ploeh blog danish software design

Normal functions form functors. An article for object-oriented programmers.

This article is an instalment in an article series about functors. In a previous article you saw, for example, how to implement the Maybe functor in C#. In this article, you'll see another functor example: Reader.

The Reader functor is similar to the Identity functor in the sense that it seems practically useless. If that's the case, then why care about it?

As I wrote about the Identity functor:

"The inutility of Identity doesn't mean that it doesn't exist. The Identity functor exists, whether it's useful or not. You can ignore it, but it still exists. In C# or F# I've never had any use for it (although I've described it before), while it turns out to be occasionally useful in Haskell, where it's built-in. The value of Identity is language-dependent."

The same holds for Reader. It exists. Furthermore, it teaches us something important about ordinary functions.

Reader interface #

Imagine the following interface:

public interface IReader<R, A>
{
    A Run(R environment);
}

An IReader object can produce a value of the type A when given a value of the type R. The input is typically called the environment. A Reader reads the environment and produces a value. A possible (although not particularly useful) implementation might be:

public class GuidToStringReader : IReader<Guid, string>
{
    private readonly string format;
 
    public GuidToStringReader(string format)
    {
        this.format = format;
    }
 
    public string Run(Guid environment)
    {
        return environment.ToString(format);
    }
}

This may be a silly example, but it illustrates that a a simple class can implement a constructed version of the interface: IReader<Guid, string>. It also demonstrates that a class can take further arguments via its constructor.

While the IReader interface only takes a single input argument, we know that an argument list is isomorphic to a parameter object or tuple. Thus, IReader is equivalent to every possible function type - up to isomorphism, assuming that unit is also a value.

While the practical utility of the Reader functor may not be immediately apparent, it's hard to argue that it isn't ubiquitous. Every method is (with a bit of hand-waving) a Reader.

Functor #

You can turn the IReader interface into a functor by adding an appropriate Select method:

public static IReader<R, B> Select<A, B, R>(this IReader<R, A> reader, Func<A, B> selector)
{
    return new FuncReader<R, B>(r => selector(reader.Run(r)));
}
 
private sealed class FuncReader<R, A> : IReader<R, A>
{
    private readonly Func<R, A> func;
 
    public FuncReader(Func<R, A> func)
    {
        this.func = func;
    }
 
    public A Run(R environment)
    {
        return func(environment);
    }
}

The implementation of Select requires a private class to capture the projected function. FuncReader is, however, an implementation detail.

When you Run a Reader, the output is a value of the type A, and since selector is a function that takes an A value as input, you can use the output of Run as input to selector. Thus, the return type of the lambda expression r => selector(reader.Run(r)) is B. Therefore, Select returns an IReader<R, B>.

Here's an example of using the Select method to project an IReader<Guid, string> to IReader<Guid, int>:

[Fact]
public void WrappedFunctorExample()
{
    IReader<Guid, string> r = new GuidToStringReader("N");
 
    IReader<Guid, int> projected = r.Select(s => s.Count(c => c.IsDigit()));
 
    var input = new Guid("{CAB5397D-3CF9-40BB-8CBD-B3243B7FDC23}");
    Assert.Equal(16, projected.Run(input));
}

The expected result is 16 because the input Guid contains 16 digits (the numbers from 0 to 9). Count them if you don't believe me.

As usual, you can also use query syntax:

[Fact]
public void QuerySyntaxFunctorExample()
{
    var projected =
        from s in new GuidToStringReader("N")
        select TimeSpan.FromMinutes(s.Length);
 
    var input = new Guid("{FE2AB9C6-DDB1-466C-8AAA-C70E02F964B9}");
 
    Assert.Equal(32, projected.Run(input).TotalMinutes);
}

The actual computation shown here makes little sense, since the result will always be 32, but it illustrates that arbitrary projections are possible.

Raw functions #

The IReader<R, A> interface isn't really necessary. It was just meant as an introduction to make things a bit easier for object-oriented programmers. You can write a similar Select extension method for any Func<R, A>:

public static Func<R, B> Select<A, B, R>(this Func<R, A> func, Func<A, B> selector)
{
    return r => selector(func(r));
}

Compare this implementation to the one above. It's essentially the same lambda expression, but now Select returns the raw function instead of wrapping it in a class.

In the following, I'll use raw functions instead of the IReader interface.

First functor law #

The Select method obeys the first functor law. As usual, it's proper computer-science work to actually prove this, but you can write some tests to demonstrate the first functor law for the IReader<R, A> interface. In this article, you'll see parametrised tests written with xUnit.net. First, the first functor law:

[Theory]
[InlineData("")]
[InlineData("foo")]
[InlineData("bar")]
[InlineData("corge")]
[InlineData("antidisestablishmentarianism")]
public void FirstFunctorLaw(string input)
{
    T id<T>(T x) => x;
    Func<string, int> f = s => s.Length;
 
    Func<string, int> actual = f.Select(id);
 
    Assert.Equal(f(input), actual(input));
}

The 'original' Reader f (for function) takes a string as input and returns its length. The id function (which isn't built-in in C#) is implemented as a local function. It returns whichever input it's given.

Since id returns any input without modifying it, it'll also return any number produced by f without modification.

To evaluate whether f is equal to f.Select(id), the assertion calls both functions with the same input. If the functions have equal behaviour, they ought to return the same output.

The above test cases all pass.

Second functor law #

Like the above example, you can also write a parametrised test that demonstrates that a function (Reader) obeys the second functor law:

[Theory]
[InlineData("")]
[InlineData("foo")]
[InlineData("bar")]
[InlineData("corge")]
[InlineData("antidisestablishmentarianism")]
public void SecondFunctorLaw(string input)
{
    Func<string, int> h = s => s.Length;
    Func<int, bool> g = i => i % 2 == 0;
    Func<bool, char> f = b => b ? 't' : 'f';
 
    Assert.Equal(
        h.Select(g).Select(f)(input),
        h.Select(i => f(g(i)))(input));
}

You can't easily compare two different functions for equality, so, like above, this test defines equality as the functions producing the same result when you invoke them.

Again, while the test doesn't prove anything, it demonstrates that for the five test cases, it doesn't matter if you project the 'original' Reader h in one or two steps.

Haskell #

In Haskell, normal functions a -> b are already Functor instances, which means that you can easily replicate the functions from the SecondFunctorLaw test:

> h = length
> g i = i `mod` 2 == 0
> f b = if b then 't' else 'f'
> (fmap f $ fmap g $ h) "ploeh"
'f'

Here f, g, and h are equivalent to their above C# namesakes, while the last line composes the functions stepwise and calls the composition with the input string "ploeh". In Haskell you generally read code from right to left, so this composition corresponds to h.Select(g).Select(f).

Conclusion #

Functions give rise to functors, usually known collectively as the Reader functor. Even in Haskell where this fact is ingrained into the fabric of the language, I rarely make use of it. It just is. In C#, it's likely to be even less useful for practical programming purposes.

That a function a -> b forms a functor, however, is an important insight into just what a function actually is. It describes an essential property of functions. In itself this may still seem underwhelming, but mixed with some other properties (that I'll describe in a future article) it can produce some profound insights. So stay tuned.

Next: The IO functor.

On long-standing controversies, biases, and failures of communication.

If you can stay out of politics, Twitter can be a great place to engage in robust discussions. I mostly follow and engage with people in the programming community, and every so often find myself involved in a discussion about one of several long-standing controversies. No, not the tabs-versus-spaces debate, but other debates such as functional versus object-oriented programming, dynamic versus static typing, or oral versus written collaboration.

It happened again the past week, but while this article is a reaction, it's not about the specific debacle. Thus, I'm not going to link to the tweets in question.

These discussion usually leave me wondering why people with decades of industry experience seem to have such profound disagreements.

I might be wrong #

Increasingly, I find myself disagreeing with my heroes. This isn't a comfortable position. Could I be wrong?

I've definitely been wrong before. For example, in my article Types + Properties = Software, I wrote about type systems:

"To the far right, we have a hypothetical language with such a strong type system that, indeed, if it compiles, it works."

To the right, in this context, means more statically typed. While the notion is natural, the sentence is uninformed. When I wrote the article, I hadn't yet read Charles Petzold's excellent Annotated Turing. Although I had heard about the halting problem before reading the book, I hadn't internalised it. I wasn't able to draw inferences based on that labelled concept.

After I read the book, I've come to understand that general-purpose static type system can never prove unequivocally that a generic program works. That's what Church, Turing, and Gödel proved.

I've been writing articles on this blog since January 2009. To date, I've published 582 posts. Some are bound to be misinformed, outdated, or just plain wrong. Due to the sheer volume, I make no consistent effort to retroactively monitor and correct my past self. (I'm happy to engage with specific posts. If you feel that an old post is misleading, erroneous, or the like, please leave a comment.)

For good measure, despite my failure to understand the implications of the halting problem, I'm otherwise happy with the article series Types + Properties = Software. You shouldn't consider this particular example a general condemnation of it. It's just an example of a mistake I made. This time, I'm aware of it, but there are bound to be plenty of other examples where I don't even realise it.

