must return the same, unifying type, which is often a string or something similar that can be shown in a user interface: </p> <p> <pre>&gt; <span style="color:#2b91af;">IEither</span>&lt;<span style="color:#2b91af;">TimeSpan</span>,&nbsp;<span style="color:blue;">int</span>&gt;&nbsp;e&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Left</span>&lt;<span style="color:#2b91af;">TimeSpan</span>,&nbsp;<span style="color:blue;">int</span>&gt;(<span style="color:#2b91af;">TimeSpan</span>.FromMinutes(3)); &gt; e.Match(ts&nbsp;=&gt;&nbsp;ts.ToString(),&nbsp;i&nbsp;=&gt;&nbsp;i.ToString()) "00:03:00" &gt; <span style="color:#2b91af;">IEither</span>&lt;<span style="color:#2b91af;">TimeSpan</span>,&nbsp;<span style="color:blue;">int</span>&gt;&nbsp;e&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Right</span>&lt;<span style="color:#2b91af;">TimeSpan</span>,&nbsp;<span style="color:blue;">int</span>&gt;(42); &gt; e.Match(ts&nbsp;=&gt;&nbsp;ts.ToString(),&nbsp;i&nbsp;=&gt;&nbsp;i.ToString()) "42"</pre> </p> <p> You'll often see examples like the above that turns both left and right cases into strings or something that can be represented as a unifying response type, but this is in no way required. If you can come up with a unifying type, you can convert both cases to that type: </p> <p> <pre>&gt; <span style="color:#2b91af;">IEither</span>&lt;<span style="color:#2b91af;">Guid</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;e&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Left</span>&lt;<span style="color:#2b91af;">Guid</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#2b91af;">Guid</span>.NewGuid()); &gt; e.Match(g&nbsp;=&gt;&nbsp;g.ToString().Count(c&nbsp;=&gt;&nbsp;<span style="color:#a31515;">&#39;a&#39;</span>&nbsp;&lt;=&nbsp;c),&nbsp;s&nbsp;=&gt;&nbsp;s.Length) 12 &gt; <span style="color:#2b91af;">IEither</span>&lt;<span style="color:#2b91af;">Guid</span>,&nbsp;<span style="color:blue;">string</span>&gt;&nbsp;e&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Right</span>&lt;<span style="color:#2b91af;">Guid</span>,&nbsp;<span style="color:blue;">string</span>&gt;(<span style="color:#a31515;">&quot;foo&quot;</span>); &gt; e.Match(g&nbsp;=&gt;&nbsp;g.ToString().Count(c&nbsp;=&gt;&nbsp;<span style="color:#a31515;">&#39;a&#39;</span>&nbsp;&lt;=&nbsp;c),&nbsp;s&nbsp;=&gt;&nbsp;s.Length) 3</pre> </p> <p> In the two above examples, you use two different functions that both reduce respectively <code>Guid</code> and <code>string</code> values to a number. The function that turns <code>Guid</code> values into a number counts how many of the hexadecimal digits that are greater than or equal to A (10). The other function simply returns the length of the <code>string</code>, if it's there. That example makes little sense, but the <code>Match</code> method doesn't care about that. </p> <p> In practical use, Either is often used for error handling. The <a href="/2018/06/11/church-encoded-either">article on the Church encoding of Either</a> contains an example. </p> <h3 id="99e1027823114e95bebf81c08d35779f"> Either F-Algebra <a href="#99e1027823114e95bebf81c08d35779f" title="permalink">#</a> </h3> <p> As in the <a href="/2019/05/27/list-catamorphism">previous article</a>, I'll use <code>Fix</code> and <code>cata</code> as explained in <a href="https://bartoszmilewski.com">Bartosz Milewski</a>'s excellent <a href="https://bartoszmilewski.com/2017/02/28/f-algebras/">article on F-Algebras</a>. </p> <p> While F-Algebras and fixed points are mostly used for recursive data structures, you can also define an F-Algebra for a non-recursive data structure. You already saw an example of that in the articles about <a href="/2019/05/06/boolean-catamorphism">Boolean catamorphism</a> and <a href="/2019/05/20/maybe-catamorphism">Maybe catamorphism</a>. The difference between e.g. Maybe values and Either is that both cases of Either carry a value. You can model this as a <code>Functor</code> with both a carrier type and two type arguments for the data that Either may contain: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;EitherF&nbsp;l&nbsp;r&nbsp;c&nbsp;=&nbsp;LeftF&nbsp;l&nbsp;|&nbsp;RightF&nbsp;r&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;(<span style="color:blue;">EitherF</span>&nbsp;l&nbsp;r)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;&nbsp;(LeftF&nbsp;l)&nbsp;=&nbsp;LeftF&nbsp;l &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;(RightF&nbsp;r)&nbsp;=&nbsp;RightF&nbsp;r</pre> </p> <p> I chose to call the 'data types' <code>l</code> (for <em>left</em>) and <code>r</code> (for <em>right</em>), and the carrier type <code>c</code> (for <em>carrier</em>). As was also the case with <code>BoolF</code> and <code>MaybeF</code>, the <code>Functor</code> instance ignores the map function because the carrier type is missing from both the <code>LeftF</code> case and the <code>RightF</code> case. Like the <code>Functor</code> instances for <code>BoolF</code> and <code>MaybeF</code>, it'd seem that nothing happens, but at the type level, this is still a translation from <code>EitherF l r c</code> to <code>EitherF l r c1</code>. Not much of a function, perhaps, but definitely an <em>endofunctor</em>. </p> <p> As was also the case when deducing the Maybe and List catamorphisms, Haskell isn't too happy about defining instances for a type like <code>Fix (EitherF l r)</code>. To address that problem, you can introduce a <code>newtype</code> wrapper: </p> <p> <pre><span style="color:blue;">newtype</span>&nbsp;EitherFix&nbsp;l&nbsp;r&nbsp;=&nbsp;EitherFix&nbsp;{&nbsp;unEitherFix&nbsp;::&nbsp;Fix&nbsp;(EitherF&nbsp;l&nbsp;r)&nbsp;}&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> You can define <code>Functor</code>, <code>Applicative</code>, <code>Monad</code>, etc. instances for this type without resorting to any funky GHC extensions. Keep in mind that ultimately, the purpose of all this code is just to figure out what the catamorphism looks like. This code isn't intended for actual use. </p> <p> A pair of helper functions make it easier to define <code>EitherFix</code> values: </p> <p> <pre><span style="color:#2b91af;">leftF</span>&nbsp;::&nbsp;l&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">EitherFix</span>&nbsp;l&nbsp;r leftF&nbsp;=&nbsp;EitherFix&nbsp;.&nbsp;Fix&nbsp;.&nbsp;LeftF <span style="color:#2b91af;">rightF</span>&nbsp;::&nbsp;r&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">EitherFix</span>&nbsp;l&nbsp;r rightF&nbsp;=&nbsp;EitherFix&nbsp;.&nbsp;Fix&nbsp;.&nbsp;RightF</pre> </p> <p> With those functions, you can create <code>EitherFix</code> values: </p> <p> <pre>Prelude Data.UUID Data.UUID.V4 Fix Either&gt; leftF &lt;$&gt; nextRandom EitherFix {unEitherFix = Fix (LeftF e65378c2-0d6e-47e0-8bcb-7cc29d185fad)} Prelude Data.UUID Data.UUID.V4 Fix Either&gt; rightF "foo" EitherFix {unEitherFix = Fix (RightF "foo")}</pre> </p> <p> That's all you need to identify the catamorphism. </p> <h3 id="67d13d05a8564f3481e181d0c32b3165"> Haskell catamorphism <a href="#67d13d05a8564f3481e181d0c32b3165" title="permalink">#</a> </h3> <p> At this point, you have two out of three elements of an F-Algebra. You have an endofunctor (<code>EitherF l r</code>), and an object <code>c</code>, but you still need to find a morphism <code>EitherF l r c -&gt; c</code>. Notice that the algebra you have to find is the function that reduces the functor to its <em>carrier type</em> <code>c</code>, not the 'data types' <code>l</code> and <code>r</code>. This takes some time to get used to, but that's how catamorphisms work. This doesn't mean, however, that you get to ignore <code>l</code> and <code>r</code>, as you'll see. </p> <p> As in the previous articles, start by writing a function that will become the catamorphism, based on <code>cata</code>: </p> <p> <pre>eitherF&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unEitherFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;(LeftF&nbsp;l)&nbsp;=&nbsp;<span style="color:blue;">undefined</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(RightF&nbsp;r)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> While this compiles, with its <code>undefined</code> implementations, it obviously doesn't do anything useful. I find, however, that it helps me think. How can you return a value of the type <code>c</code> from the <code>LeftF</code> case? You could pass an argument to the <code>eitherF</code> function: </p> <p> <pre>eitherF&nbsp;fl&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unEitherFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;(LeftF&nbsp;l)&nbsp;=&nbsp;fl&nbsp;l &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(RightF&nbsp;r)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> While you could, technically, pass an argument of the type <code>c</code> to <code>eitherF</code> and then return that value from the <code>LeftF</code> case, that would mean that you would ignore the <code>l</code> value. This would be incorrect, so instead, make the argument a function and call it with <code>l</code>. Likewise, you can deal with the <code>RightF</code> case in the same way: </p> <p> <pre><span style="color:#2b91af;">eitherF</span>&nbsp;::&nbsp;(l&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;(r&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">EitherFix</span>&nbsp;l&nbsp;r&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c eitherF&nbsp;fl&nbsp;fr&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unEitherFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;(LeftF&nbsp;l)&nbsp;=&nbsp;fl&nbsp;l &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(RightF&nbsp;r)&nbsp;=&nbsp;fr&nbsp;r</pre> </p> <p> This works. Since <code>cata</code> has the type <code>Functor f =&gt; (f a -&gt; a) -&gt; Fix f -&gt; a</code>, that means that <code>alg</code> has the type <code>f a -&gt; a</code>. In the case of <code>EitherF</code>, the compiler infers that the <code>alg</code> function has the type <code>EitherF l r c -&gt; c</code>, which is just what you need! </p> <p> You can now see what the carrier type <code>c</code> is for. It's the type that the algebra extracts, and thus the type that the catamorphism returns. </p> <p> This, then, is the catamorphism for Either. As has been consistent so far, it's a pair, but now made from two functions. As you've seen repeatedly, this isn't the only possible catamorphism, since you can, for example, trivially flip the arguments to <code>eitherF</code>. </p> <p> I've chosen the representation shown here because it's isomorphic to the <code>either</code> function from Haskell's built-in <code>Data.Either</code> module, which calls the function the "Case analysis for the <code>Either</code> type". Both of these functions (<code>eitherF</code> and <code>either</code>) are equivalent to the above C# <code>Match</code> method. </p> <h3 id="77731d63c79543b0994c06721521b6f3"> Basis <a href="#77731d63c79543b0994c06721521b6f3" title="permalink">#</a> </h3> <p> You can implement most other useful functionality with <code>eitherF</code>. Here's the <code>Bifunctor</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Bifunctor</span>&nbsp;<span style="color:blue;">EitherFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;bimap&nbsp;f&nbsp;s&nbsp;=&nbsp;eitherF&nbsp;(leftF&nbsp;.&nbsp;f)&nbsp;(rightF&nbsp;.&nbsp;s)</pre> </p> <p> From that instance, the <code>Functor</code> instance trivially follows: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;(<span style="color:blue;">EitherFix</span>&nbsp;l)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;=&nbsp;second</pre> </p> <p> On top of <code>Functor</code> you can add <code>Applicative</code>: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Applicative</span>&nbsp;(<span style="color:blue;">EitherFix</span>&nbsp;l)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;pure&nbsp;=&nbsp;rightF &nbsp;&nbsp;f&nbsp;&lt;*&gt;&nbsp;x&nbsp;=&nbsp;eitherF&nbsp;leftF&nbsp;(&lt;$&gt;&nbsp;x)&nbsp;f</pre> </p> <p> Notice that the <code>&lt;*&gt;</code> implementation is similar to to the <code>&lt;*&gt;</code> implementation for <code>MaybeFix</code>. The same is the case for the <code>Monad</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Monad</span>&nbsp;(<span style="color:blue;">EitherFix</span>&nbsp;l)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;x&nbsp;&gt;&gt;=&nbsp;f&nbsp;=&nbsp;eitherF&nbsp;leftF&nbsp;f&nbsp;x</pre> </p> <p> Not only is <code>EitherFix</code> <code>Foldable</code>, it's <code>Bifoldable</code>: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Bifoldable</span>&nbsp;<span style="color:blue;">EitherFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;bifoldMap&nbsp;=&nbsp;eitherF</pre> </p> <p> Notice, interestingly, that <code>bifoldMap</code> is identical to <code>eitherF</code>. </p> <p> The <code>Bifoldable</code> instance enables you to trivially implement the <code>Foldable</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Foldable</span>&nbsp;(<span style="color:blue;">EitherFix</span>&nbsp;l)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;foldMap&nbsp;=&nbsp;bifoldMap&nbsp;mempty</pre> </p> <p> You may find the presence of <code>mempty</code> puzzling, since <code>bifoldMap</code> (or <code>eitherF</code>; they're identical) takes as arguments two functions. Is <code>mempty</code> a function? </p> <p> Yes, <code>mempty</code> can be a function. Here, it is. There's a <code>Monoid</code> instance for any function <code>a -&gt; m</code>, where <code>m</code> is a <code>Monoid</code> instance, and <code>mempty</code> is the identity for that monoid. That's the instance in use here. </p> <p> Just as <code>EitherFix</code> is <code>Bifoldable</code>, it's also <code>Bitraversable</code>: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Bitraversable</span>&nbsp;<span style="color:blue;">EitherFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;bitraverse&nbsp;fl&nbsp;fr&nbsp;=&nbsp;eitherF&nbsp;(<span style="color:blue;">fmap</span>&nbsp;leftF&nbsp;.&nbsp;fl)&nbsp;(<span style="color:blue;">fmap</span>&nbsp;rightF&nbsp;.&nbsp;fr)</pre> </p> <p> You can comfortably implement the <code>Traversable</code> instance based on the <code>Bitraversable</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Traversable</span>&nbsp;(<span style="color:blue;">EitherFix</span>&nbsp;l)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;sequenceA&nbsp;=&nbsp;bisequenceA&nbsp;.&nbsp;first&nbsp;pure</pre> </p> <p> Finally, you can implement conversions to and from the standard <code>Either</code> type, using <code>ana</code> as the dual of <code>cata</code>: </p> <p> <pre><span style="color:#2b91af;">toEither</span>&nbsp;::&nbsp;<span style="color:blue;">EitherFix</span>&nbsp;l&nbsp;r&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">Either</span>&nbsp;l&nbsp;r toEither&nbsp;=&nbsp;eitherF&nbsp;Left&nbsp;Right <span style="color:#2b91af;">fromEither</span>&nbsp;::&nbsp;<span style="color:#2b91af;">Either</span>&nbsp;a&nbsp;b&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">EitherFix</span>&nbsp;a&nbsp;b fromEither&nbsp;=&nbsp;EitherFix&nbsp;.&nbsp;ana&nbsp;coalg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;coalg&nbsp;&nbsp;(Left&nbsp;l)&nbsp;=&nbsp;&nbsp;LeftF&nbsp;l &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;coalg&nbsp;(Right&nbsp;r)&nbsp;=&nbsp;RightF&nbsp;r</pre> </p> <p> This demonstrates that <code>EitherFix</code> is isomorphic to <code>Either</code>, which again establishes that <code>eitherF</code> and <code>either</code> are equivalent. </p> <h3 id="00a274a2317548538521d4a171191230"> Relationships <a href="#00a274a2317548538521d4a171191230" title="permalink">#</a> </h3> <p> In this series, you've seen various examples of catamorphisms of structures that have no folds, catamorphisms that coincide with folds, and now a catamorphism that is more general than the fold. The introduction to the series included this diagram: </p> <p> <img src="/content/binary/catamorphism-and-fold-relations.png" alt="Catamorphisms and folds as sets, for various sum types."> </p> <p> This shows that Boolean values and Peano numbers have catamorphisms, but no folds, whereas for Maybe and List, the fold and the catamorphism is the same. For Either, however, the fold is a special case of the catamorphism. The fold for Either 'pretends' that the left side doesn't exist. Instead, the left value is interpreted as a missing right value. Thus, in order to fold Either values, you must supply a 'fallback' value that will be used in case an Either value isn't a <em>right</em> value: </p> <p> <pre>Prelude Fix Either&gt; e = rightF LT :: EitherFix Integer Ordering Prelude Fix Either&gt; foldr (const . show) "" e "LT" Prelude Fix Either&gt; e = leftF 42 :: EitherFix Integer Ordering Prelude Fix Either&gt; foldr (const . show) "" e ""</pre> </p> <p> In a GHCi session like the above, you can create two Either values of the same type. The <em>right</em> case is an <code>Ordering</code> value, while the <em>left</em> case is an <code>Integer</code> value. </p> <p> With <code>foldr</code>, there's no way to access the <em>left</em> case. While you can access and transform the right <code>Ordering</code> value, the number <code>42</code> is simply ignored during the fold. Instead, the default value <code>""</code> is returned. </p> <p> Contrast this with the catamorphism, which can access both cases: </p> <p> <pre>Prelude Fix Either&gt; e = rightF LT :: EitherFix Integer Ordering Prelude Fix Either&gt; eitherF show show e "LT" Prelude Fix Either&gt; e = leftF 42 :: EitherFix Integer Ordering Prelude Fix Either&gt; eitherF show show e "42"</pre> </p> <p> In a session like this, you recreate the same values, but using the catamorphism <code>eitherF</code>, you can now access and transform both the <em>left</em> and the <em>right</em> cases. In other words, the catamorphism enables you to perform operations not possible with the fold. </p> <p> It's interesting, however, to note that while the fold is a specialisation of the catamorphism, the <em>bifold</em> is identical to the catamorphism. </p> <h3 id="7512bca753b747ad9accde04bf13b6ca"> Summary <a href="#7512bca753b747ad9accde04bf13b6ca" title="permalink">#</a> </h3> <p> The catamorphism for Either is a pair of functions. One function transforms the <em>left</em> case, while the other function transforms the <em>right</em> case. For any Either value, only one of those functions will be used. </p> <p> When I originally encountered the concept of a <em>catamorphism</em>, I found it difficult to distinguish between catamorphism and fold. My problem was, I think, that the tutorials I ran into mostly used linked lists to demonstrate how, <a href="/2019/05/27/list-catamorphism">in that case</a>, the fold <em>is</em> the catamorphism. It turns out, however, that this isn't always the case. A catamorphism is a general abstraction. A fold, on the other hand, seems to me to be mostly related to collections. </p> <p> In this article you saw the first example of a catamorphism that can do more than the fold. For Either, the fold is just a special case of the catamorphism. You also saw, however, how the catamorphism was identical to the <em>bifold</em>. Thus, it's still not entirely clear how these concepts relate. Therefore, in the next article, you'll get an example of a <a href="https://bartoszmilewski.com/2014/01/14/functors-are-containers">container</a> where there's no bifold, and where the catamorphism is, indeed, a generalisation of the fold. </p> <p> <strong>Next:</strong> <a href="/2019/06/10/tree-catamorphism">Tree catamorphism</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. List catamorphism https://blog.ploeh.dk/2019/05/27/list-catamorphism 2019-05-27T06:10:00+00:00 Mark Seemann <div id="post"> <p> <em>The catamorphism for a list is the same as its fold.