Heroes #

I don't have a formal degree in computer science. As so many others of my age, I began my software career by tinkering with computers and (later) programming. The first five years of my professional career, there wasn't much methodology to the way I approached software development. Self-taught often means that you have to learn everything the hard way.

This changed when I first heard about test-driven development (TDD). I credit Martin Fowler with that. Around the same time I also read Design Patterns. Armed with those two techniques, I was able to rescue a failed software project and bring it to successful completion. I even received an (internal) award for it.

While there's more to skilled programming than test-driven development and patterns, it wasn't a bad place to start. Where, before, I had nothing that even resembled a methodology, now I had a set of practices I could use. This gave me an opportunity to experiment and observe. A few years later, I'd already started to notice some recurring beneficial patterns in the code that I wrote, as well as some limits of TDD.

Still, that was a decade where I voraciously read, attended conferences, and tried to learn from my heroes. I hope that they won't mind that I list them here:

• Martin Fowler
• Kent Beck
• Robert C. Martin
• Michael Feathers
• Dan North

There's a reason I list those particular heroes. I should include a few more of whom I wasn't aware in my formative years, but whom I've since come to highly respect: Brian Marick and Jessica Kerr.

Why do I mention these heroes of mine?

Bias #

Humans aren't as rational as we'd like to think. We all have plenty of cognitive biases. I'm aware of a few of mine, but I expect most of them to be hidden from me. Sometimes, it's easier to spot the bias in others. Perhaps, by spotting the bias in others, it reveals something about oneself?

I increasingly find myself disagreeing with my heroes. One example is the long-standing controversy about static versus dynamic typing.

I hope I'm not misrepresenting anyone, but the heroes I enumerate above seem to favour dynamic typing over static typing - some more strongly than others. This worries me.

These are people who've taught me a lot; whose opinion I deeply respect, and yet I fail to grasp the benefit of dynamic typing. What are the benefits they gain from their preferred languages that I'm blind to? What am I missing?

Whenever I find myself disagreeing with my heroes, I can't help question my own judgment. Am I biased? Yes, obviously, but in which way? What bias prohibits me from seeing the benefits that are so obvious to them?

It's too easy to jump to conclusions - to erect a dichotomy:

• My heroes are right, and I'm wrong
• My heroes are all wrong, and I'm right

I'm hoping that there's a more nuanced position to take - that the above is a false dichotomy.

What's the problem? #

Perhaps we're really talking past each other. Perhaps we're trying to solve different problems, and thereby arrive at different solutions.

I can only guess at the kinds of problems that my heroes think of when they prefer dynamic languages, and I don't want to misrepresent them. What I can do, however, is outline the kind of problem that I typically have in mind.

I've spent much of my career trying to balance sustainability with correctness. I consider correctness as a prerequisite for all code. As Gerald Weinberg implies, if a program doesn't have to work, anything goes. Thus, sustainability is a major focus for me: how do we develop software that can sustain our organisation now and in the future? How do we structure and organise code so that future change is possible?

Whenever I get into debates, that's implicitly the problem on my mind. It'd probably improve communication if I stated this explicitly going into every debate, but sometimes, I get dragged sideways into a debacle... I do, however, speculate that much disagreement may stem from such implicit assumptions. I bring my biases and implicit problem statements into any discussion. I consider it only human if my interlocutors do the same, but their biases and implicit problem understanding may easily be different than mine. What are they, I wonder?

This seems to happen not only in the debate about dynamic versus static types. I get a similar sense when I discuss collaboration. Most of my heroes seem to advocate for high-band face-to-face collaboration, while I favour asynchronous, written communication. Indeed, I admit that my bias is speaking. I self-identify as a contrarian introvert (although, again, we should be careful not turning introversion versus extroversion into binary classification).

Still, even when I try to account for my bias, I get the sense that my opponents and I may actually try to accomplish a common goal, but by addressing two contrasting problems.

I think and hope that, ultimately, we're trying to accomplish the same goal: to deliver and sustain business capability.

I do get the sense that the proponents of more team co-location, more face-to-face collaboration are coming at the problem from a different direction than I am. Perhaps the problem they're trying to solve is micro-management, red tape, overly bureaucratic processes, and a lack of developer autonomy. I can certainly see that if that's the problem, talking to each other is likely to improve the situation. I'd recommend that too, in such a situation.

Perhaps it's a local Danish (or perhaps Northern European) phenomenon, but that's not the kind of problem I normally encounter. Most organisations who ask for my help essentially have no process. Everything is ad hoc, nothing is written down, deployment is a manual process, and there are meetings and interruptions all the time. Since nothing is written down, decisions aren't recorded, so team members and stakeholders keep having the same meetings over and over. Again, little gets done, but for an entirely different reason than too much bureaucracy. I see this more frequently than too much red tape, so I tend to recommend that people start leaving behind some sort of written trail of what they've been doing. Pull request reviews, for example, are great for that, and I see no incongruity between agile and pull requests.

Shaped by scars #

The inimitable Richard Campbell has phrased our biases as the scars we collect during our careers. If you've deleted the production database one too many times, you develop routines and practices to avoid doing that in the future. If you've mostly worked in an organisation that stifled progress by subjecting you to Brazil-levels of bureaucracy, it's understandable if you develop a preference for less formal methods. And if, like me, you've mostly seen dysfunction manifest as a lack of beneficial practices, you develop a taste for written over oral communication.

Does it go further than that? Are we also shaped by our successes, just like we are shaped by our scars?

The first time I had professional success with a methodology was when I discovered TDD. This made me a true believer in TDD. I'm still a big proponent of TDD, but since I learned what algebraic data types can do in terms of modelling, I see no reason to write a run-time test if I instead can get the compiler to enforce a rule.

In a recent discussion, some of my heroes expressed the opinion that they don't need fancy functional-programming concepts and features to write good code. I'm sure that they don't.

My heroes have written code for decades. While I have met bad programmers with decades of experience, most programmers who last that long ultimately become good programmers. I'm not so worried about them.

The people who need my help are typically younger teams. Statistically, there just aren't that many older programmers around.

When I recommend certain practices or programming techniques, those recommendations are aimed at anyone who care to listen. Usually, I find that the audience who engage with me is predominantly programmers with five to ten years of professional experience.

Anecdotal evidence #

This is a difficult discussion to have. I think that another reason that we keep returning to the same controversies is that we mostly rely on anecdotal evidence. As we progress through our careers, we observe what works and what doesn't, but it's likely that confirmation bias makes us remember the results that we already favour, whereas we conveniently forget about the outcomes that don't fit our preferred narrative.

Could we settle these discussions with more science? Alas, that's difficult.

I can't think of anything better, then, than keep having the same discussions over and over. I try hard to overcome my biases to understand the other side, and now and then, I learn something that I find illuminating. It doesn't seem to be a particularly efficient way to address these difficult questions, but I don't know what else to do. What's the alternative to discussion? To not discuss? To not exchange ideas?

That seems worse.

Conclusion #

In this essay, I've tried to come to grips with an increasing cognitive incongruity that I'm experiencing. I find myself disagreeing with my heroes on a regular basis, and that makes me uncomfortable. Could it be that I'm erecting an echo chamber for myself?

The practices that I follow and endorse work well for me, but could I be stuck in a local maximum?

This essay has been difficult to write. I'm not sure that I've managed to convey my doubts and motivations. Should I have named my heroes, only to describe how I disagree with them? Will it be seen as aggressive?

I hope not. I'm writing about my heroes with reverence and gratitude for all that they've taught me. I mean no harm.

At the same time, I'm struggling with reconciling that they rarely seem to agree with me these days. Perhaps they never did.

Comments

When you need discriminated unions, but all you have are objects.

After I learned that the Visitor design pattern is isomorphic to sum types (AKA discriminated unions), I wanted to try how easy it is to carry out a translation in practice. For that reason, I decided to translate my go-to F# implementation of the Tennis kata to C#, using the Visitor design pattern everywhere I'd use a discriminated union in F#.

The resulting C# code shows that it is, indeed, possible to 'mechanically' translate discriminated unions to the Visitor design pattern. Given that the Visitor pattern requires multiple interfaces and classes to model just a single discriminated union, it's no surprise that the resulting code is much more complex. As a solution to the Tennis kata itself, all this complexity is unwarranted. On the other hand, as an exercise in getting by with the available tools, it's quite illustrative. If all you have is C# (or a similar language), but you really need discriminated unions, the solution is ready at hand. It'll be complex, but not complicated.

The main insight of this exercise is that translating any discriminated union to a Visitor is, indeed, possible. You can best internalise such insights, however, if you actually do the work yourself. Thus, in this article, I'll only show a few highlights from my own exercise. I highly recommend that you try it yourself.

Typical F# solution #

You can see my typical F# solution in great detail in my article series Types + Properties = Software. To be clear, there are many ways to implement the Tennis kata, even in F#, and the one shown in the articles is neither overly clever nor too boring. As implementations go, this one is quite pedestrian.