</em> </p> <p> This article is part of an <a href="/2019/04/29/catamorphisms">article series about catamorphisms</a>. A catamorphism is a <a href="/2017/10/04/from-design-patterns-to-category-theory">universal abstraction</a> that describes how to digest a data structure into a potentially more compact value. </p> <p> This article presents the catamorphism for (linked) lists, and other collections in general. It also shows how to identify it. The beginning of this article presents the catamorphism in C#, with an example. The rest of the article describes how to deduce the catamorphism. This part of the article presents my work in <a href="https://www.haskell.org">Haskell</a>. Readers not comfortable with Haskell can just read the first part, and consider the rest of the article as an optional appendix. </p> <p> The C# part of the article will discuss <code>IEnumerable&lt;T&gt;</code>, while the Haskell part will deal specifically with linked lists. Since C# is a less strict language anyway, we have to make some concessions when we consider how concepts translate. In my experience, the functionality of <code>IEnumerable&lt;T&gt;</code> closely mirrors that of Haskell lists. </p> <h3 id="3190f7181a954b6388d77f61a1dbb928"> C# catamorphism <a href="#3190f7181a954b6388d77f61a1dbb928" title="permalink">#</a> </h3> <p> The .NET base class library defines this <a href="https://docs.microsoft.com/dotnet/api/system.linq.enumerable.aggregate">Aggregate</a> overload: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">TAccumulate</span>&nbsp;Aggregate&lt;<span style="color:#2b91af;">TSource</span>,&nbsp;<span style="color:#2b91af;">TAccumulate</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">TSource</span>&gt;&nbsp;source, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">TAccumulate</span>&nbsp;seed, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">TAccumulate</span>,&nbsp;<span style="color:#2b91af;">TSource</span>,&nbsp;<span style="color:#2b91af;">TAccumulate</span>&gt;&nbsp;func);</pre> </p> <p> This is the catamorphism for linked lists (and, I'll conjecture, for <code>IEnumerable&lt;T&gt;</code> in general). The <a href="/2019/04/29/catamorphisms">introductory article</a> already used it to show several motivating examples, of which I'll only repeat the last: </p> <p> <pre>&gt; <span style="color:blue;">new</span>[]&nbsp;{&nbsp;42,&nbsp;1337,&nbsp;2112,&nbsp;90125,&nbsp;5040,&nbsp;7,&nbsp;1984&nbsp;} . .Aggregate(<span style="color:#2b91af;">Angle</span>.Identity,&nbsp;(a,&nbsp;i)&nbsp;=&gt;&nbsp;a.Add(<span style="color:#2b91af;">Angle</span>.FromDegrees(i))) [{ Angle = 207° }]</pre> </p> <p> In short, the catamorphism is, similar to the previous catamorphisms covered in this article series, a pair made from an initial value and a function. This has been true for both the <a href="/2019/05/13/peano-catamorphism">Peano catamorphism</a> and the <a href="/2019/05/20/maybe-catamorphism">Maybe catamorphism</a>. An initial value is just a value in all three cases, but you may notice that the function in question becomes increasingly elaborate. For <code>IEnumerable&lt;T&gt;</code>, it's a function that takes two values. One of the values are of the type of the input list, i.e. for <code>IEnumerable&lt;TSource&gt;</code> it would be <code>TSource</code>. By elimination you can deduce that this value must come from the input list. The other value is of the type <code>TAccumulate</code>, which implies that it could be the <code>seed</code>, or the result from a previous call to <code>func</code>. </p> <h3 id="46afb325e03743d9ac2c2cf391607f82"> List F-Algebra <a href="#46afb325e03743d9ac2c2cf391607f82" title="permalink">#</a> </h3> <p> As in the <a href="/2019/05/20/maybe-catamorphism">previous article</a>, I'll use <code>Fix</code> and <code>cata</code> as explained in <a href="https://bartoszmilewski.com">Bartosz Milewski</a>'s excellent <a href="https://bartoszmilewski.com/2017/02/28/f-algebras/">article on F-Algebras</a>. The <code>ListF</code> type comes from his article as well, although I've renamed the type arguments: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;ListF&nbsp;a&nbsp;c&nbsp;=&nbsp;NilF&nbsp;|&nbsp;ConsF&nbsp;a&nbsp;c&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;(<span style="color:blue;">ListF</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;NilF&nbsp;&nbsp;=&nbsp;NilF &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;f&nbsp;(ConsF&nbsp;a&nbsp;c)&nbsp;=&nbsp;ConsF&nbsp;a&nbsp;$&nbsp;f&nbsp;c</pre> </p> <p> Like I did with <code>MaybeF</code>, I've named the 'data' type argument <code>a</code>, and the carrier type <code>c</code> (for <em>carrier</em>). Once again, notice that the <code>Functor</code> instance maps over the carrier type <code>c</code>; not over the 'data' type <code>a</code>. </p> <p> As was also the case when deducing the Maybe catamorphism, Haskell isn't too happy about defining instances for a type like <code>Fix (ListF a)</code>. To address that problem, you can introduce a <code>newtype</code> wrapper: </p> <p> <pre><span style="color:blue;">newtype</span>&nbsp;ListFix&nbsp;a&nbsp;=&nbsp;ListFix&nbsp;{&nbsp;unListFix&nbsp;::&nbsp;Fix&nbsp;(ListF&nbsp;a)&nbsp;}&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> You can define <code>Functor</code>, <code>Applicative</code>, <code>Monad</code>, etc. instances for this type without resorting to any funky GHC extensions. Keep in mind that ultimately, the purpose of all this code is just to figure out what the catamorphism looks like. This code isn't intended for actual use. </p> <p> A few helper functions make it easier to define <code>ListFix</code> values: </p> <p> <pre><span style="color:#2b91af;">nilF</span>&nbsp;::&nbsp;<span style="color:blue;">ListFix</span>&nbsp;a nilF&nbsp;=&nbsp;ListFix&nbsp;$&nbsp;Fix&nbsp;NilF <span style="color:#2b91af;">consF</span>&nbsp;::&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;a consF&nbsp;x&nbsp;=&nbsp;ListFix&nbsp;.&nbsp;Fix&nbsp;.&nbsp;ConsF&nbsp;x&nbsp;.&nbsp;unListFix</pre> </p> <p> With those functions, you can create <code>ListFix</code> linked lists: </p> <p> <pre>Prelude Fix List&gt; nilF ListFix {unListFix = Fix NilF} Prelude Fix List&gt; consF 42 $ consF 1337 $ consF 2112 nilF ListFix {unListFix = Fix (ConsF 42 (Fix (ConsF 1337 (Fix (ConsF 2112 (Fix NilF))))))}</pre> </p> <p> The first example creates an empty list, whereas the second creates a list of three integers, corresponding to <code>[42,1337,2112]</code>. </p> <p> That's all you need to identify the catamorphism. </p> <h3 id="db44e85b27fb40d0b570c53cf4ce1843"> Haskell catamorphism <a href="#db44e85b27fb40d0b570c53cf4ce1843" title="permalink">#</a> </h3> <p> At this point, you have two out of three elements of an F-Algebra. You have an endofunctor (<code>ListF</code>), and an object <code>a</code>, but you still need to find a morphism <code>ListF a c -&gt; c</code>. Notice that the algebra you have to find is the function that reduces the functor to its <em>carrier type</em> <code>c</code>, not the 'data type' <code>a</code>. This takes some time to get used to, but that's how catamorphisms work. This doesn't mean, however, that you get to ignore <code>a</code>, as you'll see. </p> <p> As in the previous article, start by writing a function that will become the catamorphism, based on <code>cata</code>: </p> <p> <pre>listF&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unListFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;NilF&nbsp;&nbsp;=&nbsp;<span style="color:blue;">undefined</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(ConsF&nbsp;h&nbsp;t)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> While this compiles, with its <code>undefined</code> implementations, it obviously doesn't do anything useful. I find, however, that it helps me think. How can you return a value of the type <code>c</code> from the <code>NilF</code> case? You could pass an argument to the <code>listF</code> function: </p> <p> <pre>listF&nbsp;n&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unListFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;NilF&nbsp;&nbsp;=&nbsp;n &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(ConsF&nbsp;h&nbsp;t)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> The <code>ConsF</code> case, contrary to <code>NilF</code>, contains both a head <code>h</code> (of type <code>a</code>) and a tail <code>t</code> (of type <code>c</code>). While you could make the code compile by simply returning <code>t</code>, it'd be incorrect to ignore <code>h</code>. In order to deal with both, you'll need a function that turns both <code>h</code> and <code>t</code> into a value of the type <code>c</code>. You can do this by passing a function to <code>listF</code> and using it: </p> <p> <pre><span style="color:#2b91af;">listF</span>&nbsp;::&nbsp;(a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c listF&nbsp;f&nbsp;n&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unListFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;NilF&nbsp;&nbsp;=&nbsp;n &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(ConsF&nbsp;h&nbsp;t)&nbsp;=&nbsp;f&nbsp;h&nbsp;t</pre> </p> <p> This works. Since <code>cata</code> has the type <code>Functor f =&gt; (f a -&gt; a) -&gt; Fix f -&gt; a</code>, that means that <code>alg</code> has the type <code>f a -&gt; a</code>. In the case of <code>ListF</code>, the compiler infers that the <code>alg</code> function has the type <code>ListF a c -&gt; c</code>, which is just what you need! </p> <p> You can now see what the carrier type <code>c</code> is for. It's the type that the algebra extracts, and thus the type that the catamorphism returns. </p> <p> This, then, is the catamorphism for lists. As has been consistent so far, it's a pair made from an initial value and a function. Once more, this isn't the only possible catamorphism, since you can, for example, trivially flip the arguments to <code>listF</code>: </p> <p> <pre><span style="color:#2b91af;">listF</span>&nbsp;::&nbsp;c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;(a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c listF&nbsp;n&nbsp;f&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unListFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;NilF&nbsp;&nbsp;=&nbsp;n &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(ConsF&nbsp;h&nbsp;t)&nbsp;=&nbsp;f&nbsp;h&nbsp;t</pre> </p> <p> You can also flip the arguments of <code>f</code>: </p> <p> <pre><span style="color:#2b91af;">listF</span>&nbsp;::&nbsp;c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;(c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c listF&nbsp;n&nbsp;f&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unListFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;NilF&nbsp;&nbsp;=&nbsp;n &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(ConsF&nbsp;h&nbsp;t)&nbsp;=&nbsp;f&nbsp;t&nbsp;h</pre> </p> <p> These representations are all isomorphic to each other, but notice that the last variation is similar to the above C# <code>Aggregate</code> overload. The initial <code>n</code> value is the <code>seed</code>, and the function <code>f</code> has the same shape as <code>func</code>. Thus, I consider it reasonable to conjecture that that <code>Aggregate</code> overload is the catamorphism for <code>IEnumerable&lt;T&gt;</code>. </p> <h3 id="3ba9143e87544443b713727eb4ea60ba"> Basis <a href="#3ba9143e87544443b713727eb4ea60ba" title="permalink">#</a> </h3> <p> You can implement most other useful functionality with <code>listF</code>. The rest of this article uses the first of the variations shown above, with the type <code>(a -&gt; c -&gt; c) -&gt; c -&gt; ListFix a -&gt; c</code>. Here's the <code>Semigroup</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Semigroup</span>&nbsp;(<span style="color:blue;">ListFix</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;xs&nbsp;&lt;&gt;&nbsp;ys&nbsp;=&nbsp;listF&nbsp;consF&nbsp;ys&nbsp;xs</pre> </p> <p> The initial value passed to <code>listF</code> is <code>ys</code>, and the function to apply is simply the <code>consF</code> function, thus 'consing' the two lists together. Here's an example of the operation in action: </p> <p> <pre>Prelude Fix List&gt; consF 42 $ consF 1337 nilF &lt;&gt; (consF 2112 $ consF 1 nilF) ListFix {unListFix = Fix (ConsF 42 (Fix (ConsF 1337 (Fix (ConsF 2112 (Fix (ConsF 1 (Fix NilF))))))))}</pre> </p> <p> With a <code>Semigroup</code> instance, it's trivial to also implement the <code>Monoid</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Monoid</span>&nbsp;(<span style="color:blue;">ListFix</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;mempty&nbsp;=&nbsp;nilF</pre> </p> <p> While you <em>could</em> implement <code>mempty</code> with <code>listF</code> (<code>mempty = listF (const id) nilF nilF</code>), that'd be overcomplicated. Just because you can implement all functionality using <code>listF</code>, it doesn't mean that you should, if a simpler alternative exists. </p> <p> You can, on the other hand, use <code>listF</code> for the <code>Functor</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;f&nbsp;=&nbsp;listF&nbsp;(\h&nbsp;l&nbsp;-&gt;&nbsp;consF&nbsp;(f&nbsp;h)&nbsp;l)&nbsp;nilF</pre> </p> <p> You could write the function you pass to <code>listF</code> in a point-free fashion as <code>consF . f</code>, but I thought it'd be easier to follow what happens when written as an explicit lambda expression. The function receives a 'current value' <code>h</code>, as well as the part of the list which has already been translated <code>l</code>. Use <code>f</code> to translate <code>h</code>, and <code>consF</code> the result unto <code>l</code>. </p> <p> You can add <code>Applicative</code> and <code>Monad</code> instances in a similar fashion: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Applicative</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;pure&nbsp;x&nbsp;=&nbsp;consF&nbsp;x&nbsp;nilF &nbsp;&nbsp;fs&nbsp;&lt;*&gt;&nbsp;xs&nbsp;=&nbsp;listF&nbsp;(\f&nbsp;acc&nbsp;-&gt;&nbsp;(f&nbsp;&lt;$&gt;&nbsp;xs)&nbsp;&lt;&gt;&nbsp;acc)&nbsp;nilF&nbsp;fs <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Monad</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;xs&nbsp;&gt;&gt;=&nbsp;f&nbsp;=&nbsp;listF&nbsp;(\x&nbsp;acc&nbsp;-&gt;&nbsp;f&nbsp;x&nbsp;&lt;&gt;&nbsp;acc)&nbsp;nilF&nbsp;xs</pre> </p> <p> What may be more interesting, however, is the <code>Foldable</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Foldable</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">foldr</span>&nbsp;=&nbsp;listF</pre> </p> <p> The demonstrates that <code>listF</code> and <code>foldr</code> is the same. </p> <p> Next, you can also add a <code>Traversable</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Traversable</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;sequenceA&nbsp;=&nbsp;listF&nbsp;(\x&nbsp;acc&nbsp;-&gt;&nbsp;consF&nbsp;&lt;$&gt;&nbsp;x&nbsp;&lt;*&gt;&nbsp;acc)&nbsp;(pure&nbsp;nilF)</pre> </p> <p> Finally, you can implement conversions to and from the standard list <code>[]</code> type, using <code>ana</code> as the dual of <code>cata</code>: </p> <p> <pre><span style="color:#2b91af;">toList</span>&nbsp;::&nbsp;<span style="color:blue;">ListFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;[a] toList&nbsp;=&nbsp;listF&nbsp;<span style="color:#2b91af;">(:)</span>&nbsp;<span style="color:blue;">[]</span> <span style="color:#2b91af;">fromList</span>&nbsp;::&nbsp;[a]&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">ListFix</span>&nbsp;a fromList&nbsp;=&nbsp;ListFix&nbsp;.&nbsp;ana&nbsp;coalg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;coalg&nbsp;&nbsp;&nbsp;<span style="color:blue;">[]</span>&nbsp;&nbsp;=&nbsp;NilF &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;coalg&nbsp;(h:t)&nbsp;=&nbsp;ConsF&nbsp;h&nbsp;t</pre> </p> <p> This demonstrates that <code>ListFix</code> is isomorphic to <code>[]</code>, which again establishes that <code>listF</code> and <code>foldr</code> are equivalent. </p> <h3 id="616c31c72b1f43cca647760d9fa8b226"> Summary <a href="#616c31c72b1f43cca647760d9fa8b226" title="permalink">#</a> </h3> <p> The catamorphism for lists is a pair made from an initial value and a function. One variation is equal to <code>foldr</code>. Like Maybe, the catamorphism is the same as the fold. </p> <p> In C#, this function corresponds to the <code>Aggregate</code> extension method identified above. </p> <p> You've now seen two examples where the catamorphism coincides with the fold. You've also seen examples (<a href="/2019/05/06/boolean-catamorphism">Boolean catamorphism</a> and <a href="/2019/05/13/peano-catamorphism">Peano catamorphism</a>) where there's a catamorphism, but no fold at all. In the next article, you'll see an example of a <a href="https://bartoszmilewski.com/2014/01/14/functors-are-containers">container</a> that has both catamorphism and fold, but where the catamorphism is more general than the fold. </p> <p> <strong>Next:</strong> <a href="/2019/06/03/either-catamorphism">Either catamorphism</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Maybe catamorphism https://blog.ploeh.dk/2019/05/20/maybe-catamorphism 2019-05-20T06:04:00+00:00 Mark Seemann <div id="post"> <p> <em>The catamorphism for Maybe is just a simplification of its fold.</em> </p> <p> This article is part of an <a href="/2019/04/29/catamorphisms">article series about catamorphisms</a>. A catamorphism is a <a href="/2017/10/04/from-design-patterns-to-category-theory">universal abstraction</a> that describes how to digest a data structure into a potentially more compact value. </p> <p> This article presents the catamorphism for <a href="/2018/03/26/the-maybe-functor">Maybe</a>, as well as how to identify it. The beginning of this article presents the catamorphism in C#, with examples. The rest of the article describes how to deduce the catamorphism. This part of the article presents my work in <a href="https://www.haskell.org">Haskell</a>. Readers not comfortable with Haskell can just read the first part, and consider the rest of the article as an optional appendix. </p> <p> <em>Maybe</em> is a <a href="https://bartoszmilewski.com/2014/01/14/functors-are-containers">data container</a> that models the absence or presence of a value. <a href="/2015/11/13/null-has-no-type-but-maybe-has">Contrary to null, Maybe has a type</a>, so offers a sane and reasonable way to model that situation. </p> <h3 id="1feeee3382ff44d182e0a28a33f8f80a"> C# catamorphism <a href="#1feeee3382ff44d182e0a28a33f8f80a" title="permalink">#</a> </h3> <p> This article uses <a href="/2018/06/04/church-encoded-maybe">Church-encoded Maybe</a>. Other, <a href="/2018/03/26/the-maybe-functor">alternative implementations of Maybe are possible</a>. The catamorphism for Maybe is the <code>Match</code> method: </p> <p> <pre><span style="color:#2b91af;">TResult</span>&nbsp;Match&lt;<span style="color:#2b91af;">TResult</span>&gt;(<span style="color:#2b91af;">TResult</span>&nbsp;nothing,&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;just);</pre> </p> <p> Like the <a href="/2019/05/13/peano-catamorphism">Peano catamorphism</a>, the Maybe catamorphism is a pair of a value and a function. The <code>nothing</code> value corresponds to the absence of data, whereas the <code>just</code> function handles the presence of data. </p> <p> Given, for example, a Maybe containing a number, you can use <code>Match</code> to <a href="/2019/02/04/how-to-get-the-value-out-of-the-monad">get the value out of the Maybe</a>: </p> <p> <pre>&gt; <span style="color:#2b91af;">IMaybe</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;maybe&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Just</span>&lt;<span style="color:blue;">int</span>&gt;(42); &gt; maybe.Match(0,&nbsp;x&nbsp;=&gt;&nbsp;x) 42 &gt; <span style="color:#2b91af;">IMaybe</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;maybe&nbsp;=&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Nothing</span>&lt;<span style="color:blue;">int</span>&gt;(); &gt; maybe.Match(0,&nbsp;x&nbsp;=&gt;&nbsp;x) 0</pre> </p> <p> The functionality is, however, more useful than a simple <em>get-value-or-default</em> operation. Often, you don't have a good default value for the type potentially wrapped in a Maybe object. In the core of your application architecture, it may not be clear how to deal with, say, the absence of a <code>Reservation</code> object, whereas at the boundary of your system, it's evident how to convert both absence and presence of data into a unifying type, such as an HTTP response: </p> <p> <pre><span style="color:#2b91af;">Maybe</span>&lt;<span style="color:#2b91af;">Reservation</span>&gt;&nbsp;maybe&nbsp;=&nbsp;<span style="color:green;">//&nbsp;...</span> <span style="color:blue;">return</span>&nbsp;maybe &nbsp;&nbsp;&nbsp;&nbsp;.Select(r&nbsp;=&gt;&nbsp;Repository.Create(r)) &nbsp;&nbsp;&nbsp;&nbsp;.Match&lt;<span style="color:#2b91af;">IHttpActionResult</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;nothing:&nbsp;Content( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">HttpStatusCode</span>.InternalServerError, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">HttpError</span>(<span style="color:#a31515;">&quot;Couldn&#39;t&nbsp;accept.&quot;</span>)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;just:&nbsp;id&nbsp;=&gt;&nbsp;Ok(id));</pre> </p> <p> This enables you to avoid special cases, such as null <code>Reservation</code> objects, or magic numbers like <code>-1</code> to indicate the absence of <code>id</code> values. At the boundary of an HTTP-based application, you know that you must return an HTTP response. That's the unifying type, so you can return <code>200 OK</code> with a reservation ID in the response body when data is present, and <code>500 Internal Server Error</code> when data is absent. </p> <h3 id="87d91da8944f4eb5b8b24e9ea20d3e1b"> Maybe F-Algebra <a href="#87d91da8944f4eb5b8b24e9ea20d3e1b" title="permalink">#</a> </h3> <p> As in the <a href="/2019/05/13/peano-catamorphism">previous article</a>, I'll use <code>Fix</code> and <code>cata</code> as explained in <a href="https://bartoszmilewski.com">Bartosz Milewski</a>'s excellent <a href="https://bartoszmilewski.com/2017/02/28/f-algebras/">article on F-Algebras</a>. </p> <p> While F-Algebras and fixed points are mostly used for recursive data structures, you can also define an F-Algebra for a non-recursive data structure. You already saw an example of that in the article about <a href="/2019/05/06/boolean-catamorphism">Boolean catamorphism</a>. The difference between Boolean values and Maybe is that the <em>just</em> case of Maybe carries a value. You can model this as a <code>Functor</code> with both a carrier type and a type argument for the data that Maybe may contain: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;MaybeF&nbsp;a&nbsp;c&nbsp;=&nbsp;NothingF&nbsp;|&nbsp;JustF&nbsp;a&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;(<span style="color:blue;">MaybeF</span>&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;&nbsp;NothingF&nbsp;=&nbsp;NothingF &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;(JustF&nbsp;x)&nbsp;=&nbsp;JustF&nbsp;x</pre> </p> <p> I chose to call the 'data type' <code>a</code> and the carrier type <code>c</code> (for <em>carrier</em>). As was also the case with <code>BoolF</code>, the <code>Functor</code> instance ignores the map function because the carrier type is missing from both the <code>NothingF</code> case and the <code>JustF</code> case. Like the <code>Functor</code> instance for <code>BoolF</code>, it'd seem that nothing happens, but at the type level, this is still a translation from <code>MaybeF a c</code> to <code>MaybeF a c1</code>. Not much of a function, perhaps, but definitely an <em>endofunctor</em>. </p> <p> In the previous articles, it was possible to work directly with the fixed points of both functors; i.e. <code>Fix BoolF</code> and <code>Fix NatF</code>. Haskell isn't happy about attempts to define various instances for <code>Fix (MaybeF a)</code>, so in order to make this easier, you can define a <code>newtype</code> wrapper: </p> <p> <pre><span style="color:blue;">newtype</span>&nbsp;MaybeFix&nbsp;a&nbsp;= &nbsp;&nbsp;MaybeFix&nbsp;{&nbsp;unMaybeFix&nbsp;::&nbsp;Fix&nbsp;(MaybeF&nbsp;a)&nbsp;}&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>)</pre> </p> <p> In order to make it easier to work with <code>MaybeFix</code> you can add helper functions to create values: </p> <p> <pre><span style="color:#2b91af;">nothingF</span>&nbsp;::&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;a nothingF&nbsp;=&nbsp;MaybeFix&nbsp;$&nbsp;Fix&nbsp;NothingF <span style="color:#2b91af;">justF</span>&nbsp;::&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;a justF&nbsp;=&nbsp;MaybeFix&nbsp;.&nbsp;Fix&nbsp;.&nbsp;JustF</pre> </p> <p> You can now create <code>MaybeFix</code> values to your heart's content: </p> <p> <pre>Prelude Fix Maybe&gt; justF 42 MaybeFix {unMaybeFix = Fix (JustF 42)} Prelude Fix Maybe&gt; nothingF MaybeFix {unMaybeFix = Fix NothingF}</pre> </p> <p> That's all you need to identify the catamorphism. </p> <h3 id="24db4c715d1f4540bd8f87604819952f"> Haskell catamorphism <a href="#24db4c715d1f4540bd8f87604819952f" title="permalink">#</a> </h3> <p> At this point, you have two out of three elements of an F-Algebra. You have an endofunctor (<code>MaybeF</code>), and an object <code>a</code>, but you still need to find a morphism <code>MaybeF a c -&gt; c</code>. Notice that the algebra you have to find is the function that reduces the functor to its <em>carrier type</em> <code>c</code>, not the 'data type' <code>a</code>. This takes some time to get used to, but that's how catamorphisms work. This doesn't mean, however, that you get to ignore <code>a</code>, as you'll see. </p> <p> As in the previous article, start by writing a function that will become the catamorphism, based on <code>cata</code>: </p> <p> <pre>maybeF&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unMaybeFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;NothingF&nbsp;=&nbsp;<span style="color:blue;">undefined</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(JustF&nbsp;x)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> While this compiles, with its <code>undefined</code> implementations, it obviously doesn't do anything useful. I find, however, that it helps me think. How can you return a value of the type <code>c</code> from the <code>NothingF</code> case? You could pass an argument to the <code>maybeF</code> function: </p> <p> <pre>maybeF&nbsp;n&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unMaybeFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;NothingF&nbsp;=&nbsp;n &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(JustF&nbsp;x)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> The <code>JustF</code> case, contrary to <code>NothingF</code>, already contains a value, and it'd be incorrect to ignore it. On the other hand, <code>x</code> is a value of type <code>a</code>, and you need to return a value of type <code>c</code>. You'll need a function to perform the conversion, so pass such a function as an argument to <code>maybeF</code> as well: </p> <p> <pre><span style="color:#2b91af;">maybeF</span>&nbsp;::&nbsp;c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;(a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c maybeF&nbsp;n&nbsp;f&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unMaybeFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;NothingF&nbsp;=&nbsp;n &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(JustF&nbsp;x)&nbsp;=&nbsp;f&nbsp;x</pre> </p> <p> This works. Since <code>cata</code> has the type <code>Functor f =&gt; (f a -&gt; a) -&gt; Fix f -&gt; a</code>, that means that <code>alg</code> has the type <code>f a -&gt; a</code>. In the case of <code>MaybeF</code>, the compiler infers that the <code>alg</code> function has the type <code>MaybeF a c -&gt; c</code>, which is just what you need! </p> <p> You can now see what the carrier type <code>c</code> is for. It's the type that the algebra extracts, and thus the type that the catamorphism returns. </p> <p> Notice that <code>maybeF</code>, like the above C# <code>Match</code> method, takes as arguments a pair of a value and a function (together with the Maybe value itself). Those are two representations of the same idea. Furthermore, this is nearly identical to the <code>maybe</code> function in Haskell's <code>Data.Maybe</code> module. I found if fitting, therefore, to name the function <code>maybeF</code>. </p> <h3 id="d8a0eed800de48a994085c419b7b5379"> Basis <a href="#d8a0eed800de48a994085c419b7b5379" title="permalink">#</a> </h3> <p> You can implement most other useful functionality with <code>maybeF</code>. Here's the <code>Functor</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;f&nbsp;=&nbsp;maybeF&nbsp;nothingF&nbsp;(justF&nbsp;.&nbsp;f)</pre> </p> <p> Since <code>fmap</code> should be a structure-preserving map, you'll have to map the <em>nothing</em> case to the <em>nothing</em> case, and <em>just</em> to <em>just</em>. When calling <code>maybeF</code>, you must supply a value for the <em>nothing</em> case and a function to deal with the <em>just</em> case. The <em>nothing</em> case is easy to handle: just use <code>nothingF</code>. </p> <p> In the <em>just</em> case, first apply the function <code>f</code> to map from <code>a</code> to <code>b</code>, and then use <code>justF</code> to wrap the <code>b</code> value in a new <code>MaybeFix</code> container to get <code>MaybeFix b</code>. </p> <p> <code>Applicative</code> is a little harder, but not much: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Applicative</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;pure&nbsp;=&nbsp;justF &nbsp;&nbsp;f&nbsp;&lt;*&gt;&nbsp;x&nbsp;=&nbsp;maybeF&nbsp;nothingF&nbsp;(&lt;$&gt;&nbsp;x)&nbsp;f</pre> </p> <p> The <code>pure</code> function is just <em>justF</em> (pun intended). The <em>apply</em> operator <code>&lt;*&gt;</code> is more complex. </p> <p> Both <code>f</code> and <code>x</code> surrounding <code>&lt;*&gt;</code> are <code>MaybeFix</code> values: <code>f</code> is <code>MaybeFix (a -&gt; b)</code>, and <code>x</code> is <code>MaybeFix a</code>. While it's becoming increasingly clear that you can use a catamorphism like <code>maybeF</code> to implement most other functionality, to which <code>MaybeFix</code> value should you apply it? To <code>f</code> or <code>x</code>? </p> <p> Both are possible, but the code looks (in my opinion) more readable if you apply it to <code>f</code>. Again, when <code>f</code> is <em>nothing</em>, return <code>nothingF</code>. When <code>f</code> is <em>just</em>, use the functor instance to map <code>x</code> (using the infix <code>fmap</code> alias <code>&lt;$&gt;</code>). </p> <p> The <code>Monad</code> instance, on the other hand, is almost trivial: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Monad</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;x&nbsp;&gt;&gt;=&nbsp;f&nbsp;=&nbsp;maybeF&nbsp;nothingF&nbsp;f&nbsp;x</pre> </p> <p> As usual, map <em>nothing</em> to <em>nothing</em> by supplying <code>nothingF</code>. <code>f</code> is already a function that returns a <code>MaybeFix b</code> value, so just use that. </p> <p> The <code>Foldable</code> instance is likewise trivial (although, as you'll see below, you can make it even more trivial): </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Foldable</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;foldMap&nbsp;=&nbsp;maybeF&nbsp;mempty</pre> </p> <p> The <code>foldMap</code> function must return a <code>Monoid</code> instance, so for the <em>nothing</em> case, simply return the identity, <em>mempty</em>. Furthermore, <code>foldMap</code> takes a function <code>a -&gt; m</code>, but since the <code>foldMap</code> implementation is <a href="https://en.wikipedia.org/wiki/Tacit_programming">point-free</a>, you can't 'see' that function as an argument. </p> <p> Finally, for the sake of completeness, here's the <code>Traversable</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Traversable</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;sequenceA&nbsp;=&nbsp;maybeF&nbsp;(pure&nbsp;nothingF)&nbsp;(justF&nbsp;&lt;$&gt;)</pre> </p> <p> In the <em>nothing</em> case, you can put <code>nothingF</code> into the desired <code>Applicative</code> with <code>pure</code>. In the <em>just</em> case you can take advantage of the desired <code>Applicative</code> being also a <code>Functor</code> by simply mapping the inner value(s) with <code>justF</code>. </p> <p> Since the <code>Applicative</code> instance for <code>MaybeFix</code> equals <code>pure</code> to <code>justF</code>, you could alternatively write the <code>Traversable</code> instance like this: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Traversable</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;sequenceA&nbsp;=&nbsp;maybeF&nbsp;(pure&nbsp;nothingF)&nbsp;(pure&nbsp;&lt;$&gt;)</pre> </p> <p> I like this alternative less, since I find it confusing. The two appearances of <code>pure</code> relate to two different types. The <code>pure</code> in <code>pure nothingF</code> has the type <code>MaybeFix a -&gt; f (MaybeFix a)</code>, while the <code>pure</code> in <code>pure&nbsp;&lt;$&gt;</code> has the type <code>a -&gt; MaybeFix a</code>! </p> <p> Both implementations work the same, though: </p> <p> <pre>Prelude Fix Maybe&gt; sequenceA (justF ("foo", 42)) ("foo",MaybeFix {unMaybeFix = Fix (JustF 42)})</pre> </p> <p> Here, I'm using the <code>Applicative</code> instance of <code>(,) String</code>. </p> <p> Finally, you can implement conversions to and from the standard <code>Maybe</code> type, using <code>ana</code> as the dual of <code>cata</code>: </p> <p> <pre><span style="color:#2b91af;">toMaybe</span>&nbsp;::&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&nbsp;a toMaybe&nbsp;=&nbsp;maybeF&nbsp;Nothing&nbsp;<span style="color:blue;">return</span> <span style="color:#2b91af;">fromMaybe</span>&nbsp;::&nbsp;<span style="color:#2b91af;">Maybe</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;a fromMaybe&nbsp;=&nbsp;MaybeFix&nbsp;.