My emphasis with that kind of solution to the Tennis kata is on readability, correctness, and testability. Going for a functional architecture automatically addresses the testability concern. In F#, I endeavour to use the excellent type system to make illegal states unrepresentable. Discriminated unions are essential ingredients in that kind of design.

In F#, I'd typically model a Tennis game score as a discriminated union like this:

type Score =
| Points of PointsData
| Forty of FortyData
| Deuce
| Advantage of Player
| Game of Player

That's not the only discriminated union involved in the implementation. Player is also a discriminated union, and both PointsData and FortyData compose a third discriminated union called Point:

type Point = Love | Fifteen | Thirty

Please refer to the older article for full details of the F# 'domain model'.

This was the sort of design I wanted to try to translate to C#, using the Visitor design pattern in place of discriminated unions.

Player #

In F# you must declare all types and functions before you can use them. To newcomers, this often looks like a constraint, but is actually one of F#'s greatest strengths. Since other types transitively use the Player discriminated union, this is the first type you have to define in an F# code file:

type Player = PlayerOne | PlayerTwo

This one is fairly straightforward to translate to C#. You might reach for an enum, but those aren't really type-safe in C#, since they're little more than glorified integers. Using a discriminated union is safer, so define a Visitor:

public interface IPlayerVisitor<T>
{
    T VisitPlayerOne();
 
    T VisitPlayerTwo();
}

A Visitor interface is where you enumerate the cases of the discriminated union - in this example player one and player two. You can idiomatically prefix each method with Visit as I've done here, but that's optional.

Once you've defined the Visitor, you can declare the 'actual' type you're modelling: the player:

public interface IPlayer
{
    T Accept<T>(IPlayerVisitor<T> visitor);
}

Those are only the polymorphic types required to model the discriminated union. You also need to add concrete classes for each of the cases. Here's PlayerOne:

public struct PlayerOne : IPlayer
{
    public T Accept<T>(IPlayerVisitor<T> visitor)
    {
        return visitor.VisitPlayerOne();
    }
}

This is an example of the Visitor pattern's central double dispatch mechanism. Clients of the IPlayer interface will dispatch execution to the Accept method, which again dispatches execution to the visitor.

I decided to make PlayerOne a struct because it holds no data. Perhaps I also ought to have sealed it, or, as Design Patterns likes to suggest, make it a Singleton.

Hardly surprising, PlayerTwo looks almost identical to PlayerOne.

Apart from a single-case discriminated union (which is degenerate), a discriminated union doesn't get any simpler than Player. It has only two cases and carries no data. Even so, it takes four types to translate it to a Visitor: two interfaces and two concrete classes. This highlights how the Visitor pattern adds significant complexity.

And it only gets worse with more complex discriminated unions.

Score #

I'm going to leave the translation of Point as an exercise. It's similar to the translation of Player, but instead of two cases, it enumerates three cases. Instead, consider how to enumerate the cases of Score. First, add a Visitor:

public interface IScoreVisitor<T>
{
    T VisitPoints(IPoint playerOnePoint, IPoint playerTwoPoint);
 
    T VisitForty(IPlayer player, IPoint otherPlayerPoint);
 
    T VisitDeuce();
 
    T VisitAdvantage(IPlayer advantagedPlayer);
 
    T VisitGame(IPlayer playerWhoWonGame);
}

Notice that these methods take arguments, apart from VisitDeuce. I could have made that member a read-only property instead, but for consistency's sake, I kept it as a method.

All the other methods take arguments that are, in their own right, Visitors.

In addition to the Visitor interface, you also need an interface to model the score itself:

public interface IScore
{
    T Accept<T>(IScoreVisitor<T> visitor);
}

This one defines, as usual, just a single Accept method.

Since IScoreVisitor enumerates five distinct cases, you must also add five concrete implementations of IScore. Here's Forty:

public struct Forty : IScore
{
    private readonly IPlayer player;
    private readonly IPoint otherPlayerPoint;
 
    public Forty(IPlayer player, IPoint otherPlayerPoint)
    {
        this.player = player;
        this.otherPlayerPoint = otherPlayerPoint;
    }
 
    public T Accept<T>(IScoreVisitor<T> visitor)
    {
        return visitor.VisitForty(player, otherPlayerPoint);
    }
}

I'm leaving other concrete classes as an exercise to the reader. All of them are similar, though, in that they all implement IScore and unconditionally dispatch to 'their' method on IScoreVisitor - Forty calls VisitForty, Points calls VisitPoints, and so on. Each concrete implementation has a distinct constructor, though, since what they need to dispatch to the Visitor differs.

Deuce, being degenerate, doesn't have to explicitly define a constructor:

public struct Deuce : IScore
{
    public T Accept<T>(IScoreVisitor<T> visitor)
    {
        return visitor.VisitDeuce();
    }
}

The C# compiler automatically adds a parameterless constructor when none is defined. An alternative implementation would, again, be to make Deuce a Singleton.

In all, it takes seven types (two interfaces and five concrete classes) to model Score - a type that requires only a few lines of code in F# (six lines in my code, but you could format it more densely if you want to compromise readability).

Keeping score #

In order to calculate the score of a game, I also translated the score function. I put that in an extension method so as to not 'pollute' the IScore interface:

public static IScore BallTo(this IScore score, IPlayer playerWhoWinsBall)
{
    return score.Accept(new ScoreVisitor(playerWhoWinsBall));
}

Given an IScore value, there's little you can do with it, apart from calling its Accept method. In order to do that, you'll need an IScoreVisitor, which I defined as a private nested class:

private class ScoreVisitor : IScoreVisitor<IScore>
{
    private readonly IPlayer playerWhoWinsBall;
 
    public ScoreVisitor(IPlayer playerWhoWinsBall)
    {
        this.playerWhoWinsBall = playerWhoWinsBall;
    }
 
    // Implementation goes here...

Some of the methods are trivial to implement, like VisitDeuce:

public IScore VisitDeuce()
{
    return new Advantage(playerWhoWinsBall);
}

Most others are more complicated. Keep in mind that the method arguments (IPlayer, IPoint) are Visitors in their own right, so in order to do anything useful with them, you'll have to call their Accept methods with a corresponding, specialised Visitor.

Pattern matching #

I quickly realised that this would become too tedious, even for an exercise, so I leveraged my knowledge that the Visitor pattern is isomorphic to a Church encoding. Instead of defining umpteen specialised Visitors, I just defined a generic Match method for each Visitor-based object. I put those in extension methods as well. Here's the Match method for IPlayer:

public static T Match<T>(this IPlayer player, T playerOne, T playerTwo)
{
    return player.Accept(new MatchVisitor<T>(playerOne, playerTwo));
}

The implementation is based on a private nested MatchVisitor:

private class MatchVisitor<T> : IPlayerVisitor<T>
{
    private readonly T playerOneValue;
    private readonly T playerTwoValue;
 
    public MatchVisitor(T playerOneValue, T playerTwoValue)
    {
        this.playerOneValue = playerOneValue;
        this.playerTwoValue = playerTwoValue;
    }
 
    public T VisitPlayerOne()
    {
        return playerOneValue;
    }
 
    public T VisitPlayerTwo()
    {
        return playerTwoValue;
    }
}

This enables pattern matching, upon which you can implement other reusable methods, such as Other:

public static IPlayer Other(this IPlayer player)
{
    return player.Match<IPlayer>(
        playerOne: new PlayerTwo(),
        playerTwo: new PlayerOne());
}

It's also useful to be able to compare two players and return two alternative values depending on the result:

public static T Equals<T>(
    this IPlayer p1,
    IPlayer p2,
    T same,
    T different)
{
    return p1.Match(
        playerOne: p2.Match(
            playerOne: same,
            playerTwo: different),
        playerTwo: p2.Match(
            playerOne: different,
            playerTwo: same));
}

You can add similar Match and Equals extension methods for IPoint, which enables you to implement all the methods of the ScoreVisitor class. Here's VisitForty as an example:

public IScore VisitForty(IPlayer player, IPoint otherPlayerPoint)
{
    return playerWhoWinsBall.Equals(player,
        same: new Game(player),
        different: otherPlayerPoint.Match<IScore>(
            love: new Forty(player, new Fifteen()),
            fifteen: new Forty(player, new Thirty()),
            thirty: new Deuce()));
}

If playerWhoWinsBall.Equals(player the implementation matches on same, and returns new Game(player). Otherwise, it matches on different, in which case it then has to Match on otherPlayerPoint to figure out what to return.

Again, I'll leave the remaining code as an exercise to the reader.

Tests #

While all this code is written in C#, it's all pure functions. Thus, it's intrinsically testable. Knowing that, I could have added tests after writing the 'production code', but that's so boring, so I still developed this code base using test-driven development. Here's an example of a test:

[Theory, MemberData(nameof(Player))]
public void TransitionFromDeuce(IPlayer player)
{
    var sut = new Deuce();
 
    var actual = sut.BallTo(player);
 
    var expected = new Advantage(player);
    Assert.Equal(expected, actual);
}

I wrote all tests as properties for all. The above test uses xUnit.net 2.4.0. The Player data source is a private MemberData defined as a read-only property:

public static IEnumerable<object[]> Player
{
    get
    {
        yield return new object[] { new PlayerOne() };
        yield return new object[] { new PlayerTwo() };
    }
}

Other tests are defined in a likewise manner.