&nbsp;ana&nbsp;coalg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;coalg&nbsp;&nbsp;Nothing&nbsp;=&nbsp;NothingF &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;coalg&nbsp;(Just&nbsp;x)&nbsp;=&nbsp;JustF&nbsp;x</pre> </p> <p> This demonstrates that <code>MaybeFix</code> is isomorphic to <code>Maybe</code>, which again establishes that <code>maybeF</code> and <code>maybe</code> are equivalent. </p> <h3 id="2ec047e5122b4750a10cbe2012285524"> Alternatives <a href="#2ec047e5122b4750a10cbe2012285524" title="permalink">#</a> </h3> <p> As usual, the above <code>maybeF</code> isn't the only possible catamorphism. A trivial variation is to flip its arguments, but other variations exist. </p> <p> It's a recurring observation that a catamorphism is just a generalisation of a <em>fold</em>. In the above code, the <code>Foldable</code> instance already looked as simple as anyone could desire, but another variation of a catamorphism for Maybe is this gratuitously embellished definition: </p> <p> <pre><span style="color:#2b91af;">maybeF</span>&nbsp;::&nbsp;(a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;c maybeF&nbsp;f&nbsp;n&nbsp;=&nbsp;cata&nbsp;alg&nbsp;.&nbsp;unMaybeFix &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;NothingF&nbsp;=&nbsp;n &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(JustF&nbsp;x)&nbsp;=&nbsp;f&nbsp;x&nbsp;n</pre> </p> <p> This variation redundantly passes <code>n</code> as an argument to <code>f</code>, thereby changing the type of <code>f</code> to <code>a -&gt; c -&gt; c</code>. There's no particular motivation for doing this, apart from establishing that this catamorphism is exactly the same as the fold: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Foldable</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">foldr</span>&nbsp;=&nbsp;maybeF</pre> </p> <p> You can still implement the other instances as well, but the rest of the code suffers in clarity. Here's a few examples: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;f&nbsp;=&nbsp;maybeF&nbsp;(<span style="color:blue;">const</span>&nbsp;.&nbsp;justF&nbsp;.&nbsp;f)&nbsp;nothingF <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Applicative</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;pure&nbsp;=&nbsp;justF &nbsp;&nbsp;f&nbsp;&lt;*&gt;&nbsp;x&nbsp;=&nbsp;maybeF&nbsp;(<span style="color:blue;">const</span>&nbsp;.&nbsp;(&lt;$&gt;&nbsp;x))&nbsp;nothingF&nbsp;f <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Monad</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;x&nbsp;&gt;&gt;=&nbsp;f&nbsp;=&nbsp;maybeF&nbsp;(<span style="color:blue;">const</span>&nbsp;.&nbsp;f)&nbsp;nothingF&nbsp;x</pre> </p> <p> I find that the need to compose with <code>const</code> does nothing to improve the readability of the code, so this variation is mostly, I think, of academic interest. It does show, though, that the catamorphism of Maybe is isomorphic to its fold, as the diagram in the overview article indicated: </p> <p> <img src="/content/binary/catamorphism-and-fold-relations.png" alt="Catamorphisms and folds as sets, for various sum types."> </p> <p> You can demonstrate that this variation, too, is isomorphic to <code>Maybe</code> with a set of conversion: </p> <p> <pre><span style="color:#2b91af;">toMaybe</span>&nbsp;::&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">Maybe</span>&nbsp;a toMaybe&nbsp;=&nbsp;maybeF&nbsp;(<span style="color:blue;">const</span>&nbsp;.&nbsp;<span style="color:blue;">return</span>)&nbsp;Nothing <span style="color:#2b91af;">fromMaybe</span>&nbsp;::&nbsp;<span style="color:#2b91af;">Maybe</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">MaybeFix</span>&nbsp;a fromMaybe&nbsp;=&nbsp;MaybeFix&nbsp;.&nbsp;ana&nbsp;coalg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;coalg&nbsp;&nbsp;Nothing&nbsp;=&nbsp;NothingF &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;coalg&nbsp;(Just&nbsp;x)&nbsp;=&nbsp;JustF&nbsp;x</pre> </p> <p> Only <code>toMaybe</code> has changed, compared to above; the <code>fromMaybe</code> function remains the same. The only change to <code>toMaybe</code> is that the arguments have been flipped, and <code>return</code> is now composed with <code>const</code>. </p> <p> Since (according to <a href="http://amzn.to/13tGJ0f">Conceptual Mathematics</a>) isomorphisms are transitive this means that the two variations of <code>maybeF</code> are isomorphic. The latter, more complex, variation of <code>maybeF</code> is identical <code>foldr</code>, so we can consider the simpler, more frequently encountered variation a simplification of <em>fold</em>. </p> <h3 id="f88757d425a04e97956d89270b32c0c0"> Summary <a href="#f88757d425a04e97956d89270b32c0c0" title="permalink">#</a> </h3> <p> The catamorphism for Maybe is the same as its Church encoding: a pair made from a default value and a function. In Haskell's base library, this is simply called <code>maybe</code>. In the above C# code, it's called <code>Match</code>. </p> <p> This function is total, and you can implement any other functionality you need with it. I therefore consider it the canonical representation of Maybe, which is also why it annoys me that most Maybe implementations come equipped with partial functions like <code>fromJust</code>, or F#'s <code>Option.get</code>. Those functions shouldn't be part of a sane and reasonable Maybe API. You shouldn't need them. </p> <p> In this series of articles about catamorphisms, you've now seen the first example of catamorphism and fold coinciding. In the next article, you'll see another such example - probably the most well-known catamorphism example of them all. </p> <p> <strong>Next:</strong> <a href="/2019/05/27/list-catamorphism">List catamorphism</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Peano catamorphism https://blog.ploeh.dk/2019/05/13/peano-catamorphism 2019-05-13T05:10:00+00:00 Mark Seemann <div id="post"> <p> <em>The catamorphism for Peano numbers involves a base value and a successor function.</em> </p> <p> This article is part of an <a href="/2019/04/29/catamorphisms">article series about catamorphisms</a>. A catamorphism is a <a href="/2017/10/04/from-design-patterns-to-category-theory">universal abstraction</a> that describes how to digest a data structure into a potentially more compact value. </p> <p> This article presents the catamorphism for <a href="https://en.wikipedia.org/wiki/Natural_number">natural numbers</a>, as well as how to identify it. The beginning of the article presents the catamorphism in C#, with examples. The rest of the article describes how I deduced the catamorphism. This part of the article presents my work in <a href="https://www.haskell.org">Haskell</a>. Readers not comfortable with Haskell can just read the first part, and consider the rest of the article as an optional appendix. </p> <h3 id="742f4b9c0c014152a933961c10f98b66"> C# catamorphism <a href="#742f4b9c0c014152a933961c10f98b66" title="permalink">#</a> </h3> <p> In this article, I model natural numbers using <a href="https://en.wikipedia.org/wiki/Peano_axioms">Peano's model</a>, and I'll reuse the <a href="/2018/05/28/church-encoded-natural-numbers">Church-encoded implementation you've seen before</a>. The catamorphism for <code>INaturalNumber</code> is: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">T</span>&nbsp;Cata&lt;<span style="color:#2b91af;">T</span>&gt;(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;n,&nbsp;<span style="color:#2b91af;">T</span>&nbsp;zero,&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;&nbsp;succ) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;n.Match(zero,&nbsp;p&nbsp;=&gt;&nbsp;p.Cata(succ(zero),&nbsp;succ)); }</pre> </p> <p> Notice that this is an extension method on <code>INaturalNumber</code>, taking two other arguments: a <code>zero</code> argument which will be returned when the number is <em>zero</em>, and a successor function to return the 'next' value based on a previous value. </p> <p> The <code>zero</code> argument is the easiest to understand. It's simply passed to <code>Match</code> so that this is the value that <code>Cata</code> returns when <code>n</code> is <em>zero</em>. </p> <p> The other argument to <code>Match</code> must be a <code>Func&lt;INaturalNumber, T&gt;</code>; that is, a function that takes an <code>INaturalNumber</code> as input and returns a value of the type <code>T</code>. You can supply such a function by using a lambda expression. This expression receives a predecessor <code>p</code> as input, and has to return a value of the type <code>T</code>. The only function available in this context, however, is <code>succ</code>, which has the type <code>Func&lt;T, T&gt;</code>. How can you make that work? </p> <p> As is often the case when programming with generics, it pays to <em>follow the types</em>. A <code>Func&lt;T, T&gt;</code> requires an input of the type <code>T</code>. Do you have any <code>T</code> objects around? </p> <p> The only available <code>T</code> object is <code>zero</code>, so you could call <code>succ(zero)</code> to produce another <code>T</code> value. While you could return that immediately, that'd ignore the predecessor <code>p</code>, so that's not going to work. Another option, which is the one that works, is to recursively call <code>Cata</code> with <code>succ(zero)</code> as the <code>zero</code> value, and <code>succ</code> as the second argument. </p> <p> What this accomplishes is that <code>Cata</code> keeps recursively calling itself until <code>n</code> is <em>zero</em>. The <code>zero</code> object, however, will be the result of repeated applications of <code>succ(zero)</code>. In other words, <code>succ</code> will be called as many times as the natural number. If <code>n</code> is 7, then <code>succ</code> will be called seven times, the first time with the original <code>zero</code> value, the next time with the result of <code>succ(zero)</code>, the third time with the result of <code>succ(succ(zero))</code>, and so on. If the number is 42, <code>succ</code> will be called 42 times. </p> <h3 id="633dae2048cd45ebaa17962710048c67"> Arithmetic <a href="#633dae2048cd45ebaa17962710048c67" title="permalink">#</a> </h3> <p> You can implement all the functionality you saw in the article on Church-encoded natural numbers. You can start gently by converting a Peano number into a normal C# <code>int</code>: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:blue;">int</span>&nbsp;Count(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;n) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;n.Cata(0,&nbsp;x&nbsp;=&gt;&nbsp;1&nbsp;+&nbsp;x); }</pre> </p> <p> You can play with the functionality in <em>C# Interactive</em> to get a feel for how it works: </p> <p> <pre>&gt; <span style="color:#2b91af;">NaturalNumber</span>.Eight.Count() 8 &gt; <span style="color:#2b91af;">NaturalNumber</span>.Five.Count() 5</pre> </p> <p> The <code>Count</code> extension method uses <code>Cata</code> to count the level of recursion. The <code>zero</code> value is, not surprisingly, <code>0</code>, and the successor function simply adds one to the previous number. Since the successor function runs as many times as encoded by the Peano number, and since the initial value is <code>0</code>, you get the integer value of the number when <code>Cata</code> exits. </p> <p> A useful building block you can write using <code>Cata</code> is a function to increment a natural number by one: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;Increment(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;n) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;n.Cata(One,&nbsp;p&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Successor</span>(p)); }</pre> </p> <p> This, again, works as you'd expect: </p> <p> <pre>&gt; <span style="color:#2b91af;">NaturalNumber</span>.Zero.Increment().Count() 1 &gt; <span style="color:#2b91af;">NaturalNumber</span>.Eight.Increment().Count() 9</pre> </p> <p> With the <code>Increment</code> method and <code>Cata</code>, you can easily implement addition: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;Add(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;x,&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;y) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;x.Cata(y,&nbsp;p&nbsp;=&gt;&nbsp;p.Increment()); }</pre> </p> <p> The trick here is to use <code>y</code> as the <code>zero</code> case for <code>x</code>. In other words, if <code>x</code> is <em>zero</em>, then <code>Add</code> should return <code>y</code>. If <code>x</code> isn't <em>zero</em>, then <code>Increment</code> it as many times as the number encodes, but starting at <code>y</code>. In other words, start with <code>y</code> and <code>Increment</code> <code>x</code> times. </p> <p> The catamorphism makes it much easier to implement arithmetic operation. Just consider multiplication, which wasn't the simplest implementation in the previous article. Now, it's as simple as this: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;Multiply(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;x,&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;y) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;x.Cata(Zero,&nbsp;p&nbsp;=&gt;&nbsp;p.Add(y)); }</pre> </p> <p> Start at <code>0</code> and simply <code>Add(y)</code> <code>x</code> times. </p> <p> <pre>&gt; <span style="color:#2b91af;">NaturalNumber</span>.Nine.Multiply(<span style="color:#2b91af;">NaturalNumber</span>.Four).Count() 36</pre> </p> <p> Finally, you can also implement some common predicates: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IChurchBoolean</span>&nbsp;IsZero(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;n) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;n.Cata&lt;<span style="color:#2b91af;">IChurchBoolean</span>&gt;(<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ChurchTrue</span>(),&nbsp;_&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ChurchFalse</span>()); } <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IChurchBoolean</span>&nbsp;IsEven(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;n) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;n.Cata&lt;<span style="color:#2b91af;">IChurchBoolean</span>&gt;(<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ChurchTrue</span>(),&nbsp;b&nbsp;=&gt;&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ChurchNot</span>(b)); } <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">IChurchBoolean</span>&nbsp;IsOdd(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">INaturalNumber</span>&nbsp;n) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">ChurchNot</span>(n.IsEven()); }</pre> </p> <p> Particularly <code>IsEven</code> is elegant: It considers <code>zero</code> even, so simply uses a <code>new ChurchTrue()</code> object for that case. In all other cases, it alternates between <em>false</em> and <em>true</em> by negating the predecessor. </p> <p> <pre>&gt; <span style="color:#2b91af;">NaturalNumber</span>.Three.IsEven().ToBool() false</pre> </p> <p> It seems convincing that we can use <code>Cata</code> to implement all the other functionality we need. That seems to be a characteristic of a catamorphism. Still, how do we know that <code>Cata</code> is, in fact, the catamorphism for natural numbers? </p> <h3 id="05dbca489b8a49be830df87e13bfcae3"> Peano F-Algebra <a href="#05dbca489b8a49be830df87e13bfcae3" title="permalink">#</a> </h3> <p> As in the <a href="/2019/05/06/boolean-catamorphism">previous article</a>, I'll use <code>Fix</code> and <code>cata</code> as explained in <a href="https://bartoszmilewski.com">Bartosz Milewski</a>'s excellent <a href="https://bartoszmilewski.com/2017/02/28/f-algebras/">article on F-Algebras</a>. The <code>NatF</code> type comes from his article as well: </p> <p> <pre><span style="color:blue;">data</span>&nbsp;NatF&nbsp;a&nbsp;=&nbsp;ZeroF&nbsp;|&nbsp;SuccF&nbsp;a&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;<span style="color:blue;">NatF</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;ZeroF&nbsp;=&nbsp;ZeroF &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;f&nbsp;(SuccF&nbsp;x)&nbsp;=&nbsp;SuccF&nbsp;$&nbsp;f&nbsp;x</pre> </p> <p> You can use the fixed point of this functor to define numbers with a shared type. Here's just the first ten: </p> <p> <pre><span style="color:#2b91af;">zeroF</span>,&nbsp;<span style="color:#2b91af;">oneF</span>,&nbsp;<span style="color:#2b91af;">twoF</span>,&nbsp;<span style="color:#2b91af;">threeF</span>,&nbsp;<span style="color:#2b91af;">fourF</span>,&nbsp;<span style="color:#2b91af;">fiveF</span>,&nbsp;<span style="color:#2b91af;">sixF</span>,&nbsp;<span style="color:#2b91af;">sevenF</span>,&nbsp;<span style="color:#2b91af;">eightF</span>,&nbsp;<span style="color:#2b91af;">nineF</span>&nbsp;::&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span> zeroF&nbsp;&nbsp;=&nbsp;Fix&nbsp;ZeroF oneF&nbsp;&nbsp;&nbsp;=&nbsp;Fix&nbsp;$&nbsp;SuccF&nbsp;zeroF twoF&nbsp;&nbsp;&nbsp;=&nbsp;Fix&nbsp;$&nbsp;SuccF&nbsp;oneF threeF&nbsp;=&nbsp;Fix&nbsp;$&nbsp;SuccF&nbsp;twoF fourF&nbsp;&nbsp;=&nbsp;Fix&nbsp;$&nbsp;SuccF&nbsp;threeF fiveF&nbsp;&nbsp;=&nbsp;Fix&nbsp;$&nbsp;SuccF&nbsp;fourF sixF&nbsp;&nbsp;&nbsp;=&nbsp;Fix&nbsp;$&nbsp;SuccF&nbsp;fiveF sevenF&nbsp;=&nbsp;Fix&nbsp;$&nbsp;SuccF&nbsp;sixF eightF&nbsp;=&nbsp;Fix&nbsp;$&nbsp;SuccF&nbsp;sevenF nineF&nbsp;&nbsp;=&nbsp;Fix&nbsp;$&nbsp;SuccF&nbsp;eightF</pre> </p> <p> That's all you need to identify the catamorphism. </p> <h3 id="f0e66c873a034830a4b069229971299a"> Haskell catamorphism <a href="#f0e66c873a034830a4b069229971299a" title="permalink">#</a> </h3> <p> At this point, you have two out of three elements of an F-Algebra. You have an endofunctor (<code>NatF</code>), and an object <code>a</code>, but you still need to find a morphism <code>NatF a -&gt; a</code>. </p> <p> As in the previous article, start by writing a function that will become the catamorphism, based on <code>cata</code>: </p> <p> <pre>natF&nbsp;=&nbsp;cata&nbsp;alg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;ZeroF&nbsp;=&nbsp;<span style="color:blue;">undefined</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(SuccF&nbsp;predecessor)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> While this compiles, with its <code>undefined</code> implementations, it obviously doesn't do anything useful. I find, however, that it helps me think. How can you return a value of the type <code>a</code> from the <code>ZeroF</code> case? You could pass an argument to the <code>natF</code> function: </p> <p> <pre>natF&nbsp;z&nbsp;=&nbsp;cata&nbsp;alg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;ZeroF&nbsp;=&nbsp;z &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(SuccF&nbsp;predecessor)&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> In the <code>SuccF</code> case, <code>predecessor</code> is already of the polymorphic type <code>a</code>, so instead of returning a constant value, you can supply a function as an argument to <code>natF</code> and use it in that case: </p> <p> <pre><span style="color:#2b91af;">natF</span>&nbsp;::&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;(a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a natF&nbsp;z&nbsp;next&nbsp;=&nbsp;cata&nbsp;alg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;ZeroF&nbsp;=&nbsp;z &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;(SuccF&nbsp;predecessor)&nbsp;=&nbsp;next&nbsp;predecessor</pre> </p> <p> This works. Since <code>cata</code> has the type <code>Functor f =&gt; (f a -&gt; a) -&gt; Fix f -&gt; a</code>, that means that <code>alg</code> has the type <code>f a -&gt; a</code>. In the case of <code>NatF</code>, the compiler infers that the <code>alg</code> function has the type <code>NatF a -&gt; a</code>, which is just what you need! </p> <p> For good measure, I should point out that, as usual, the above <code>natF</code> function isn't the only possible catamorphism. Trivially, you can flip the order of the arguments, and this would also be a catamorphism. These two alternatives are isomorphic. </p> <p> The <code>natF</code> function identifies the Peano number catamorphism, which is equivalent to the C# representation in the beginning of the article. I called the function <code>natF</code>, because there's a tendency in Haskell to name the 'case analysis' or catamorphism after the type, just with a lower-case initial letter. </p> <h3 id="e78ae79059e14f4b92f525393cc74861"> Basis <a href="#e78ae79059e14f4b92f525393cc74861" title="permalink">#</a> </h3> <p> A catamorphism can be used to implement most (if not all) other useful functionality, like all of the above C# functionality. In fact, I wrote the Haskell code first, and then translated my implementations into the above C# extension methods. This means that the following functions apply the same reasoning: </p> <p> <pre><span style="color:#2b91af;">evenF</span>&nbsp;::&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span> evenF&nbsp;=&nbsp;natF&nbsp;trueF&nbsp;notF <span style="color:#2b91af;">oddF</span>&nbsp;::&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span> oddF&nbsp;=&nbsp;notF&nbsp;.&nbsp;evenF <span style="color:#2b91af;">incF</span>&nbsp;::&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span> incF&nbsp;=&nbsp;natF&nbsp;oneF&nbsp;$&nbsp;Fix&nbsp;.&nbsp;SuccF <span style="color:#2b91af;">addF</span>&nbsp;::&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span> addF&nbsp;x&nbsp;y&nbsp;=&nbsp;natF&nbsp;y&nbsp;incF&nbsp;x <span style="color:#2b91af;">multiplyF</span>&nbsp;::&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span> multiplyF&nbsp;x&nbsp;y&nbsp;=&nbsp;natF&nbsp;zeroF&nbsp;(addF&nbsp;y)&nbsp;x</pre> </p> <p> Here are some GHCi usage examples: </p> <p> <pre>Prelude Boolean Nat&gt; evenF eightF Fix TrueF Prelude Boolean Nat&gt; toNum $ multiplyF sevenF sixF 42</pre> </p> <p> The <code>toNum</code> function corresponds to the above <code>Count</code> C# method. It is, again, based on <code>cata</code>. You can use <code>ana</code> to convert the other way: </p> <p> <pre><span style="color:#2b91af;">toNum</span>&nbsp;::&nbsp;<span style="color:blue;">Num</span>&nbsp;a&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a toNum&nbsp;=&nbsp;natF&nbsp;0&nbsp;(+&nbsp;1) <span style="color:#2b91af;">fromNum</span>&nbsp;::&nbsp;(<span style="color:blue;">Eq</span>&nbsp;a,&nbsp;<span style="color:blue;">Num</span>&nbsp;a)&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">NatF</span> fromNum&nbsp;=&nbsp;ana&nbsp;coalg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;coalg&nbsp;0&nbsp;=&nbsp;ZeroF &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;coalg&nbsp;x&nbsp;=&nbsp;SuccF&nbsp;$&nbsp;x&nbsp;-&nbsp;1</pre> </p> <p> This demonstrates that <code>Fix NatF</code> is isomorphic to <code>Num</code> instances, such as <code>Integer</code>. </p> <h3 id="d09e79446af14875a42f66869e10f33a"> Summary <a href="#d09e79446af14875a42f66869e10f33a" title="permalink">#</a> </h3> <p> The catamorphism for Peano numbers is a pair consisting of a zero value and a successor function. The most common description of catamorphisms that I've found emphasise how a catamorphism is like a <em>fold;</em> an operation you can use to reduce a data structure like a list or a tree to a single value. This is what happens here, but even so, the <code>Fix NatF</code> type isn't a <code>Foldable</code> instance. The reason is that while <code>NatF</code> is a polymorphic type, its fixed point <code>Fix NatF</code> isn't. Haskell's <code>Foldable</code> type class requires foldable containers to be polymorphic (what C# programmers would call 'generic'). </p> <p> When I first ran into the concept of a <em>catamorphism</em>, it was invariably described as a 'generalisation of fold'. The examples shown were always how the catamorphism for linked list is the same as its <em>fold</em>. I found such explanations unhelpful, because I couldn't understand how those two concepts differ. </p> <p> The purpose with this article series is to show just how much more general the abstraction of a catamorphism is. In this article you saw how an infinitely recursive data structure like Peano numbers have a catamorphism, even though it isn't a parametrically polymorphic type. In the next article, though, you'll see the first example of a polymorphic type where the catamorphism coincides with the fold. </p> <p> <strong>Next:</strong> <a href="/2019/05/20/maybe-catamorphism">Maybe catamorphism</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Boolean catamorphism https://blog.ploeh.dk/2019/05/06/boolean-catamorphism 2019-05-06T12:30:00+00:00 Mark Seemann <div id="post"> <p> <em>The catamorphism for Boolean values is just the common ternary operator.</em> </p> <p> This article is part of an <a href="/2019/04/29/catamorphisms">article series about catamorphisms</a>. A catamorphism is a <a href="/2017/10/04/from-design-patterns-to-category-theory">universal abstraction</a> that describes how to digest a data structure into a potentially more compact value. </p> <p> This article presents the catamorphism for Boolean values, as well as how you identify it. The beginning of this article presents the catamorphism in C#, with a simple example. The rest of the article describes how I deduced the catamorphism. That part of the article presents my work in <a href="https://www.haskell.org">Haskell</a>. Readers not comfortable with Haskell can just read the first part, and consider the rest of the article as an optional appendix. </p> <h3 id="35155b758274445cbe57f75d730a4eb6"> C# catamorphism <a href="#35155b758274445cbe57f75d730a4eb6" title="permalink">#</a> </h3> <p> The catamorphism for Boolean values is the familiar <a href="https://en.wikipedia.org/wiki/%3F:">ternary conditional operator</a>: </p> <p> <pre>&gt; <span style="color:#2b91af;">DateTime</span>.Now.Day&nbsp;%&nbsp;2&nbsp;==&nbsp;0&nbsp;?&nbsp;<span style="color:#a31515;">&quot;Even&nbsp;date&quot;</span>&nbsp;:&nbsp;<span style="color:#a31515;">&quot;Odd&nbsp;date&quot;</span> "Odd date"</pre> </p> <p> Given a Boolean expression, you basically provide two values: one to use in case the Boolean expression is <em>true</em>, and one to use in case it's <em>false</em>. </p> <p> For <a href="/2018/05/24/church-encoded-boolean-values">Church-encoded Boolean values</a>, the catamorphism looks like this: </p> <p> <pre><span style="color:#2b91af;">T</span>&nbsp;Match&lt;<span style="color:#2b91af;">T</span>&gt;(<span style="color:#2b91af;">T</span>&nbsp;trueCase,&nbsp;<span style="color:#2b91af;">T</span>&nbsp;falseCase);</pre> </p> <p> This is an instance method where you must, again, supply two alternatives. When the instance represents <em>true</em>, you'll get the left-most value <code>trueCase</code>; otherwise, when the instance represents <em>false</em>, you'll get the right-most value <code>falseCase</code>. </p> <p> The catamorphism turns out to be the same as the <a href="/2018/05/22/church-encoding">Church encoding</a>. This seems to be a recurring pattern. </p> <h3 id="cd81cb92ed2d42f8bc0ad5adbde4b014"> Alternatives <a href="#cd81cb92ed2d42f8bc0ad5adbde4b014" title="permalink">#</a> </h3> <p> To be accurate, there's more than one catamorphism for Boolean values. It's only by convention that the value corresponding to <em>true</em> goes on the left, and the <em>false</em> value goes to the right. You could flip the arguments, and it would still be a catamorphism. This is, in fact, what Haskell's <code>Data.Bool</code> module does: </p> <p> <pre>Prelude Data.Bool&gt; bool "Odd date" "Even date" $ even date "Odd date"</pre> </p> <p> The <a href="http://hackage.haskell.org/package/base/docs/Data-Bool.html">module documentation</a> calls this the <em>"Case analysis for the <code>Bool</code> type"</em>, instead of a catamorphism, but the two representations are isomorphic: <blockquote> "This is equivalent to <code>if p then y else x</code>; that is, one can think of it as an if-then-else construct with its arguments reordered." </blockquote> This is another recurring result. There's typically more than one catamorphism, but the alternatives are isomorphic. In this article series, I'll mostly present the alternative that strikes me as the one you'll encounter most frequently. </p> <h3 id="60235fb428d14785a5aeea440c05cce5"> Fix <a href="#60235fb428d14785a5aeea440c05cce5" title="permalink">#</a> </h3> <p> In this and future articles, I'll derive the catamorphism from an F-Algebra. For an introduction to F-Algebras and fixed points, I'll refer you to <a href="https://bartoszmilewski.com">Bartosz Milewski</a>'s excellent <a href="https://bartoszmilewski.com/2017/02/28/f-algebras/">article on the topic</a>. In it, he presents a generic data type for a fixed point, as well as polymorphic functions for catamorphisms and anamorphisms. While they're available in his article, I'll repeat them here for good measure: </p> <p> <pre><span style="color:blue;">newtype</span>&nbsp;Fix&nbsp;f&nbsp;=&nbsp;Fix&nbsp;{&nbsp;unFix&nbsp;::&nbsp;f&nbsp;(Fix&nbsp;f)&nbsp;} <span style="color:#2b91af;">cata</span>&nbsp;::&nbsp;<span style="color:blue;">Functor</span>&nbsp;f&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;(f&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;f&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a cata&nbsp;alg&nbsp;=&nbsp;alg&nbsp;.&nbsp;<span style="color:blue;">fmap</span>&nbsp;(cata&nbsp;alg)&nbsp;.&nbsp;unFix <span style="color:#2b91af;">ana</span>&nbsp;::&nbsp;<span style="color:blue;">Functor</span>&nbsp;f&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;(a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;f&nbsp;a)&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;f ana&nbsp;coalg&nbsp;=&nbsp;Fix&nbsp;.&nbsp;<span style="color:blue;">fmap</span>&nbsp;(ana&nbsp;coalg)&nbsp;.&nbsp;coalg</pre> </p> <p> This should be recognisable from Bartosz Milewski's article. With one small exception, this is just a copy of the code shown there. </p> <h3 id="6b0fdc2b04e540d19322f0e30c00e86c"> Boolean F-Algebra <a href="#6b0fdc2b04e540d19322f0e30c00e86c" title="permalink">#</a> </h3> <p> While F-Algebras and fixed points are mostly used for recursive data structures, you can also define an F-Algebra for a non-recursive data structure. As data types go, they don't get much simpler than Boolean values, which are just two mutually exclusive cases. In order to make a <code>Functor</code> out of the definition, though, you can equip it with a <em>carrier type:</em> </p> <p> <pre><span style="color:blue;">data</span>&nbsp;BoolF&nbsp;a&nbsp;=&nbsp;TrueF&nbsp;|&nbsp;FalseF&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>,&nbsp;<span style="color:#2b91af;">Read</span>) <span style="color:blue;">instance</span>&nbsp;<span style="color:blue;">Functor</span>&nbsp;<span style="color:blue;">BoolF</span>&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;&nbsp;TrueF&nbsp;=&nbsp;&nbsp;TrueF &nbsp;&nbsp;<span style="color:blue;">fmap</span>&nbsp;_&nbsp;FalseF&nbsp;=&nbsp;FalseF</pre> </p> <p> The <code>Functor</code> instance simply ignores the carrier type and just returns <code>TrueF</code> and <code>FalseF</code>, respectively. It'd seem that nothing happens, but at the type level, this is still a translation from <code>BoolF a</code> to <code>BoolF b</code>. Not much of a function, perhaps, but definitely an <em>endofunctor</em>. </p> <p> Another note that may be in order here, as well as for all future articles in this series, is that you'll notice that most types and custom functions come with the <code>F</code> suffix. This is simply a suffix I've added to avoid conflicts with built-in types, values, and functions, such as <code>Bool</code>, <code>True</code>, <code>and</code>, and so on. The <code>F</code> is for <em>F-Algebra</em>. </p> <p> You can lift these values into <code>Fix</code> in order to make it fit with the <code>cata</code> function: </p> <p> <pre><span style="color:#2b91af;">trueF</span>,&nbsp;<span style="color:#2b91af;">falseF</span>&nbsp;::&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span> trueF&nbsp;&nbsp;=&nbsp;Fix&nbsp;&nbsp;TrueF falseF&nbsp;=&nbsp;Fix&nbsp;FalseF</pre> </p> <p> That's all you need to identify the catamorphism. </p> <h3 id="a28e972fc7eb45038427cff258c0c8f2"> Haskell catamorphism <a href="#a28e972fc7eb45038427cff258c0c8f2" title="permalink">#</a> </h3> <p> At this point, you have two out of three elements of an F-Algebra. You have an endofunctor (<code>BoolF</code>), and an object <code>a</code>, but you still need to find a morphism <code>BoolF a -&gt; a</code>. At first glance, this seems impossible, because neither <code>TrueF</code> nor <code>FalseF</code> actually contain a value of the type <code>a</code>. How, then, can you conjure an <code>a</code> value out of thin air? </p> <p> The <code>cata</code> function has the answer. </p> <p> What you can do is to start writing the function that will become the catamorphism, basing it on <code>cata</code>: </p> <p> <pre>boolF&nbsp;=&nbsp;cata&nbsp;alg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;TrueF&nbsp;=&nbsp;<span style="color:blue;">undefined</span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;FalseF&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> While this compiles, with its <code>undefined</code> implementations, it obviously doesn't do anything useful. I find, however, that it helps me think. How can you return a value of the type <code>a</code> from the <code>TrueF</code> case? You could pass an argument to the <code>boolF</code> function: </p> <p> <pre>boolF&nbsp;x&nbsp;=&nbsp;cata&nbsp;alg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;TrueF&nbsp;=&nbsp;x &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;FalseF&nbsp;=&nbsp;<span style="color:blue;">undefined</span></pre> </p> <p> That seems promising, so do that for the <code>FalseF</code> case as well: </p> <p> <pre><span style="color:#2b91af;">boolF</span>&nbsp;::&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a boolF&nbsp;x&nbsp;y&nbsp;=&nbsp;cata&nbsp;alg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;alg&nbsp;&nbsp;TrueF&nbsp;=&nbsp;x &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alg&nbsp;FalseF&nbsp;=&nbsp;y</pre> </p> <p> This works. Since <code>cata</code> has the type <code>Functor f =&gt; (f a -&gt; a) -&gt; Fix f -&gt; a</code>, that means that <code>alg</code> has the type <code>f a -&gt; a</code>. In the case of <code>BoolF</code>, the compiler infers that the <code>alg</code> function has the type <code>BoolF a -&gt; a</code>, which is just what you need! </p> <p> The <code>boolF</code> function identifies the Boolean catamorphism, which is equivalent to representations in the beginning of the article. I called the function <code>boolF</code>, because there's a tendency in Haskell to name the 'case analysis' or catamorphism after the type, just with a lower-case initial letter. </p> <p> You can use the <code>boolF</code> function just like the above ternary operator: </p> <p> <pre>Prelude Boolean Nat&gt; boolF "Even date" "Odd date" $ evenF dateF "Odd date"</pre> </p> <p> Here, I've also used <code>evenF</code> from the <code>Nat</code> module shown in the next article in the series. </p> <p> From the above definition of <code>boolF</code>, it should also be clear that you can arrive at the alternative catamorphism defined by <code>Data.Bool.bool</code> by simply flipping <code>x</code> and <code>y</code>. </p> <h3 id="54886f1be8684dd4a5909e4d50b7d5dc"> Basis <a href="#54886f1be8684dd4a5909e4d50b7d5dc" title="permalink">#</a> </h3> <p> A catamorphism can be used to implement most (if not all) other useful functionality. For Boolean values, that would be the standard Boolean operations <em>and</em>, <em>or</em>, and <em>not:</em> </p> <p> <pre><span style="color:#2b91af;">andF</span>&nbsp;::&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span> andF&nbsp;x&nbsp;y&nbsp;=&nbsp;boolF&nbsp;y&nbsp;falseF&nbsp;x <span style="color:#2b91af;">orF</span>&nbsp;::&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span> orF&nbsp;x&nbsp;y&nbsp;=&nbsp;boolF&nbsp;trueF&nbsp;y&nbsp;x <span style="color:#2b91af;">notF</span>&nbsp;::&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span> notF&nbsp;=&nbsp;boolF&nbsp;falseF&nbsp;trueF</pre> </p> <p> They work as you'd expect them to work: </p> <p> <pre>Prelude Boolean&gt; andF trueF falseF Fix FalseF Prelude Boolean&gt; orF trueF falseF Fix TrueF Prelude Boolean&gt; orF (notF trueF) falseF Fix FalseF</pre> </p> <p> You can also implement conversion to and from the built-in <code>Bool</code> type: </p> <p> <pre><span style="color:#2b91af;">toBool</span>&nbsp;::&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:#2b91af;">Bool</span> toBool&nbsp;=&nbsp;boolF&nbsp;True&nbsp;False <span style="color:#2b91af;">fromBool</span>&nbsp;::&nbsp;<span style="color:#2b91af;">Bool</span>&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Fix</span>&nbsp;<span style="color:blue;">BoolF</span> fromBool&nbsp;=&nbsp;ana&nbsp;coalg &nbsp;&nbsp;<span style="color:blue;">where</span>&nbsp;coalg&nbsp;&nbsp;True&nbsp;=&nbsp;TrueF &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;coalg&nbsp;False&nbsp;=&nbsp;FalseF</pre> </p> <p> This demonstrates that <code>Fix BoolF</code> is isomorphic to <code>Bool</code>. </p> <h3 id="327411e72cea46bbb1f6fe167738c7b2"> Summary <a href="#327411e72cea46bbb1f6fe167738c7b2" title="permalink">#</a> </h3> <p> The catamorphism for Boolean values is a function, method, or operator akin to the familiar ternary conditional operator. The most common descriptions of catamorphisms that I've found emphasise how a catamorphism is like a <em>fold;</em> an operation you can use to reduce a data structure like a list or a tree to a single value. In that light, it may be surprising that something as simple as Boolean values have an associated catamorphism. </p> <p> Since <code>Fix BoolF</code> is isomorphic to <code>Bool</code>, you may wonder what the point is. Why define this data type, and implement functions like <code>andF</code>, <code>orF</code>, and <code>notF</code>? </p> <p> The code presented here is nothing but an analysis tool. It's a way to identify the catamorphism for Boolean values. </p> <p> <strong>Next:</strong> <a href="/2019/05/13/peano-catamorphism">Peano catamorphism</a>. </p> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Catamorphisms https://blog.ploeh.dk/2019/04/29/catamorphisms 2019-04-29T18:31:00+00:00 Mark Seemann <div id="post"> <p> <em>A catamorphism is a general abstraction that enables you to handle multiple values, for example in order to reduce them to a single value.