Conclusion #

Once you've tried to solve problems with algebraic data types it can be frustrating to return to languages that don't have sum types (discriminated unions). There's no need to despair, though. Sum types are isomorphic to the Visitor design pattern, so it is possible to solve problems in the same way.

The resulting code can easily seem complex because a simple discriminated union translates to multiple C# files. Another option is to use Church encoding, but many developers who consider themselves object-oriented often balk at such constructs. When it comes to policy, the Visitor design pattern offers the advantage that it's described in Design Patterns. While it may be a bit of an underhanded tactic, it effectively deals with some teams' resistance to ideas originating in functional programming.

The exercise outlined in this article demonstrates that translating from algebraic data types to patterns-based object-oriented code is not only possible, but (once you get the hang of it) unthinking and routine. It's entirely automatable, so you could even imagine defining a DSL for defining sum types and transpiling them to C#.

But really, you don't have to, because such a DSL already exists.

Comments

// type Bool = True | False
interface IBoolVisitor<R>
{
    R True();
    R False();
}

static class BoolUtils
{
    public static Func<IBoolVisitor<R>, R> True<R>() => sym => sym.True();
    public static Func<IBoolVisitor<R>, R> False<R>() => sym => sym.False();

    private class BoolMatcher<R> : IBoolVisitor<R>
    {
        Func<R> OnTrue { get; }
        Func<R> OnFalse { get; }

        public BoolMatcher(Func<R> onTrue, Func<R> onFalse)
        {
            OnTrue = onTrue;
            OnFalse = onFalse;
        }

        public R True() => OnTrue();
        public R False() => OnFalse();
    }

    public static R Match<R>(this Func<IBoolVisitor<R>, R> value, Func<R> onTrue, Func<R> onFalse) => value(new BoolMatcher<R>(onTrue, onFalse));
}

static class FunctionsAttempt1
{
    public static string ShowBool(Func<IBoolVisitor<string>, string> value)
    {
        return value.Match(() => "True", () => "False");
    }

    public static int BoolToInt(Func<IBoolVisitor<int>, int> value)
    {
        return value.Match(() => 1, () => 0);
    }

    public static string CombineBoolFunctions<R>(Func<IBoolVisitor<R>, R> value)
    {
        return ShowBool(value) + ": " + BoolToInt(value).ToString();
    }
}

static class FunctionsAttempt2
{
    public static string ShowBool(Func<>, R> value)
    {
        return value.Match(() => "True", () => "False");
    }

    public static int BoolToInt(Func<>, R> value)
    {
        return value.Match(() => 1, () => 0);
    }

    public static string CombineBoolFunctions(Func<>, R> value)
    {
        return ShowBool(value) + ": " + BoolToInt(value).ToString();
    }
}

public abstract record Player
{
	private Player() {}
	
	public sealed record One : Player;
	public sealed record Two : Player;
}

public abstract record Point
{
	private Point() {}
	
	public sealed record Love: Point;
	public sealed record Fifteen: Point;
	public sealed record Thirty: Point;
}

public abstract record Score
{
	private Score() {}
	
	public sealed record Points(Point PlayerOnePoints, Point PlayerTwoPoints);
	public sealed record Forty(Player Player, Point OtherPlayersPoint);
	public sealed record Deuce;
	public sealed record Advantage(Player Player);
	public sealed record Game(Player Player);
}
			

Why functional programming matters.

This article is mostly excerpts from my book Code That Fits in Your Head. The overall message is too important to exclusively hide away in a book, though, which is the reason I also publish it here.

The illustrations are preliminary. While writing the manuscript, I experimented with hand-drawn figures, but Addison-Wesley prefers 'professional' illustrations. In the published book, the illustrations shown here will be replaced by cleaner, more readable, but also more boring figures.

Nested composition #

Ultimately, software interacts with the real world. It paints pixels on the screen, saves data in databases, sends emails, posts on social media, controls industrial robots, etcetera. All of these are what we in the context of Command Query Separation call side effects.

Since side effects are software's raison d'être, it seems only natural to model composition around them. This is how most people tend to approach object-oriented design. You model actions.

Object-oriented composition tends to focus on composing side effects together. The Composite design pattern may be the paragon of this style of composition, but most of the patterns in Design Patterns rely heavily on composition of side effects.

As illustrated in [the following figure] this style of composition relies on nesting objects in other objects, or side effects in other side effects. Since your goal should be code that fits in your head, this is a problem.

[Figure caption:] The typical composition of objects (or, rather, methods on objects) is nesting. The more you compose, the less the composition fits in your brain. In this figure, each star indicates a side effect that you care about. Object A encapsulates one side effect, and object B two. Object C composes A and B, but also adds a fourth side effect. That's already four side effects that you need to keep in mind when trying to understand the code. This easily gets out of hand: object E composes a total of eight side effects, and F nine. Those don't fit well in your brain.

I should add here that one of the book's central tenets is that the human short-term memory can only hold a limited amount of information. Code that fits in your head is code that respects that limitation. This is a topic I've already addressed in my Humane Code video.

In the book, I use the number seven as a symbol of the this cognitive limit. Nothing I argue, however, relies on this exact number. The point is only that short-term memory is quite limited. Seven is only a shorthand for that.

The book proceeds to provide a code example that illustrates how fast nested composition accumulates complexity that exceeds the capacity of your short-term memory. You can see the code example in the book, or in the article Good names are skin-deep, which makes a slightly different criticism than the one argued in the book.

The section on nested composition goes on:

"Abstraction is the elimination of the irrelevant and the amplification of the essential"

By hiding a side effect in a Query, I've eliminated something essential. In other words, more is going on in [the book's code listing] than meets the eye. The cyclomatic complexity may be as low as 4, but there's a hidden fifth action that you ought to be aware of.

Granted, five chunks still fit in your brain, but that single hidden interaction is an extra 14% towards the budget of seven. It doesn't take many hidden side effects before the code no longer fits in your head

Sequential composition #

While nested composition is problematic, it isn't the only way to compose things. You can also compose behaviour by chaining it together, as illustrated in [the following figure].

[Figure caption:] Sequential composition of two functions. The output from Where becomes the input to Allocate.

In the terminology of Command Query Separation, Commands cause trouble. Queries, on the other hand, tend to cause little trouble. They return data which you can use as input for other Queries.

Again, the book proceeds to show code examples. You can, of course, see the code in the book, but the methods discussed are the WillAccept function shown here, the Overlaps function shown here, as well as a few other functions that I don't think that I've shown on the blog.

The entire restaurant example code base is written with that principle in mind. Consider the WillAccept method [...]. After all the Guard Clauses it first creates a new instance of the Seating class. You can think of a constructor as a Query under the condition that it has no side effects.

The next line of code filters the existingReservations using the Overlaps method [...] as a predicate. The built-in Where method is a Query.

Finally, the WillAccept method returns whether there's Any table among the availableTables that Fits the candidate.Quantity. The Any method is another built-in Query, and Fits is a predicate.

Compared to [the sequential composition figure], you can say that the Seating constructor, seating.Overlaps, Allocate, and Fits are sequentially composed.

None of these methods have side effects, which means that once WillAccept returns its Boolean value, you can forget about how it reached that result. It truly eliminates the irrelevant and amplifies the essential

Referential transparency #

There's a remaining issue that Command Query Separation fails to address: predictability. While a Query has no side effects that your brain has to keep track of, it could still surprise you if you get a new return value every time you call it - even with the same input.

This may not be quite as bad as side effects, but it'll still tax your brain. What happens if we institute an extra rule on top of Command Query Separation: that Queries must be deterministic?

This would mean that a Query can't rely on random number generators, GUID creation, the time of day, day of the month, or any other data from the environment. That would include the contents of files and databases. That sounds restrictive, so what's the benefit?

A deterministic method without side effects is referentially transparent. It's also known as a pure function. Such functions have some very desirable qualities.

One of these qualities is that pure functions readily compose. If the output of one function fits as the input for another, you can sequentially compose them. Always. There are deep mathematical reasons for that, but suffice it to say that composition is ingrained into the fabric that pure functions are made of.

Another quality is that you can replace a pure function call with its result. The function call is equal to the output. The only difference between the result and the function call is the time it takes to get it.

Think about that in terms of Robert C. Martin's definition of abstraction. Once a pure function returns, the result is all you have to care about. How the function arrived at the result is an implementation detail. Referentially transparent functions eliminate the irrelevant and amplify the essential. As [the following figure] illustrates, they collapse arbitrary complexity to a single result; a single chunk that fits in your brain.

[Figure caption:] Pure function (left) collapsing to its result (right). Regardless of complexity, a referentially transparent function call can be replaced by its output. Thus, once you know what the output is, that's the only thing you need to keep track of as you read and interpret the calling code.

On the other hand, if you want to know how the function works, you zoom in on its implementation, in the spirit of fractal architecture. That might be the WillAccept method [...]. This method is, in fact, not just Query, it's a pure function. When you look at the source code of that function, you've zoomed in on it, and the surrounding context is irrelevant. It operates exclusively on its input arguments and immutable class fields.