</em> </p> <p> This article series is part of <a href="/2017/10/04/from-design-patterns-to-category-theory">an even larger series of articles about the relationship between design patterns and category theory</a>. In another article series in this big series of articles, you learned about <a href="/2018/03/19/functors-applicatives-and-friends">functors, applicatives, and other types of data containers</a>. </p> <p> You may have heard about <em>map-reduce</em> architectures. Much software can be designed around two general types of operations: those that <em>map</em> data, and those that <em>reduce</em> data. A <a href="https://bartoszmilewski.com/2014/01/14/functors-are-containers">functor is a container of data</a> that supports structure-preserving maps. Thus, you can think of <a href="/2018/03/22/functors">functors</a> as the general abstraction for map operations (also sometimes called <em>projections</em>). Does a similar universal abstraction exist for operations that reduce data? </p> <p> Yes, that abstraction is called a <em>catamorphism</em>. </p> <h3 id="bb64d005b16b49f892c00824ef803997"> Aggregation <a href="#bb64d005b16b49f892c00824ef803997" title="permalink">#</a> </h3> <p> <em>Catamorphism</em> is an intimidating word, so let's start with an example. You often have a collection of values that you'd like to reduce to a single value. Such a collection can contain arbitrarily complex objects, but I'll keep it simple and start with a collection of numbers: </p> <p> <pre><span style="color:blue;">new</span>[]&nbsp;{&nbsp;42,&nbsp;1337,&nbsp;2112,&nbsp;90125,&nbsp;5040,&nbsp;7,&nbsp;1984&nbsp;};</pre> </p> <p> This particular list of numbers is an array, but that's not important. What comes next works for any <code>IEnumerable&lt;T&gt;</code>, including arrays. I only chose an array because the C# syntax for array creation is more compact than for other collection types. </p> <p> How do you reduce those seven numbers to a single number? That depends on what you want that number to tell you. One option is to add the numbers together. There's a specific, built-in function for that: </p> <p> <pre>&gt; <span style="color:blue;">new</span>[]&nbsp;{&nbsp;42,&nbsp;1337,&nbsp;2112,&nbsp;90125,&nbsp;5040,&nbsp;7,&nbsp;1984&nbsp;}.Sum(); 100647</pre> </p> <p> The <a href="https://docs.microsoft.com/en-us/dotnet/api/system.linq.enumerable.sum">Sum</a> extension method is a one of many built-in functions that enable you to reduce a list of numbers to a single number: <a href="https://docs.microsoft.com/en-us/dotnet/api/system.linq.enumerable.average">Average</a>, <a href="https://docs.microsoft.com/en-us/dotnet/api/system.linq.enumerable.max">Max</a>, <a href="https://docs.microsoft.com/en-us/dotnet/api/system.linq.enumerable.count">Count</a>, and so on. </p> <p> What do you do, though, if you need to reduce many values to one, and there's no existing function for that? What if, for example, you need to add all the numbers using <a href="/2018/07/16/angular-addition-monoid">modulo 360 addition</a>? </p> <p> In that case, you use <a href="https://docs.microsoft.com/en-us/dotnet/api/system.linq.enumerable.aggregate">Aggregate</a>: </p> <p> <pre>&gt; <span style="color:blue;">new</span>[]&nbsp;{&nbsp;42,&nbsp;1337,&nbsp;2112,&nbsp;90125,&nbsp;5040,&nbsp;7,&nbsp;1984&nbsp;}.Aggregate((x,&nbsp;y)&nbsp;=&gt;&nbsp;(x&nbsp;+&nbsp;y)&nbsp;%&nbsp;360) 207</pre> </p> <p> The way to interpret this result is that the initial array represents a sequence of rotations (measured in degrees), and the result is the final angle after all the rotations have completed. </p> <p> In other (functional) languages, such a 'reduce' operation is called a <em>fold</em>. The metaphor, I suppose, is that you fold multiple values together, two by two. </p> <p> A <em>fold</em> is a catamorphism, but a catamorphism is a more general abstraction. For some data structures, the catamorphism is more powerful than the fold, but for collections, there's no difference. </p> <p> There's one edge case we need to be aware of, though. What if the collection is empty? </p> <h3 id="65483950f21d453ebe4e8949eac5751f"> Aggregation of empty containers <a href="#65483950f21d453ebe4e8949eac5751f" title="permalink">#</a> </h3> <p> What happens if you attempt to aggregate an empty collection? </p> <p> <pre>&gt; <span style="color:blue;">new</span>&nbsp;<span style="color:blue;">int</span>[0].Aggregate((x,&nbsp;y)&nbsp;=&gt;&nbsp;(x&nbsp;+&nbsp;y)&nbsp;%&nbsp;360) <span style="color:red;">Sequence contains no elements + System.Linq.Enumerable.Aggregate&lt;TSource&gt;(IEnumerable&lt;TSource&gt;, Func&lt;TSource, TSource, TSource&gt;)</span></pre> </p> <p> The <code>Aggregate</code> method throws an exception because it doesn't know how to deal with empty collections. The lambda expression you supply tells the <code>Aggregate</code> method how to combine two values into one. This is, for instance, how <a href="/2017/12/11/semigroups-accumulate">semigroups accumulate</a>. </p> <p> The lambda expression handles all cases where you have two or more values. If you have only a single value, then that's no problem either: </p> <p> <pre>&gt; <span style="color:blue;">new</span>[]&nbsp;{&nbsp;1337&nbsp;}.Aggregate((x,&nbsp;y)&nbsp;=&gt;&nbsp;(x&nbsp;+&nbsp;y)&nbsp;%&nbsp;360) 1337</pre> </p> <p> In that case, the lambda expression isn't involved at all, because the single value is simply returned without modification. In this example, this could even be interpreted as being incorrect, since you'd expect the result to be 257 (<code>1337 % 360</code>). </p> <p> It's safer to use the <code>Aggregate</code> overload that takes a <em>seed</em> value: </p> <p> <pre>&gt; <span style="color:blue;">new</span>&nbsp;<span style="color:blue;">int</span>[0].Aggregate(0,&nbsp;(x,&nbsp;y)&nbsp;=&gt;&nbsp;(x&nbsp;+&nbsp;y)&nbsp;%&nbsp;360) 0</pre> </p> <p> Not only does that gracefully handle empty collections, it also gives you a 'better' result for a single value: </p> <p> <pre>&gt; <span style="color:blue;">new</span>[]&nbsp;{&nbsp;1337&nbsp;}.Aggregate(0,&nbsp;(x,&nbsp;y)&nbsp;=&gt;&nbsp;(x&nbsp;+&nbsp;y)&nbsp;%&nbsp;360) 257</pre> </p> <p> This works better because the method always starts with the <em>seed</em> value, which means that even if there's only a single value (<code>1337</code>), the lambda expression still runs (<code>(0 + 1337) % 360</code>). </p> <p> This overload of <code>Aggregate</code> has a different type, though: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">TAccumulate</span>&nbsp;Aggregate&lt;<span style="color:#2b91af;">TSource</span>,&nbsp;<span style="color:#2b91af;">TAccumulate</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">IEnumerable</span>&lt;<span style="color:#2b91af;">TSource</span>&gt;&nbsp;source, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">TAccumulate</span>&nbsp;seed, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">TAccumulate</span>,&nbsp;<span style="color:#2b91af;">TSource</span>,&nbsp;<span style="color:#2b91af;">TAccumulate</span>&gt;&nbsp;func);</pre> </p> <p> Notice that the <code>func</code> doesn't require the accumulator to have the same type as elements from the <code>source</code> collection. This enables you to translate on the fly, so to speak. You can still use binary operations like the above modulo 360 addition, because that just implies that both <code>TSource</code> and <code>TAccumulate</code> are <code>int</code>. </p> <p> With this overload, you could, for example, use <a href="/2018/07/16/angular-addition-monoid">the Angle class</a> to perform the work: </p> <p> <pre>&gt; <span style="color:blue;">new</span>[]&nbsp;{&nbsp;42,&nbsp;1337,&nbsp;2112,&nbsp;90125,&nbsp;5040,&nbsp;7,&nbsp;1984&nbsp;} . .Aggregate(<span style="color:#2b91af;">Angle</span>.Identity,&nbsp;(a,&nbsp;i)&nbsp;=&gt;&nbsp;a.Add(<span style="color:#2b91af;">Angle</span>.FromDegrees(i))) [{ Angle = 207° }]</pre> </p> <p> Now the <code>seed</code> argument is <code>Angle.Identity</code>, which implies that <code>TAccumulate</code> is <code>Angle</code>. The <code>source</code> is still a collection of numbers, so <code>TSource</code> is <code>int</code>. Hence, I called the angle <code>a</code> and the integer <code>i</code> in the lambda expression. The output is an <code>Angle</code> object that represents 207°. </p> <p> That <code>Aggregate</code> overload is the catamorphism for collections. It reduces a collection to an object. </p> <h3 id="5a964dee9b1f4cdd8427ce0a0806d65d"> Catamorphisms and folds <a href="#5a964dee9b1f4cdd8427ce0a0806d65d" title="permalink">#</a> </h3> <p> Is <em>catamorphism</em> just an intimidating word for <em>aggregate</em>, <em>accumulate</em>, <em>fold</em>, or <em>reduce?</em> </p> <p> It took me a long time to be able to tell the difference, because in many cases, it seems that there's no difference. The purpose of this article series is to make the distinction clearer. In short, a catamorphism is a more general concept. </p> <p> <img src="/content/binary/catamorphism-and-fold-relations.png" alt="Catamorphisms and folds as sets, for various sum types."> </p> <p> For some data structures, such as <a href="/2018/05/24/church-encoded-boolean-values">Boolean values</a>, or <a href="/2018/05/28/church-encoded-natural-numbers">Peano numbers</a>, the catamorphism is all there is; no fold exists. For other data structures, such as <a href="/2018/06/04/church-encoded-maybe">Maybe</a> or collections, the catamorphism and the fold coincide. Still other data structures, such as <a href="/2018/06/11/church-encoded-either">Either</a> and <a href="/2018/08/06/a-tree-functor">trees</a>, support folding, but the fold is based on the catamorphism. For those types, there are operations you can do with the catamorphism that are impossible to implement with the <em>fold</em> function. One example is that a tree's catamorphism enables you to count its leaves; you can't do that with its <em>fold</em> function. </p> <p> You'll see plenty of examples in this article series: </p> <p> <ul> <li><a href="/2019/05/06/boolean-catamorphism">Boolean catamorphism</a></li> <li><a href="/2019/05/13/peano-catamorphism">Peano catamorphism</a></li> <li><a href="/2019/05/20/maybe-catamorphism">Maybe catamorphism</a></li> <li><a href="/2019/05/27/list-catamorphism">List catamorphism</a></li> <li><a href="/2019/06/03/either-catamorphism">Either catamorphism</a></li> <li><a href="/2019/06/10/tree-catamorphism">Tree catamorphism</a></li> <li><a href="/2019/08/05/rose-tree-catamorphism">Rose tree catamorphism</a></li> <li><a href="/2019/06/24/full-binary-tree-catamorphism">Full binary tree catamorphism</a></li> <li><a href="/2019/07/08/payment-types-catamorphism">Payment types catamorphism</a></li> </ul> </p> <p> Each of these articles will contain a fair amount of <a href="https://www.haskell.org">Haskell</a> code, but even if you're an object-oriented programmer who doesn't read Haskell, you should still scan them, as I'll start each with some C# examples. The Haskell code, by the way, is <a href="https://github.com/ploeh/FAlgebras">available on GitHub</a>. </p> <h3 id="cc687d1bebed47229cbdeffdf98fd666"> Greek <a href="#cc687d1bebed47229cbdeffdf98fd666" title="permalink">#</a> </h3> <p> When encountering a word like <em>catamorphism</em>, your reaction might be: <blockquote> "Catamorphism?! What does that even mean? It's all Greek to me." </blockquote> Indeed, it's Greek, as is so much of mathematical terminology. The <em>cata</em> prefix means 'down'; lots of words start with <em>cata</em>, like <em>catastrophe</em>, <em>catalogue</em>, <em>catatonia</em>, <em>catacomb</em>, etc. </p> <p> The <em>morph</em> suffix generally means 'shape'. While the <em>cata</em> prefix appears in common words like <em>catastrophe</em>, the <em>morph</em> suffix mostly appears in more academic contexts. Programmers will probably have encountered <em>polymorphism</em> and <em>skeuomorphism</em>, not to mention <a href="/2018/01/08/software-design-isomorphisms">isomorphism</a>. While <em>morphism</em> is heavily used in mathematics, other sciences use the suffix too, like <em>dimorphism</em> in biology. </p> <p> In category theory, a <em>morphism</em> is basically just an arrow that points from one object to another. Think of it as a function. </p> <p> If a morphism is just a function, why don't we just call it that, then? Is it really necessary with this intimidating terminology? Yes and no. </p> <p> If someone had originally figured all of this out in the context of mainstream programming, he or she would probably have used friendlier names, like <em>condense</em>, <em>reduce</em>, <em>fold</em>, and so on. This would have been more encouraging, although <a href="/2017/10/05/monoids-semigroups-and-friends">I'm not sure it would have been better</a>. </p> <p> In software architecture we use many overloaded terms. For example, what's a <em>service</em>, or a <em>client?