When you zoom out again, the entire function collapses into its result. It's the only thing your brain needs to keep track of.

Regardless of complexity, a referentially transparent function reduces to a single chunk: the result that it produces. Thus, referentially transparent code is code that fits in your head.

Conclusion #

Code That Fits in Your Head isn't a book about functional programming, but it does advocate for the functional core, imperative shell (also known as an impureim sandwich) architecture, among many other techniques, practices, and heuristics that it presents.

I hope that you found the excerpt so inspiring that you consider buying the book.

Comments

Stateful computations as a functor. An example for object-oriented programmers.

This article is an instalment in an article series about functors. In a previous article, you saw how to implement the Maybe functor in C#. In this article, you'll see another functor example: State.

In functional programming, sooner or later a particular question comes up: How do you implement a stateful computation without mutating state?

You use a polymorphic function that takes the current state as input and returns the new state and a result as output. In a C-based language like C#, you can model it as an interface:

public interface IState<S, T>
{
    Tuple<T, S> Run(S state);
}

The interface is generic in both the type of state and the type of return value. Notice that the type declaration lists the state type S before the type of the value T, whereas the returned tuple lists T before S. This is quite confusing, but is how Haskell does it. Not ideal, but I've chosen to keep that convention for the benefit of readers who'd like to compare the various implementations.

This article introduces the implementation and machinery of the type. In a later article I'll show an example.

A nonsense example #

You can implement the interface by doing something useful, or, as in the following example, something fatuous like expanding (or contracting) all vowels in a word according to an integer state:

private class VowelExpander : IState<int, string>
{
    private readonly string text;
 
    public VowelExpander(string text)
    {
        this.text = text;
    }
 
    public Tuple<string, int> Run(int state)
    {
        const string vowels = "aeiouy";
        var expanded = text.SelectMany(c =>
            vowels.Contains(c) ?
                Enumerable.Repeat(c, state) :
                new[] { c });
 
        var newState = state + 1;
 
        return Tuple.Create(new string(expanded.ToArray()), newState);
    }
}

This class repeats each vowel in a string by the number indicated by the current state. It also increments the state. Here's a parametrised test that shows how various input produces different outputs:

[Theory]
[InlineData("foo", 0, "f")]
[InlineData("foo", 1, "foo")]
[InlineData("foo", 2, "foooo")]
[InlineData("bar", 0, "br")]
[InlineData("bar", 1, "bar")]
[InlineData("bar", 2, "baar")]
public void BasicUsageExample(string txt, int count, string expected)
{
    IState<int, string> s = new VowelExpander(txt);
    Tuple<string, int> t = s.Run(count);
    Assert.Equal(Tuple.Create(expected, count + 1), t);
}

That's just one, simple stateful computation. It's a silly example, but it's referentially transparent.

Functor #

You can turn the IState interface into a functor by adding an appropriate Select method:

public static IState<S, T1> Select<S, T, T1>(
    this IState<S, T> source,
    Func<T, T1> selector)
{
    return new SelectState<S, T, T1>(source, selector);
}
 
private class SelectState<S, T, T1> : IState<S, T1>
{
    private IState<S, T> source;
    private Func<T, T1> selector;
 
    public SelectState(IState<S, T> source, Func<T, T1> selector)
    {
        this.source = source;
        this.selector = selector;
    }
 
    public Tuple<T1, S> Run(S state)
    {
        Tuple<T, S> tuple = source.Run(state);
        T1 projection = selector(tuple.Item1);
        return Tuple.Create(projection, tuple.Item2);
    }
}

A functor maps from one contained type to another, in this case from T to T1, while the state type S remains the same. Notice that it's possible to change the value of the state, but not the type. Even though the State functor has two generic type arguments, it's not a bifunctor. You can pick any type you'd like for S, such as int in the above VowelExpander, but once you've picked a type for the state, you can't project it. It's possible to prove that you can't implement a lawful mapping for the S dimension of State, but if you'd like to understand it intuitively, it's a great exercise to try to implement a function from IState<S, T> to IState<S1, T>. Try it, and you'll soon learn why this is impossible.

Here's an example of using the Select method to project an IState<int, string> to IState<int, int>:

[Fact]
public void BasicSelectExample()
{
    IState<int, string> s = new VowelExpander("bar");
 
    IState<int, int> projection = s.Select(x => x.Length);
    Tuple<int, int> t = projection.Run(2);
 
    Assert.Equal(Tuple.Create(4, 3), t);
}

As usual, you can also use query syntax:

[Fact]
public void QuerySyntaxExample()
{
    IState<int, string> s = new VowelExpander("baz");
 
    IState<int, DayOfWeek> projection =
        from txt in s
        select txt.Length % 2 == 0 ? DayOfWeek.Friday : DayOfWeek.Sunday;
    Tuple<DayOfWeek, int> t = projection.Run(3);
 
    Assert.Equal(Tuple.Create(DayOfWeek.Sunday, 4), t);
}

This is, once again, a nonsensical function that only exists to show that arbitrary projections are possible.

First functor law #

The Select method obeys the first functor law. As usual, it's proper computer-science work to actually prove this, but you can write some tests to demonstrate the first functor law for the IState<S, T> interface. In this article, you'll see parametrised tests written with xUnit.net. First, the first functor law:

[Theory]
[InlineData(DayOfWeek.Monday)]
[InlineData(DayOfWeek.Tuesday)]
[InlineData(DayOfWeek.Wednesday)]
[InlineData(DayOfWeek.Thursday)]
[InlineData(DayOfWeek.Friday)]
[InlineData(DayOfWeek.Saturday)]
[InlineData(DayOfWeek.Sunday)]
public void FirstFunctorLaw(DayOfWeek day)
{
    Func<Guid, Guid> id = g => g;
    IState<DayOfWeek, Guid> s = new DayIdentifier();
 
    Assert.Equal(s.Run(day), s.Select(id).Run(day));
}

This test uses another frivolous IState implementation:

private class DayIdentifier : IState<DayOfWeek, Guid>
{
    public static readonly Guid Monday =
        new Guid("5AB18569-29C7-4041-9719-5255266B808D");
    public static readonly Guid OtherDays =
        new Guid("00553FC8-82C9-40B2-9FAA-F9ADFFD4EE66");
 
    public Tuple<Guid, DayOfWeek> Run(DayOfWeek state)
    {
        if (state == DayOfWeek.Monday)
            return Tuple.Create(Monday, DayOfWeek.Tuesday);
 
        return Tuple.Create(OtherDays, DayOfWeek.Monday);
    }
}

I only chose to write another implementation of IState to show a bit of variation, and to demonstrate that both S and T can be whichever type you need them to be.

The above test cases pass.

Second functor law #

Like the above example, you can also write a parametrised test that demonstrates that IState obeys the second functor law:

[Theory]
[InlineData( "foo", 0)]
[InlineData( "bar", 1)]
[InlineData( "baz", 2)]
[InlineData("quux", 3)]
public void SecondFunctorLaw(string txt, int i)
{
    Func<string, int> g = x => x.Length;
    Func<int, bool> f = x => x % 2 == 0;
    var s = new VowelExpander(txt);
 
    Assert.Equal(
        s.Select(g).Select(f).Run(i),
        s.Select(x => f(g(x))).Run(i));
}

This test defines two local functions, f and g. Instead of explicitly declaring the functions as Func variables, this test uses a (relatively) new C# feature called local functions.

You can't easily compare two different functions for equality, so this test defines equality as the functions producing the same result when you Run them.

Again, while the test doesn't prove anything, it demonstrates that for the five test cases, it doesn't matter if you project the state s in one or two steps.

Haskell #

In Haskell, State is available in the mtl package. You can implement the behaviour from VowelExpander like this:

expandVowels :: String -> Int -> (String, Int)
expandVowels text s =
  let vowels = "aeiouy"
      expanded = text >>= (\c -> if c `elem` vowels then replicate s c else [c])
      newState = s + 1
  in (expanded, newState)

Instead of defining an interface, you can use any function s -> (a, s), which you can elevate to the State functor using a function called state. You can then use fmap or <$> to map the value:

> runState (length <$> state (expandVowels "bar")) 2
(4,3)

You can see a more useful example of the Haskell State functor in use in the article An example of state-based testing in Haskell.

Conclusion #

A function that takes a state value as input and returns a value and a (potentially new) state value as output is a functor known as State. It can be used as a convenient way to express stateful computations as pure functions.

Next: The Reader functor.

A paper read and translated to C#.

When I have the time (and I do make this a priority) I set aside an hour every day to study. Lately I've been using these time slots to read and reproduce the code in the 2012 paper "Extensibility for the Masses. Practical Extensibility with Object Algebras" by Bruno C. d. S. Oliveira and William R. Cook. As is often common with academic papers, they don't have a single, authoritative address on the internet. You can find the paper in various places. I've used a copy hosted by University of Texas, which is the institution with which William R. Cook is associated.

While the example code in the paper is in Java, the authors claim that it translates easily to C#. I decided to give this a try, and found it to be true.

Git repository #

From the beginning I created a Git repository with an eye to publishing it in case anyone was interested in looking over my shoulder. Not only can you see the 'final' translation, but you can also follow along with each commit.

I committed each time I had something that seemed to work. When I struggled to understand how to interpret some of the code, I left detailed commit messages describing my doubts, and explaining why I had chosen to interpret things in a particular way.

Along the way I also added automated tests, because I found that the paper lacked examples. Those tests represent my own interpretation of the code in the paper, and how one is supposed to use it. In total, I wrote 75 test cases.

Help from one of the authors #

At one time I hit a snag that I couldn't readily resolve. After searching the web in vain, I posted a question on Stack Overflow. After a few days, I got an answer from Bruno C. d. S. Oliveira, one of the authors of the paper!

It turns out that some of my confusion stemmed from an otherwise inconsequential error in the paper. We shouldn't be shocked that an academic paper contains errors. One of many things I learned from reading Charles Petzold's excellent book The Annotated Turing was that later researchers found several errors in Turing's 1936 paper, but none that changed the overall conclusion. So it seems to be here as well. There's at least one confirmed error (and another one that I only suspect), but it's inconsequential and easily corrected.

It does, however, raise a point about scientific papers in general: Even if they're peer-reviewed they may contain errors. I'm much in favour of scientific knowledge, but also sceptical about some claims about computer science and software engineering.

Readability #

The paper's title claims to give extensibility to the masses, but will 'the masses' be able to read the paper? As papers go, I found this one quite readable. While other papers present their examples in Haskell, this one uses Java. If you're comfortable with Java (or C#), you should be able to follow the code examples (or my C# translation).

You won't entirely escape Greek letters or other abstruse nomenclature. This is, after all, an academic paper, so it can't be lucid all the way through. There's a section called Algebraic Signatures, F-Algebras, and Church Encodings that is definitely not for 'the masses'. I understand enough about F-algebras and Church encodings to follow the outline of this section, but I didn't find it helpful.

If you're interested in the overall article, but don't know what these words mean, I suggest you skim those parts and pay as much attention to the details as when Geordi La Forge spews technobabble. In other words, I think you can follow the rest of the article just was well, even if Church_Σ = ∀A.(T₁ → A) × ... × (T_n → A) → A makes no sense to you.

Conclusion #

Does the paper deliver on its promise? Yes and no. Formally, I think that it does. In the beginning, it establishes some criteria for a successful solution, and as far as I can tell, it can check off all of them.

It's also true that the proposed solution requires only intermediary language features. Generics and inheritance, as they're available in C# and Java, is all that's required.

On the other hand, I don't find the paper's examples compelling. Few 'mass developers' are tasked with implementing a simple expression-based language. I can't shake the feeling that most of what the paper accomplishes could be handled with tasty application of composition and the Adapter pattern.

Still, I'll keep this paper in mind if I ever have to publish a reusable and extensible, type-safe software library.

Comments

The code behind the cover of Code That Fits in Your Head

A book needs a cover, but how do you choose a cover illustration for a programming book? Software development tends to be abstract, so how to pick a compelling picture?

Occasionally, a book manages to do everything right, such as the wonderful Mazes for Programmers, which also happens to have a cover appropriate for its contents. Most publishers, however, resort to pictures of bridges, outer space, animals, or folk costumes.

For my new book, I wanted a cover that, like Mazes for Programmers, relates to its contents. Fortunately, the book contains a few concepts that easily visualise, including a concept I call fractal architecture. You can create some compelling drawings based on hex flowers nested within the petals of other hex flowers. I chose to render some of those for the book cover:

I used Observable to host the code that renders the fractal hex flowers. This enabled me to experiment with various colour schemes and effects. Here's one example:

Not only did I write the program to render the figures on Observable, I turned the notebook into a comprehensive article explaining not only how to draw the figures, but also the geometry behind it.

The present blog post is really only meant as a placeholder. The real article is over at Observable. Go there to read it.

Including an example of property-based testing without much partitioning.

A tweet from Brian Marick induced me to read a paper by Dick Hamlet and Ross Taylor called Partition Testing Does Not Inspire Confidence. In general, I find the conclusion fairly intuitive, but on the other hand hardly an argument against property-based testing.

I'll later return to why I find the conclusion intuitive, but first, I'd like to address the implied connection between partition testing and property-based testing. I'll also show a detailed example.

The source code used in this article is available on GitHub.

Not the same #

The Hamlet & Taylor paper is exclusively concerned with partition testing, which makes sense, since it's from 1990. As far as I'm aware, property-based testing wasn't invented until later.

Brian Marick extends its conclusions to property-based testing:

"I've been a grump about property-based testing because I bought into the conclusion of Hamlet&Taylor's 1990 "Partition testing does not inspire confidence""

There's certainly overlap between partition testing and property-based testing, but it's not complete. Some property-based testing isn't partition testing, or the other way around:

To be fair, the overlap may easily be larger than the figure implies, but you can certainly describes properties without having to partition a function's domain.

In fact, the canonical example of property-based testing (that reversing a list twice yields the original list: reverse (reverse xs) == xs) does not rely on partitioning. It works for all finite lists.

You may think that this is only because the case is so simple, but that's not the case. You can also avoid partitioning on the slightly more complex problem presented by the Diamond kata. In fact, the domain for that problem is so small that you don't need a property-based framework.

You may argue that the Diamond kata is another toy problem, but I've also solved a realistic, complex business problem with property-based testing without relying on partitioning. Granted, the property shown in that article doesn't sample uniformly from the entire domain of the System Under Test, but the property (there's only one) doesn't rely on partitioning. Instead, it relies on incremental tightening of preconditions to tease out the desired behaviour.

I'll show another example next.

FizzBuzz via partitioning #

When introducing equivalence classes and property-based testing in workshops, I sometimes use the FizzBuzz kata as an example. When I do this, I first introduce the concept of equivalence classes and then proceed to explain how instead of manually picking values from each partition, you can randomly sample from them:

[<Property(QuietOnSuccess = true)>]
let ``FizzBuzz.transform returns Buzz`` (number : int) =
    (number % 5 = 0 && number % 3 <> 0) ==> lazy
    let actual = FizzBuzz.transform number
    let expected = "Buzz"
    expected = actual

(That's F# code, but the rest of the code in this article is going to be Haskell.)

While this gently introduces the concept of testing based on randomly sampling inputs, it relies heavily on partitioning. The above example filters inputs so that it only runs with numbers that are divisible by five, but not by three.

As at least one workshop attendee objected, it's getting perilously close to reproducing the implementation logic in the test. It always hurts when someone directs valid criticism at you, but he was right. That's not a problem with property-based testing, though, but rather with the way I presented it.

We can do better.

FizzBuzz with proper properties #

The trick to 'true' property-based testing is identifying proper properties for the problem being solved. Granted, this can be difficult and often requires some creative thinking (which is also why I find it so enjoyable). Fortunately, certain patterns tend to recur; for example, Scott Wlaschin has a small collection of property-based testing patterns.

As the FizzBuzz kata is described, the domain for a fizzBuzz function is only the numbers from one to 100. Let's be generous, however, and expand it to all integers, since it makes no practical difference.

In Haskell, for example, we might aim for a function with this API:

fizzBuzz :: (Integral a, Show a) => a -> String

Is it possible to test-drive the correct implementation with QuickCheck without relying on partitioning?

I must admit that I can't figure out how to entirely avoid partitioning, but it's possible to bootstrap the process using only a single partition. If you know of a way to entirely do without partitioning, leave a comment.

FizzBuzz #

In order to anchor the behaviour, we have to describe how at least a single value translates to a string, for example that all multiples of both three and five translate to "FizzBuzz". It might be enough to simply state that a small number like 0 or 15 translates to "FizzBuzz", but we might as well exercise that entire partition:

testProperty "Divisible by both 3 and 5" $ \ (seed :: Int) ->
  let i = seed * 3 * 5
      actual = fizzBuzz i
  in "FizzBuzz" === actual

Here I take any integer seed and use it to produce an integer i which is guaranteed to belong to the partition that always produces the output "FizzBuzz".

Certainly, this tests only a single partition, but as Johannes Link points out, property-based testing still performs randomised testing within the partition.

The simplest implementation that passes the test is this:

fizzBuzz :: Integral a => a -> String
fizzBuzz _ = "FizzBuzz"

From here, however, it's possible to describe the rest of the problem without relying on partition testing.

At least one number in three consecutive values #

How to proceed from there long eluded me. Then it dawned on me that while it's hard to test a single call to the fizzBuzz function without relying on partitioning, you can examine the output from projecting a small range of inputs to outputs.

What happens if we pick a random number, use it as an origin to enumerate three numbers in total (that is: two more numbers), and then call fizzBuzz with each of them?

Imagine what happens if we randomly pick 10. In that case, we're going to enumerate three numbers, starting with 10: 10, 11, 12. What's the expected output of applying these three numbers to fizzBuzz? It's Buzz, 11, Fizz. Let's try a few more, and make a table of it:

Do you notice a pattern?