</em> What does <em>tier</em> mean? Is it the same as a <em>layer</em>, or is it something different? What's the <a href="http://tomasp.net/blog/2015/library-frameworks">difference between a library and a framework</a>? </p> <p> At least a word like <em>catamorphism</em> is concise. It's not in common use, so isn't overloaded and vague. </p> <p> Another, more pragmatic, concern is that whether you like it or not, the terminology is already established. Mathematicians decided to name the concept <em>catamorphism</em>. While the name may seem intimidating, I prefer to teach concepts like these using established terminology. This means that if my articles are unclear, you can do further research with other resources. That's the benefit of established terminology, whether you like the specific words or not. </p> <h3 id="c179dad7693c48c2ae592d33cb2be792"> Summary <a href="#c179dad7693c48c2ae592d33cb2be792" title="permalink">#</a> </h3> <p> You can compose entire applications based on the abstractions of <em>map</em> and <em>reduce</em>. You can see one example of such a system in my <a href="https://blog.ploeh.dk/functional-architecture-with-fsharp">A Functional Architecture with F#</a> Pluralsight course. </p> <p> The terms <em>map</em> and <em>reduce</em> may, however, not be helpful, because it may not be clear exactly what types of data you can map, and what types you can reduce. One of the most important goals of this overall article series about universal abstractions is to help you identify when such software architectures apply. This is more often that you think. </p> <p> What sort of data can you map? You can map <em>functors</em>. While hardly finite, there's a catalogue of well-known functors, of which I've covered some, but not all. That catalogue contains data containers like <a href="/2018/03/26/the-maybe-functor">Maybe</a>, <a href="/2018/08/06/a-tree-functor">Tree</a>, <a href="/2018/09/10/the-lazy-functor">lazy computations</a>, <a href="/2018/09/24/asynchronous-functors">tasks</a>, and perhaps a score more. The catalogue of (actually useful) functors has, in my experience, a manageable size. </p> <p> Likewise you could ask: What sort of data can you reduce? How do you implement that reduction? Again, there's a compact set of well-known catamorphisms. How do you reduce a collection? You use its catamorphism (which is equal to a fold). How do you reduce a tree? You use its catamorphism. How do you reduce an Either object? You use its catamorphism. </p> <p> When we learn new programming languages, new libraries, new frameworks, we gladly invest time in learning hundreds, if not thousands, of keywords, APIs, extensibility points, and so on. May I offer, for your consideration, that your mental resources are better spent learning only a handful of universal abstractions? </p> <p> <strong>Next:</strong> <a href="/2019/05/06/boolean-catamorphism">Boolean catamorphism</a>. </p> </div> <div id="comments"> <hr> <h2 id="comments-header"> Comments </h2> <div class="comment" id="673934773ecd4263a557599471eaf57c"> <div class="comment-author">Tyson Williams</div> <div class="comment-content"> <blockquote> In other (functional) languages, such a 'reduce' operation is called a <em>fold</em>. The metaphor, I suppose, is that you fold multiple values together, two by two. <br> ...the <code>Aggregate</code> overload that takes a <em>seed</em> value... </blockquote> <p> My impression is that part of the functional programming style is to avoid function overloading. Consistent with that is the naming used by Language Ext for these concepts. In Language Ext, the function with type (in F# notation) <code>seq&lt;'a&gt; -&gt; ('a -&gt; 'a -&gt; 'a) -&gt; 'a</code> is called <a href="https://github.com/louthy/language-ext/blob/master/LanguageExt.Core/DataTypes/List/Lst.Extensions.cs#L599">Reduce</a> and the function with type (in F# notation) <code>seq&lt;'a&gt; -&gt; 'b -&gt; ('b -&gt; 'a -&gt; 'b) -&gt; 'b</code> is called <a href="https://github.com/louthy/language-ext/blob/master/LanguageExt.Core/DataTypes/List/Lst.Extensions.cs#L420">Fold</a>. </p> <p> I don't know the origin of these two names, but I remember the difference by thinking about preparing food. In cooking, <a href="https://en.wikipedia.org/wiki/Reduction_(cooking)">reduction</a> increases the concentration of a liquid by boiling away some of its water. I think of the returned <code>'a</code> as being a highly concentrated form of the input sequence since every sequence element (and only those elements) was used to create that return value. In baking, <a href="https://www.wikihow.com/Fold-(Baking)">folding</a> is a technique that carefully combines two mixtures into one. This reminds me of how the seed value <code>'b</code> and the sequence of <code>'a</code> are (typically) two different types and are combined by the given function. They are not perfect analogies, but they work for me. </p> <p> On a related note, <a href="https://github.com/louthy/language-ext/issues/583">I dislike</a> that Reduce returns <code>'a</code> instead of <code>Option<'a></code>. </p> </div> <div class="comment-date">2019-07-12 12:20 UTC</div> </div> <div class="comment" id="43e48df0645b4c05992b0f599cc71d10"> <div class="comment-author"><a href="/">Mark Seemann</a></div> <div class="comment-content"> <p> Tyson, thank you for writing. As you may know, <a href="https://amzn.to/2TE8tJx">my book</a> liberally employs cooking analogies, but I admit that I've never thought about <em>reduction</em> and <em>fold</em> in that light before. Good analogies, although perhaps a bit <em>strained</em> (pun intended). </p> <p> They do work well, though, for the reasons you give. <blockquote> "the functional programming style is to avoid function overloading" </blockquote> As far as I can tell, this has more to do with the combination of <a href="https://en.wikipedia.org/wiki/Hindley%E2%80%93Milner_type_system">Hindley–Milner type inference</a> and currying you encounter in Haskell and ML-derived languages than it has to do with functional programming in itself. If I recall correctly, <a href="https://clojure.org">Clojure</a> makes copious use of overloading. </p> <p> The problem with overloading in a language like <a href="https://fsharp.org">F#</a> is that if you imagine that the function you refer to as <code>fold</code> was also called <code>reduce</code>, a partially applied expression like this would be ambiguous: </p> <p> <pre>let foo = reduce xs bar</pre> </p> <p> What is <code>bar</code>, here? If <code>reduce</code> is overloaded, is it a function, or is it a 'seed value'? </p> <p> As far as I can tell, the compiler can't infer that, so instead of compromising on type inference, the languages in question disallow function overloading. </p> <p> Notice, additionally, that F# does allow <em>method</em> overloading, for the part of the language that enables interoperability with the rest of .NET. In that part of the language, type inference rarely works anyway. I'm not an expert in how the F# compiler works, but I've always understood this to indicate that the interop part of F# isn't based on Hindley-Milner. I don't see how it could be, since the .NET/IL type system isn't a Hindley-Milner type system. </p> <p> The <code>reduce</code> function you refer to is, by the way, based on a <a href="/2017/11/27/semigroups">semigroup</a> instance. More specifically, it's simply how <a href="/2017/12/11/semigroups-accumulate">semigroups accumulate</a>. I agree that <code>reduce</code> is partial, and therefore not as pretty as one could wish, but I think a more appropriate solution is to define it on <code>NotEmptyCollection&lt;T&gt;</code>, instead of on <code>IEnumerable&lt;T&gt;</code>, as shown in <a href="/2017/12/11/semigroups-accumulate">that article</a>. </p> <p> In other words, I don't think <code>reduce</code> belongs on <code>IEnumerable&lt;T&gt;</code> at all. I know it's in both F# and Haskell, but my personal opinion is that it shouldn't be, just like Haskell's <code>head</code> function ought not to exist either... </p> </div> <div class="comment-date">2019-07-14 16:29 UTC</div> </div> </div><hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Applicative monoids https://blog.ploeh.dk/2019/04/22/applicative-monoids 2019-04-22T05:36:00+00:00 Mark Seemann <div id="post"> <p> <em>An applicative functor containing monoidal values itself forms a monoid.</em> </p> <p> This article is an instalment in <a href="/2018/10/01/applicative-functors">an article series about applicative functors</a>. An applicative functor is a <a href="https://bartoszmilewski.com/2014/01/14/functors-are-containers">data container</a> that supports combinations. If an applicative functor contains values of a type that gives rise to a <a href="/2017/10/06/monoids">monoid</a>, then the <a href="/2018/03/22/functors">functor</a> itself forms a monoid. </p> <p> In a previous article you learned that <a href="/2019/04/15/lazy-monoids">lazy computations of monoids remain monoids</a>. Furthermore, <a href="/2018/12/17/the-lazy-applicative-functor">a lazy computation is an applicative functor</a>, and it turns out that the result generalises. The result regarding lazy computation is just a special case. </p> <h3 id="3c6acb0da15b4ae8b78f5d8879b7efe3"> Monap <a href="#3c6acb0da15b4ae8b78f5d8879b7efe3" title="permalink">#</a> </h3> <p> Since version 4.11 of <a href="https://www.haskell.org">Haskell</a>'s <em>base</em> library, <code>Monoid</code> is a subset of <code>Semigroup</code>, so in order to create a <code>Monoid</code> instance, you must first define a <code>Semigroup</code> instance. </p> <p> In order to escape the need for flexible contexts, you'll have to define a wrapper <code>newtype</code> that'll be the instance. What should you call it? It's going to be an applicative functor of monoids, so perhaps something like <em>ApplicativeMonoid?</em> Nah, that's too long. <em>AppMon</em>, then? Sure, but how about flipping the terms: <em>MonApp?</em> That's better. Let's drop the last <em>p</em> and dispense with the <a href="https://en.wikipedia.org/wiki/Camel_case">Pascal case</a>: <em>Monap</em>. </p> <p> <em>Monap</em> almost looks like <em>Monad</em>, only with the last letter rotated half a revolution. This should allow for maximum confusion. </p> <p> To be clear, I normally don't advocate for droll word play when writing production code, but I occasionally do it in articles and presentations. The <em>Monap</em> in this article exists only to illustrate a point. It's not intended to be used. Furthermore, this article doesn't discuss monads at all, so the risk of confusion should, hopefully, be minimised. I may, however, regret this decision... </p> <h3 id="fb2568655a314bba8e417d06317b2690"> Applicative semigroup <a href="#fb2568655a314bba8e417d06317b2690" title="permalink">#</a> </h3> <p> First, introduce the wrapper <code>newtype</code>: </p> <p> <pre><span style="color:blue;">newtype</span>&nbsp;Monap&nbsp;f&nbsp;a&nbsp;=&nbsp;Monap&nbsp;{&nbsp;runMonap&nbsp;::&nbsp;f&nbsp;a&nbsp;}&nbsp;<span style="color:blue;">deriving</span>&nbsp;(<span style="color:#2b91af;">Show</span>,&nbsp;<span style="color:#2b91af;">Eq</span>)</pre> </p> <p> This only states that there's a type called <code>Monap</code> that wraps some higher-kinded type <code>f a</code>; that is, a container <code>f</code> of values of the type <code>a</code>. The intent is that <code>f</code> is an applicative functor, hence the use of the letter <em>f</em>, but the type itself doesn't constrain <code>f</code> to any type class. </p> <p> The <code>Semigroup</code> instance does, though: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;(<span style="color:blue;">Applicative</span>&nbsp;f,&nbsp;<span style="color:blue;">Semigroup</span>&nbsp;a)&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;<span style="color:blue;">Semigroup</span>&nbsp;(<span style="color:blue;">Monap</span>&nbsp;f&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;(Monap&nbsp;x)&nbsp;&lt;&gt;&nbsp;(Monap&nbsp;y)&nbsp;=&nbsp;Monap&nbsp;$&nbsp;liftA2&nbsp;<span style="color:#2b91af;">(&lt;&gt;)</span>&nbsp;x&nbsp;y </pre> </p> <p> This states that when <code>f</code> is a <code>Applicative</code> instance, and <code>a</code> is a <code>Semigroup</code> instance, then <code>Monap f a</code> is also a <code>Semigroup</code> instance. </p> <p> Here's an example of combining two applicative <a href="/2017/11/27/semigroups">semigroups</a>: </p> <p> <pre>λ&gt; Monap (Just (Max 42)) &lt;&gt; Monap (Just (Max 1337)) Monap {runMonap = Just (Max {getMax = 1337})}</pre> </p> <p> This example uses the <code>Max</code> semigroup container, and <code>Maybe</code> as the applicative functor. For <code>Max</code>, the <code>&lt;&gt;</code> operator returns the value that contains the highest value, which in this case is 1337. </p> <p> It even works when the applicative functor in question is <code>IO</code>: </p> <p> <pre>λ&gt; runMonap $ Monap (Sum &lt;$&gt; randomIO @Word8) &lt;&gt; Monap (Sum &lt;$&gt; randomIO @Word8) Sum {getSum = 165}</pre> </p> <p> This example uses <code>randomIO</code> to generate two random values. It uses the <code>TypeApplications</code> GHC extension to make <code>randomIO</code> generate <code>Word8</code> values. Each random number is projected into the <code>Sum</code> container, which means that <code>&lt;&gt;</code> will add the numbers together. In the above example, the result is 165, but if you evaluate the expression a second time, you'll most likely get another result: </p> <p> <pre>λ&gt; runMonap $ Monap (Sum &lt;$&gt; randomIO @Word8) &lt;&gt; Monap (Sum &lt;$&gt; randomIO @Word8) Sum {getSum = 246}</pre> </p> <p> You can also use linked list (<code>[]</code>) as the applicative functor. In this case, the result may be surprising (depending on what you expect): </p> <p> <pre>λ&gt; Monap [Product 2, Product 3] &lt;&gt; Monap [Product 4, Product 5, Product 6] Monap {runMonap = [Product {getProduct = 8},Product {getProduct = 10},Product {getProduct = 12}, Product {getProduct = 12},Product {getProduct = 15},Product {getProduct = 18}]}</pre> </p> <p> Notice that we get all the combinations of products: <em>2</em> multiplied with each element in the second list, followed by <em>3</em> multiplied by each of the elements in the second list. This shouldn't be that startling, though, since you've already, previously in this article series, seen several examples of how an applicative functor implies combinations. </p> <h3 id="33d4ff4f0fbf4e3cbf4d802b0f287f63"> Applicative monoid <a href="#33d4ff4f0fbf4e3cbf4d802b0f287f63" title="permalink">#</a> </h3> <p> With the <code>Semigroup</code> instance in place, you can now add the <code>Monoid</code> instance: </p> <p> <pre><span style="color:blue;">instance</span>&nbsp;(<span style="color:blue;">Applicative</span>&nbsp;f,&nbsp;<span style="color:blue;">Monoid</span>&nbsp;a)&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;<span style="color:blue;">Monoid</span>&nbsp;(<span style="color:blue;">Monap</span>&nbsp;f&nbsp;a)&nbsp;<span style="color:blue;">where</span> &nbsp;&nbsp;mempty&nbsp;=&nbsp;Monap&nbsp;$&nbsp;pure&nbsp;$&nbsp;mempty </pre> </p> <p> This is straightforward: you take the identity (<code>mempty</code>) of the monoid <code>a</code>, promote it to the applicative functor <code>f</code> with <code>pure</code>, and finally put that value into the <code>Monap</code> wrapper. </p> <p> This works fine as well: </p> <p> <pre>λ&gt; mempty :: Monap Maybe (Sum Integer) Monap {runMonap = Just (Sum {getSum = 0})} λ&gt; mempty :: Monap [] (Product Word8) Monap {runMonap = [Product {getProduct = 1}]}</pre> </p> <p> The identity laws also seem to hold: </p> <p> <pre>λ&gt; Monap (Right mempty) &lt;&gt; Monap (Right (Sum 2112)) Monap {runMonap = Right (Sum {getSum = 2112})} λ&gt; Monap ("foo", All False) &lt;&gt; Monap mempty Monap {runMonap = ("foo",All {getAll = False})}</pre> </p> <p> The last, right-identity example is interesting, because the applicative functor in question is a tuple. Tuples are <code>Applicative</code> instances when the first, or left, element is a <code>Monoid</code> instance. In other words, <code>f</code> is, in this case, <code>(,) String</code>. The <code>Monoid</code> instance that <code>Monap</code> sees as <code>a</code>, on the other hand, is <code>All</code>. </p> <p> Since <a href="/2017/10/30/tuple-monoids">tuples of monoids are themselves monoids</a>, however, I can get away with writing <code>Monap mempty</code> on the right-hand side, instead of the more elaborate template the other examples use: </p> <p> <pre>λ&gt; Monap ("foo", All False) &lt;&gt; Monap ("", mempty) Monap {runMonap = ("foo",All {getAll = False})}</pre> </p> <p> or perhaps even: </p> <p> <pre>λ&gt; Monap ("foo", All False) &lt;&gt; Monap (mempty, mempty) Monap {runMonap = ("foo",All {getAll = False})}</pre> </p> <p> Ultimately, all three alternatives mean the same. </p> <h3 id="6b484b60d19b4ef4a3716ceaa693cb8b"> Associativity <a href="#6b484b60d19b4ef4a3716ceaa693cb8b" title="permalink">#</a> </h3> <p> As usual, I'm not going to do the work of formally proving that the monoid laws hold for the <code>Monap</code> instances, but I'd like to share some QuickCheck properties that indicate that they do, starting with a property that verifies associativity: </p> <p> <pre><span style="color:#2b91af;">assocLaw</span>&nbsp;::&nbsp;(<span style="color:blue;">Eq</span>&nbsp;a,&nbsp;<span style="color:blue;">Show</span>&nbsp;a,&nbsp;<span style="color:blue;">Semigroup</span>&nbsp;a)&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Property</span> assocLaw&nbsp;x&nbsp;y&nbsp;z&nbsp;=&nbsp;(x&nbsp;&lt;&gt;&nbsp;y)&nbsp;&lt;&gt;&nbsp;z&nbsp;===&nbsp;x&nbsp;&lt;&gt;&nbsp;(y&nbsp;&lt;&gt;&nbsp;z)</pre> </p> <p> This property is entirely generic. It'll verify associativity for any <code>Semigroup a</code>, not only for <code>Monap</code>. You can, however, run it for various <code>Monap</code> types, as well. You'll see how this is done a little later. </p> <h3 id="51973eba155f418e8903d37f6d3938d2"> Identity <a href="#51973eba155f418e8903d37f6d3938d2" title="permalink">#</a> </h3> <p> Likewise, you can write two properties that check left and right identity, respectively. </p> <p> <pre><span style="color:#2b91af;">leftIdLaw</span>&nbsp;::&nbsp;(<span style="color:blue;">Eq</span>&nbsp;a,&nbsp;<span style="color:blue;">Show</span>&nbsp;a,&nbsp;<span style="color:blue;">Monoid</span>&nbsp;a)&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Property</span> leftIdLaw&nbsp;x&nbsp;=&nbsp;x&nbsp;===&nbsp;mempty&nbsp;&lt;&gt;&nbsp;x <span style="color:#2b91af;">rightIdLaw</span>&nbsp;::&nbsp;(<span style="color:blue;">Eq</span>&nbsp;a,&nbsp;<span style="color:blue;">Show</span>&nbsp;a,&nbsp;<span style="color:blue;">Monoid</span>&nbsp;a)&nbsp;<span style="color:blue;">=&gt;</span>&nbsp;a&nbsp;<span style="color:blue;">-&gt;</span>&nbsp;<span style="color:blue;">Property</span> rightIdLaw&nbsp;x&nbsp;=&nbsp;x&nbsp;===&nbsp;x&nbsp;&lt;&gt;&nbsp;mempty </pre> </p> <p> Again, this is entirely generic. These properties can be used to test the identity laws for any monoid, including <code>Monap</code>. </p> <h3 id="0cc5968e519c48a0816574e1dc1667fc"> Properties <a href="#0cc5968e519c48a0816574e1dc1667fc" title="permalink">#</a> </h3> <p> You can run each of these properties multiple time, for various different functors and monoids. As <code>Applicative</code> instances, I've used <code>Maybe</code>, <code>[]</code>, <code>(,) Any</code>, and <code>Identity</code>. As <code>Monoid</code> instances, I've used <code>String</code>, <code>Sum Integer</code>, <code>Max Int16</code>, and <code>[Float]</code>. Notice that a list (<code>[]</code>) is both an applicative functor as well as a monoid. In this test set, I've used it in both roles. </p> <p> <pre>tests&nbsp;= &nbsp;&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;testGroup&nbsp;<span style="color:#a31515;">&quot;Properties&quot;</span>&nbsp;[ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;testProperty&nbsp;<span style="color:#a31515;">&quot;Associativity&nbsp;law,&nbsp;Maybe&nbsp;String&quot;</span>&nbsp;(assocLaw&nbsp;@(Monap&nbsp;Maybe&nbsp;String)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;testProperty&nbsp;<span style="color:#a31515;">&quot;Left&nbsp;identity&nbsp;law,&nbsp;Maybe&nbsp;String&quot;</span>&nbsp;(leftIdLaw&nbsp;@(Monap&nbsp;Maybe&nbsp;String)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;testProperty&nbsp;<span