There's more than a single pattern, but one is that there's always at least one number among the three results. Even if you have both a Fizz and a Buzz, as is the case with 10, 11, 12, at least one of the results (11) remains a number. Think about it some more to convince yourself that this should always be the case for three consecutive numbers.

That's a property of fizzBuzz, and it holds universally (also for negative integers).

You can turn it into a QuickCheck property like this:

testProperty "At least one number in 3 consecutive values" $ \ (i :: Int) ->
  let range = [i..i+2]
      actual = fizzBuzz <$> range
  in counterexample
      (show range ++ "->" ++ show actual) $
      any (\x -> isJust (readMaybe x :: Maybe Int)) actual

This test doesn't rely on input partitioning. It works for all integers.

In this test I used QuickCheck's counterexample function to provide a helpful message in case of failure. Running the test suite against the above version of fizzBuzz yields a failure like this:

At least one number in 3 consecutive values: [Failed]
*** Failed! Falsified (after 1 test):
0
[0,1,2]->["FizzBuzz","FizzBuzz","FizzBuzz"]
(used seed -6204080625786338123)

Here we see that the sequence [0,1,2] produces the output ["FizzBuzz","FizzBuzz","FizzBuzz"], which is not only wrong, but is specifically incorrect in the sense that none of the values can be parsed as an integer.

Given the current implementation, that's hardly surprising.

Using the Devil's Advocate technique, I chose to pass both tests with this implementation:

fizzBuzz :: Integral a => a -> String
fizzBuzz i = if i `mod` 15 == 0 then "FizzBuzz" else "2112"

The property I just added doesn't check whether the number is one of the input numbers, so the implementation can get away with returning the hard-coded string "2112".

At least one Fizz in three consecutive values #

Take a look at the above table. Do you notice any other patterns?

Of each set of three results, there's always a string that starts with Fizz. Sometimes, as we see with the input 15, the output is FizzBuzz, so it's not always just Fizz, but there's always a string that starts with Fizz.

This is another universal property of the fizzBuzz function, which we can express as a test:

testProperty "At least one Fizz in 3 consecutive values" $ \ (i :: Int) ->
  let range = [i..i+2]
      actual = fizzBuzz <$> range
  in counterexample
      (show range ++ "->" ++ show actual) $
      any ("Fizz" `isPrefixOf`) actual

Again, no partitioning is required to express this property. The arbitrary parameter i is unconstrained.

To pass all tests, I implemented fizzBuzz like this:

fizzBuzz :: Integral a => a -> String
fizzBuzz i = if i `mod` 3 == 0 then "FizzBuzz" else "2112"

It doesn't look as though much changed, but notice that the modulo check changed from modulo 15 to modulo 3. While incorrect, it passes all tests.

Only one Buzz in five consecutive values #

Using the same reasoning as above, another property emerges. If, instead of looking at sequences of three input arguments, you create five values, only one of them should result in a Buzz result; e.g. 8, 9, 10, 11, 12 should result in 8, Fizz, Buzz, 11, Fizz. Sometimes, however, the Buzz value is FizzBuzz, for example when the origin is 11: 11, 12, 13, 14, 15 should produce 11, Fizz, 13, 14, FizzBuzz.

Like the above property, there's only one Buzz, but sometimes it's part of a compound word. What's clear, though, is that there should be only one result that ends with Buzz.

Not only is the idea similar to the one above, so is the test:

testProperty "Only one Buzz in 5 consecutive values" $ \ (i :: Int) ->
  let range = [i..i+4]
      actual = fizzBuzz <$> range
  in counterexample
      (show range ++ "->" ++ show actual) $
      1 == length (filter ("Buzz" `isSuffixOf`) actual)

Again, no partitioning is required to express this property.

This version of fizzBuzz passes all tests:

fizzBuzz :: Integral a => a -> String
fizzBuzz i | i `mod` 5 == 0 = "FizzBuzz"
fizzBuzz i | i `mod` 3 == 0 = "Fizz"
fizzBuzz _ = "2112"

We're not quite there yet, but we're getting closer.

At least one literal Buzz in ten consecutive values #

What's wrong with the above implementation?

It never returns Buzz. How can we express a property that forces it to do that?

We can keep going in the same vein. We know that if we sample a sufficiently large sequence of numbers, it might produce a FizzBuzz value, but even if it does, there's going to be a Buzz value five positions before and after. For example, if the input sequence contains 30 (producing FizzBuzz) then both 25 and 35 should produce Buzz.

How big a range should we sample to be certain that there's at least one Buzz?

If it's not immediately clear, try setting up a table similar to the one above:

Notice that as one Buzz drops off to the left, a new one appears to the right. Additionally, there may be more than one literal Buzz, but there's always at least one (that is, one that's exactly Buzz, and not just ending in Buzz).

That's another universal property: for any consecutive sequence of numbers of length ten, there's at least one exact Buzz. Here's how to express that as a QuickCheck property:

testProperty "At least one literal Buzz in 10 values" $ \ (i :: Int) ->
  let range = [i..i+9]
      actual = fizzBuzz <$> range
  in counterexample
      (show range ++ "->" ++ show actual) $
      elem "Buzz" actual

Again, no partitioning is required to express this property.

This version of fizzBuzz passes all tests:

fizzBuzz :: Integral a => a -> String
fizzBuzz i | i `mod` 15 == 0 = "FizzBuzz"
fizzBuzz i | i `mod`  3 == 0 = "Fizz"
fizzBuzz i | i `mod`  5 == 0 = "Buzz"
fizzBuzz _ = "2112"

What's left? Only that the number is still hard-coded.

Numbers round-trip #

How to get rid of the hard-coded number?

From one of the above properties, we know that if we pick an arbitrary consecutive sequence of three numbers, at least one of the results will be a string representation of the input number.

It's not guaranteed to be the origin, though. If the origin is, say, 3, the input sequence is 3, 4, 5, which should yield the resulting sequence Fizz, 4, Buzz.

Since we don't know which number(s) will remain, how can we check that it translates correctly? We can use a variation of a common property-based testing pattern - the one that Scott Wlaschin calls There and back again.

We can take any sequence of three outputs and try to parse them back to integers. All successfully parsed numbers must belong to the input sequence.

That's another universal property. Here's how to express that as a QuickCheck property:

testProperty "Numbers round-trip" $ \ (i :: Int) ->
  let range = [i..i+2]
 
      actual = fizzBuzz <$> range
 
      numbers = catMaybes $ readMaybe <$> actual
  in counterexample
      (show range ++ "->" ++ show actual) $
      all (`elem` range) numbers

The parsed numbers may contain one or two elements, but in both cases, all of them must be an element of range.

Again, no partitioning is required to express this property.

This version of fizzBuzz passes all tests:

fizzBuzz :: (Integral a, Show a) => a -> String
fizzBuzz i | i `mod` 15 == 0 = "FizzBuzz"
fizzBuzz i | i `mod`  3 == 0 = "Fizz"
fizzBuzz i | i `mod`  5 == 0 = "Buzz"
fizzBuzz i = show i

This looks good. Let's call it a day.

Not so fast, though.

Redundant property? #

With the new round-trip property, isn't the property titled At least one number in 3 consecutive values redundant?

You might think so, but it's not. What happens if we remove it?

If you remove the At least one number in 3 consecutive values property, the Devil's Advocate can corrupt fizzBuzz like this:

fizzBuzz :: (Integral a, Show a) => a -> String
fizzBuzz i | i `mod` 15 == 0 = "FizzBuzz"
fizzBuzz i | i `mod`  3 == 0 = "Fizz"
fizzBuzz i | i `mod`  5 == 0 = "Buzz"
fizzBuzz _ = "Pop"

This passes all tests if you remove At least one number in 3 consecutive values. Why doesn't the Numbers round-trip property fail?

It doesn't fail because with the above implementation of fizzBuzz its numbers list is always empty. This property doesn't require numbers to be non-empty. It doesn't have to, because that's the job of the At least one number in 3 consecutive values property. Thus, that property isn't redundant. Leave it in.

Intuition behind the paper #

What about the results from the Hamlet & Taylor paper? Are the conclusions in the paper wrong?

They may be, but that's not my take. Rather, the way I understand the paper, it says that partition testing isn't much more efficient at detecting errors than pure random sampling.

I've been using the rather schizophrenic version of the Devil's Advocate technique (the one that I call Gollum style) for so long that this conclusion rings true for me.