style="color:#a31515;">&quot;Right&nbsp;identity&nbsp;law,&nbsp;Maybe&nbsp;String&quot;</span>&nbsp;(rightIdLaw&nbsp;@(Monap&nbsp;Maybe&nbsp;String)), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;testProperty&nbsp;<span style="color:#a31515;">&quot;Associativity&nbsp;law,&nbsp;[Sum&nbsp;Integer]&quot;</span>&nbsp;(assocLaw&nbsp;@(Monap&nbsp;<span style="color:blue;">[]</span>&nbsp;(Sum&nbsp;Integer))), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;testProperty&nbsp;<span style="color:#a31515;">&quot;Left&nbsp;identity&nbsp;law,&nbsp;[Sum&nbsp;Integer]&quot;</span>&nbsp;(leftIdLaw&nbsp;@(Monap&nbsp;<span style="color:blue;">[]</span>&nbsp;(Sum&nbsp;Integer))), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;testProperty&nbsp;<span style="color:#a31515;">&quot;Right&nbsp;identity&nbsp;law,&nbsp;[Sum&nbsp;Integer]&quot;</span>&nbsp;(rightIdLaw&nbsp;@(Monap&nbsp;<span style="color:blue;">[]</span>&nbsp;(Sum&nbsp;Integer))), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;testProperty&nbsp;<span style="color:#a31515;">&quot;Associativity&nbsp;law,&nbsp;(Any,&nbsp;Max&nbsp;Int8)&quot;</span>&nbsp;(assocLaw&nbsp;@(Monap&nbsp;(<span style="color:#2b91af;">(,)</span>&nbsp;Any)&nbsp;(Max&nbsp;Int8))), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;testProperty&nbsp;<span style="color:#a31515;">&quot;Left&nbsp;identity&nbsp;law,&nbsp;(Any,&nbsp;Max&nbsp;Int8)&quot;</span>&nbsp;(leftIdLaw&nbsp;@(Monap&nbsp;(<span style="color:#2b91af;">(,)</span>&nbsp;Any)&nbsp;(Max&nbsp;Int8))), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;testProperty&nbsp;<span style="color:#a31515;">&quot;Right&nbsp;identity&nbsp;law,&nbsp;(Any,&nbsp;Max&nbsp;Int8)&quot;</span>&nbsp;(rightIdLaw&nbsp;@(Monap&nbsp;(<span style="color:#2b91af;">(,)</span>&nbsp;Any)&nbsp;(Max&nbsp;Int8))), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;testProperty&nbsp;<span style="color:#a31515;">&quot;Associativity&nbsp;law,&nbsp;Identity&nbsp;[Float]&quot;</span>&nbsp;(assocLaw&nbsp;@(Monap&nbsp;Identity&nbsp;[Float])), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;testProperty&nbsp;<span style="color:#a31515;">&quot;Left&nbsp;identity&nbsp;law,&nbsp;Identity&nbsp;[Float]&quot;</span>&nbsp;(leftIdLaw&nbsp;@(Monap&nbsp;Identity&nbsp;[Float])), &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;testProperty&nbsp;<span style="color:#a31515;">&quot;Right&nbsp;identity&nbsp;law,&nbsp;Identity&nbsp;[Float]&quot;</span>&nbsp;(rightIdLaw&nbsp;@(Monap&nbsp;Identity&nbsp;[Float])) &nbsp;&nbsp;&nbsp;&nbsp;] &nbsp;&nbsp;] </pre> </p> <p> All of these properties pass. </p> <h3 id="af35f2986a734a16be80590c86d0432d"> Summary <a href="#af35f2986a734a16be80590c86d0432d" title="permalink">#</a> </h3> <p> It seems that any applicative functor that contains monoidal values itself forms a monoid. The <code>Monap</code> type presented in this article only exists to demonstrate this conjecture; it's not intended to be <em>used</em>. </p> <p> If it holds, I think it's an interesting result, because it further enables you to reason about the properties of complex systems, based on the properties of simpler systems. </p> <p> <strong>Next: </strong> <a href="/2018/12/24/bifunctors">Bifunctors</a>. </p> </div> <div id="comments"> <hr> <h2 id="comments-header"> Comments </h2> <div class="comment" id="9a3acf3cf4174e178dc9349e11fee488"> <div class="comment-author">Tyson Williams</div> <div class="comment-content"> <blockquote> It seems that any applicative functor that contains monoidal values itself forms a monoid. </blockquote> <p> Is it necessary for the functor to be applicative? Do you know of a functor that contains monoidal values for which itself does <em>not</em> form a monoid? </p> </div> <div class="comment-date">2019-05-13 11:28 UTC</div> </div> <div class="comment" id="a164909adb884cd78b309a81029a2dd8"> <div class="comment-author"><a href="/">Mark Seemann</a></div> <div class="comment-content"> <p> Tyson, thank you for writing. Yes, it's necessary for the functor to be applicative, because you need the applicative combination operator <code>&lt;*&gt;</code> in order to implement the combination. In C#, you'd need an <code>Apply</code> method <a href="/2018/10/01/applicative-functors/#cef395ee19644f30bfd1ad7a84b6f912">as shown here</a>. </p> <p> Technically, the monoidal <code>&lt;&gt;</code> operator for <code>Monap</code> is, as you can see, implemented with a call to <code>liftA2</code>. In Haskell, you can implement an instance of <code>Applicative</code> by implementing either <code>liftA2</code> or <code>&lt;*&gt;</code>, as well as <code>pure</code>. You usually see <code>Applicative</code> described by <code>&lt;*&gt;</code>, which is what I've done in <a href="/2018/10/01/applicative-functors">my article series on applicative functors</a>. If you do that, you can define <code>liftA2</code> by a combination of <code>&lt;*&gt;</code> and <code>fmap</code> (the <code>Select</code> method that defines functors). </p> <p> If you want to put this in C# terms, you need both <code>Select</code> and <code>Apply</code> in order to be able to lift a monoid into a functor. </p> <p> Is there a functor that contains monoidal values that itself doesn't form a monoid? </p> <p> Yes, indeed. In order to answer that question, we 'just' need to identify a functor that's <em>not</em> an applicative functor. Tuples are good examples. </p> <p> A <a href="https://blog.ploeh.dk/2018/12/31/tuple-bifunctor/#d918d0271c33406ba3047ef162212100">tuple forms a functor</a>, but in general nothing more than that. Consider a tuple where the first element is a <code>Guid</code>. It's a functor, but can you implement the following function? </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:#2b91af;">Guid</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;Apply&lt;<span style="color:#2b91af;">TResult</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:#2b91af;">Guid</span>,&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&gt;&nbsp;selector, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:#2b91af;">Guid</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;&nbsp;source) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">throw</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">NotImplementedException</span>(<span style="color:#a31515;">&quot;What&nbsp;would&nbsp;you&nbsp;write&nbsp;here?&quot;</span>); }</pre> </p> <p> You can pull the <code>T</code> value out of <code>source</code> and project it to a <code>TResult</code> value with <code>selector</code>, but you'll need to put it back in a <code>Tuple&lt;Guid, TResult&gt;</code>. Which <code>Guid</code> value are you going to use for that tuple? </p> <p> There's no clear answer to that question. </p> <p> More specifically, consider <code>Tuple&lt;Guid, int&gt;</code>. This is a functor that contains monoidal values. Let's say that we want to use the addition monoid over integers. How would you implement the following method? </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:#2b91af;">Guid</span>,&nbsp;<span style="color:blue;">int</span>&gt;&nbsp;Add(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:#2b91af;">Guid</span>,&nbsp;<span style="color:blue;">int</span>&gt;&nbsp;x,&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:#2b91af;">Guid</span>,&nbsp;<span style="color:blue;">int</span>&gt;&nbsp;y) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">throw</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">NotImplementedException</span>(<span style="color:#a31515;">&quot;What&nbsp;would&nbsp;you&nbsp;write&nbsp;here?&quot;</span>); }</pre> </p> <p> Again, you run into the issue that while you can pull the integers out of the tuples and add them together, there's no clear way to figure out which <code>Guid</code> value to put into the tuple that contains the sum. </p> <p> The issue particularly with tuples is that there's no general way to combine the leftmost values of the tuples. If there is - that is, if leftmost values form a monoid - then the tuple is also an applicative functor. For example, <code>Tuple&lt;string, int&gt;</code> is applicative and forms a monoid over addition: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&nbsp;Apply&lt;<span style="color:#2b91af;">TResult</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;( &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:#2b91af;">Func</span>&lt;<span style="color:#2b91af;">T</span>,&nbsp;<span style="color:#2b91af;">TResult</span>&gt;&gt;&nbsp;selector, &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;&nbsp;source) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:#2b91af;">Tuple</span>.Create( &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;selector.Item1&nbsp;+&nbsp;source.Item1, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;selector.Item2(source.Item2)); } <span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">int</span>&gt;&nbsp;Add(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">int</span>&gt;&nbsp;x,&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:blue;">int</span>&gt;&nbsp;y) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:#2b91af;">Tuple</span>.Create(x.Item1&nbsp;+&nbsp;y.Item1,&nbsp;x.Item2&nbsp;+&nbsp;y.Item2); }</pre> </p> <p> You can also implement <code>Add</code> with <code>Apply</code>, but you're going to need two <code>Apply</code> overloads to make it work. </p> <p> Incidentally, unbeknownst to me, the <code>Ap</code> wrapper was added to Haskell's <code>Data.Monoid</code> module 12 days before I wrote this article. In all but name, it's equivalent to the <code>Monap</code> wrapper presented here. </p> </div> <div class="comment-date">2019-05-14 20:44 UTC</div> </div> <div class="comment" id="8900297a3b484ac0adfb8574a66cff87"> <div class="comment-author">Tyson Williams</div> <div class="comment-content"> <blockquote> <p> ...if leftmost values form a monoid - then the tuple is also an applicative functor. For example, <code>Tuple&lt;string, int&gt;</code> is applicative... </p> </blockquote> <p> I want to add some prepositional phrases to our statements like I <a href="https://blog.ploeh.dk/2019/01/07/either-bifunctor/#79f5d74763e34cb0997a7a79df1e05f0">commented here</a> to help claify things. I don't think that <code>Tuple&lt;string, int&gt;</code> can be applicative because there are no type parameters in which it could be applicative. Were you trying to say that <code>Tuple&lt;string, B&gt;</code> is applicative in <code>B</code>? This seems to match your <code>Apply</code> function, which doesn't depend on <code>int</code>. </p> </div> <div class="comment-date">2019-05-30 05:02 UTC</div> </div> <div class="comment" id="daa6444c8fb0484a8bf1accdb3cc544b"> <div class="comment-author"><a href="/">Mark Seemann</a></div> <div class="comment-content"> <p> Tyson, you're quite right; good catch. My wording was incorrect (I was probably either tired or in a hurry when I wrote that), but fortunately, the code looks like it's correct. </p> <p> That you for pointing out my error. </p> </div> <div class="comment-date">2019-05-30 13:00 UTC</div> </div> <div class="comment" id="075df592e63948e695851d7ae20842ea"> <div class="comment-author">Tyson Williams</div> <div class="comment-content"> <blockquote> <p> ...<code>Tuple&lt;string, int&gt;</code> is applicative and forms a monoid over addition... </p> </blockquote> <p> I do agree with the monoid part, where "addition" means string concatenation for the first item and integer addition for the second item. </p> <blockquote> <p> <code>Tuple&lt;string, B&gt;</code> is applicative in <code>B</code> </p> </blockquote> <p> Now I am trying to improve my understanding of this statement. In Haskell, my understanding the definition of the <a href="https://en.wikibooks.org/wiki/Haskell/Applicative_functors#The_Applicative_class">Applicative type class</a> applied to <code>Tuple&lt;string, B&gt;</code> requires a function <code>pure</code> from <code>B</code> to <code>Tuple&lt;string, B&gt;</code>. What it the definition of this funciton? Does it use the empty string in order to make an instance of <code>Tuple&lt;string, B&gt;</code>? If so, what is the justification for this? Or maybe my reasoning here is mistaken. </p> </div> <div class="comment-date">2019-05-31 12:52 UTC</div> </div> <div class="comment" id="ce69d4440f314827a8ec441511d0ce71"> <div class="comment-author"><a href="/">Mark Seemann</a></div> <div class="comment-content"> <p> Tyson, thank you for writing. In Haskell, it's true that applicative functors must also define <code>pure</code>. In this article series, I've glosssed over that constraint, since I'm not aware of any data containers that can lawfully implement <code>&lt;*&gt;</code> or <code>liftA2</code>, but <em>can't</em> define <code>pure</code>. </p> <p> The applicative instance for tuples is, however, constrained: </p> <p> <pre>Monoid a =&gt; Applicative ((,) a)</pre> </p> <p> The construct <code>((,) a)</code> means any tuple where the first element has the generic type <code>a</code>. The entire expression means that tuples are applicative functors when the first element forms a monoid; that's the restriction on <code>a</code>. The definition of <code>pure</code>, then, is: </p> <p> <pre>pure x = (mempty, x)</pre> </p> <p> That is, use the monoidal identity (<code>mempty</code>) for the first element, and use <code>x</code> as the second element. For strings, since the identity for string concatenation is the empty string, yes, it does exactly mean that <code>pure</code> for <code>Tuple&lt;string, B&gt;</code> would return a tuple with the empty string, and the input argument as the second element: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">Tuple</span>&lt;<span style="color:blue;">string</span>,&nbsp;<span style="color:#2b91af;">T</span>&gt;&nbsp;Pure&lt;<span style="color:#2b91af;">T</span>&gt;(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">T</span>&nbsp;x) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:#2b91af;">Tuple</span>.Create(<span style="color:#a31515;">&quot;&quot;</span>,&nbsp;x); }</pre> </p> <p> That's the behaviour you get from Haskell as well: </p> <p> <pre>Prelude Data.Monoid&gt; pure 42 :: (String, Int) ("",42)</pre> </p> </div> <div class="comment-date">2019-05-31 14:59 UTC</div> </div> </div> <hr> This blog is totally free, but if you like it, please consider <a href="https://blog.ploeh.dk/support">supporting it</a>. Lazy monoids https://blog.ploeh.dk/2019/04/15/lazy-monoids 2019-04-15T13:54:00+00:00 Mark Seemann <div id="post"> <p> <em>Lazy monoids are monoids. An article for object-oriented programmers.</em> </p> <p> This article is part of a <a href="/2017/10/06/monoids">series about monoids</a>. In short, a <em>monoid</em> is an associative binary operation with a neutral element (also known as <em>identity</em>). Previous articles have shown how more complex monoids arise from simpler monoids, such as <a href="/2017/10/30/tuple-monoids">tuple monoids</a>, <a href="/2017/11/06/function-monoids">function monoids</a>, and <a href="/2018/04/03/maybe-monoids">Maybe monoids</a>. This article shows another such result: how lazy computations of monoids itself form monoids. </p> <p> You'll see how simple this is through a series of examples. Specifically, you'll revisit several of the examples you've already seen in this article series. </p> <h3 id="a715cff45376401db9863a095a5e156d"> Lazy addition <a href="#a715cff45376401db9863a095a5e156d" title="permalink">#</a> </h3> <p> Perhaps the most intuitive monoid is <em>addition</em>. Lazy addition forms a monoid as well. In C#, you can implement this with a simple extension method: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;Add(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;x,&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;y) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;(()&nbsp;=&gt;&nbsp;x.Value&nbsp;+&nbsp;y.Value); }</pre> </p> <p> This <code>Add</code> method simply adds two lazy integers together in a lazy computation. You use it like any other extension method: </p> <p> <pre><span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;x&nbsp;=&nbsp;<span style="color:green;">//&nbsp;...</span> <span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;y&nbsp;=&nbsp;<span style="color:green;">//&nbsp;...</span> <span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;sum&nbsp;=&nbsp;x.Add(y);</pre> </p> <p> I'll spare you the tedious listing of <a href="https://fscheck.github.io/FsCheck">FsCheck</a>-based properties that demonstrate that the monoid laws hold. We'll look at an example of such a set of properties later in this article, for one of the other monoids. </p> <h3 id="eda5d39029904d70995ebee84570cf60"> Lazy multiplication <a href="#eda5d39029904d70995ebee84570cf60" title="permalink">#</a> </h3> <p> Not surprisingly, I hope, you can implement multiplication over lazy numbers in the same way: </p> <p> <pre><span style="color:blue;">public</span>&nbsp;<span style="color:blue;">static</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;Multiply(<span style="color:blue;">this</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;x,&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;y) { &nbsp;&nbsp;&nbsp;&nbsp;<span style="color:blue;">return</span>&nbsp;<span style="color:blue;">new</span>&nbsp;<span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;(()&nbsp;=&gt;&nbsp;x.Value&nbsp;*&nbsp;y.Value); }</pre> </p> <p> Usage is similar to lazy addition: </p> <p> <pre><span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;x&nbsp;=&nbsp;<span style="color:green;">//&nbsp;...</span> <span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;y&nbsp;=&nbsp;<span style="color:green;">//&nbsp;...</span> <span style="color:#2b91af;">Lazy</span>&lt;<span style="color:blue;">int</span>&gt;&nbsp;product&nbsp;=&nbsp;x.Multiply(y);</pre> </p> <p> As is the case with lazy addition, this <code>Multiply</code> method currently only works with lazy <code>int</code> values. If you also want it to work with <code>long</code>, <code>s