Consider a truly adversarial developer from Hell. He or she could subvert fizzBuzz like this:

fizzBuzz :: (Integral a, Show a) => a -> String
fizzBuzz 18641 = "Pwnd"
fizzBuzz i | i `mod` 15 == 0 = "FizzBuzz"
fizzBuzz i | i `mod`  3 == 0 = "Fizz"
fizzBuzz i | i `mod`  5 == 0 = "Buzz"
fizzBuzz i = show i

The test suite is very unlikely to detect the error - even if you ask it to run each property a million times:

$ stack test --ta "-a 1000000"
FizzBuzzHaskellPropertyBased> test (suite: FizzBuzz-test, args: -a 1000000)

Divisible by both 3 and 5: [OK, passed 1000000 tests]
At least one number in 3 consecutive values: [OK, passed 1000000 tests]
At least one Fizz in 3 consecutive values: [OK, passed 1000000 tests]
Only one Buzz in 5 consecutive values: [OK, passed 1000000 tests]
At least one literal Buzz in 10 values: [OK, passed 1000000 tests]
Numbers round-trip: [OK, passed 1000000 tests]

         Properties  Total
 Passed  6           6
 Failed  0           0
 Total   6           6

FizzBuzzHaskellPropertyBased> Test suite FizzBuzz-test passed

How many times should we run these properties before we'd expect the At least one number in 3 consecutive values property to detect the error?

In Haskell, Int is an integer type with at least the range [-2^29 .. 2^29-1] - that is, from -536,870,912 to 536,870,911, for a total range of 1,073,741,823 numbers.

In order to detect the error, the At least one number in 3 consecutive values property needs to hit 18,641, which it can only do if QuickCheck supplies an i value of 18,639, 18,640, or 18,641. That's three values out of 1,073,741,823.

If we assume a uniform distribution, the chance of detecting the error is 3 / 1,073,741,823, or approximately one in 333 million.

Neither property-based testing nor randomised testing is likely to detect this kind of error. That's basically the intuition that makes sense to me when reading the Hamlet & Taylor paper. If you don't know where to look, partition testing isn't going to help you detect errors like the above.

I can live with that. After all, the value I get out of property-based testing is as a variation on test-driven development, rather than only quality assurance. It enables me to incrementally flesh out a problem in a way that example-based testing sometimes can't.

Conclusion #

There's a big overlap between partition testing and property-based testing. Often, identifying equivalence classes is the first step to expressing a property. A conspicuous example can be seen in my article series Types + Properties = Software, which shows a detailed walk-through of the Tennis kata done with FsCheck. For a hybrid approach, see Roman numerals via property-based TDD.

In my experience, it's much easier to partition a domain into equivalence classes than it is to describe universal properties of a system. Thus, many properties I write tend to be a kind of partition testing. On the other hand, it's more satisfying when you can express universal properties. I'm not sure that it's always possible, but I find that when it is, it better decouples the tests from implementation details.

Based on the FizzBuzz example shown here, you may find it unappealing that there's more test code than 'production code'. Clearly, for a problem like FizzBuzz, this is unwarranted. That, however, wasn't the point with the example. The point was to show an easily digestible example of universal properties. For a more realistic example, I'll refer you to the scheduling problem I also linked to earlier. While the production code ultimately turned out to be compact, it's far from trivial.

If it hurts, do it more often.

The agile software development movement has been instrumental in uncovering better ways of developing software. Under that umbrella term you find such seminal ideas as test-driven development, continuous delivery, responding to change, and so on. Yet, as we're entering the third decade of agile software development, most organisations still struggle to take in the ethos and adopt its practices.

Change is hard. A typical reaction from a development organisation would be:

"We tried it, but it doesn't work for us."

The usual agile response to such foot-draggery is: if it hurts, do it more often.

I'd like to suggest exactly that remedy for a perceived problem that many agile proponents seem to believe exist.

Pull request problems and solutions #

I like collaborating with other developers via pull requests, but I also admit that my personality may influence that preference. I like deep contemplation; I like doing things on my own time, to my own schedule; I like having the freedom to self-organise. Yes, I fit the description of an introvert. Pull requests do enable contemplation, they do let me self-organise.

Yet, I'm quite aware that there are plenty of problems with pull requests. First, many pull requests are poorly done. They are too big, bundle together unrelated things and introduce noise. I wrote an article titled 10 tips for better Pull Requests to help developers create better pull requests. With a little practice, you can write small, focused pull requests.

If it hurts, do it more often.

Another problem with pull requests is in the review process. The most common criticism of the pull request process with its review phase is that it takes too long. I discuss this problem, and a remedy, in my coming book Code That Fits in Your Head:

"The problem with typical approaches is illustrated by [the figure below]. A developer submits a piece of work for review. Then much time passes before the review takes place.

"[Figure caption:] How not to do code reviews: let much time pass between completion of a feature and the review. (The smaller boxes to the right of the review indicates improvements based on the initial review, and a subsequent review of the improvements.)

"[The figure below] illustrates an obvious solution to the problem. Reduce the wait time. Make code reviews part of the daily rhythm of the organisation.

"[Figure caption:] Reduce the wait time between feature completion and code review. A review will typically spark some improvements, and a smaller review of those improvements. These activities are indicated by the smaller boxes to the right of the review.

"Most people already have a routine that they follow. You should make code reviews part of that routine. You can do that on an individual level, or you can structure your team around a daily rhythm. Many teams already have a daily stand-up. Such a regularly occurring event creates an anchor around which the day revolves. Typically, lunchtime is another natural break in work.

"Consider, for example, setting aside half an hour each morning, as well as half an hour after lunch, for reviews.

"Keep in mind that you should make only small sets of changes. Sets that represent less than half a day's work. If you do that, and all team members review those small changes twice a day, the maximum wait time will be around four hours."

If it hurts, do it more often.

An underlying problem is that people often feel that they don't have the time to review their colleagues' code. This is a real problem. If you are being measured (formally or informally) on your 'individual contribution', then anything you do to help your team looks like a waste of time. This is, in my opinion, an organisational problem, rather than a problem with doing reviews.

It's also a problem that pair programming doesn't solve.

Pull requests versus pair programming #

You can make the pull request process work, but should you? Isn't pair (or ensemble) programming better?

Pair programming can also be effective. I discuss that too in my new book. What works best, I believe, is a question of trade-offs. What's more important to you? Latency or throughput?

In other words, while I have a personality-based preference for the contemplative, asynchronous pull request process, I readily admit that pair or ensemble programming may be better in many situations.

I suspect, however, that many proponents of pair programming are as driven by their personality-based preference as I am, but that since they might be extroverts, they favour close personal interaction over contemplation.

In any case, use what works for you, but be wary of unequivocal claims that one way is clearly better than the other. We have scant scientific knowledge about software development. Most of what I write, and what industry luminaries write, is based on anecdotal evidence. This also applies to this discussion of pull requests versus pair programming.

Conclusion #

I have anecdotal evidence that pull requests can work in an 'agile' setting. One team had implemented continuous deployment and used pull requests because more than half the team members were working from remote (this was before the COVID-19 pandemic). Pull requests were small and reviews could typically be done in five-ten minutes. Knowing this, reviews were prompt and frequent. Turnaround-time was good.

I also have anecdotal evidence that ensemble programming works well. To me, it solves a completely different problem, but I've used it to great effect for knowledge transfer.

Programmers more extrovert than me report anecdotal evidence that pair programming is best, and I accept that this is true - for them. I do not, however, accept it as a universal truth. Neither do I claim that my personal preference for pull request is incontrovertibly best.

Comments

The expanded universe.

It gives me great pleasure to announce that my new book Code That Fits in Your Head will be out in September 2021. The subtitle is Heuristics for Software Engineering.

Many regular readers have been expecting me to write a book about functional programming, and while that may also some day happen, this book is neither about object-oriented nor functional programming per se. Rather, it takes a step back and looks at various software development techniques and practices that I've found myself teaching for years. It covers both coding, troubleshooting, software design, team work, refactoring, and architecture.

The target audience is all the hard-working enterprise developers in our industry. I estimate that there's great overlap with the general readership of this blog. In other words, if you find this blog useful, I hope that you'll also find the book useful.

As the title suggests, the theme is working effectively with code in a way that acknowledges the limitations of the human brain. This is a theme I've already explored in my Humane Code video, but in the book I expand the scope.

Expanded universe #

I've structured the book around a realistic sample application. You'll see how to bootstrap a code base, but also how to work effectively with existing code. Along with the book, you'll get access to a complete Git repository with more than 500 commits and more than 6,000 lines of lovingly crafted code.

While I was developing the sample code, I solved many interesting problems. The best and most universal examples I used in the book, but many didn't make the cut. The book aims broadly at programmers comfortable with a C-based programming language: Java, C#, JavaScript, C++, and so on. Some of the problems I solved along the way were specific to .NET, so I found them a poor fit for the book. I didn't want these great lessons to go to waste, so instead I've been blogging about them.

These are the articles based on the code base from the book:

• Closing database connections during test teardown
• Using the nameof C# keyword with ASP.NET 3 IUrlHelper
• An ASP.NET Core URL Builder
• Adding REST links as a cross-cutting concern
• EnsureSuccessStatusCode as an assertion
• Fortunately, I don't squash my commits
• Monomorphic functors
• Signing URLs with ASP.NET
• Checking signed URLs with ASP.NET
• Redirect legacy URLs
• Name by role
• Branching tests
• Waiting to happen
• Parametrised test primitive obsession code smell
• Self-hosted integration tests in ASP.NET
• ASP.NET POCO Controllers: an experience report
• When properties are easier than examples
• Pendulum swing: internal by default
• Pendulum swing: sealed by default
• Consider including identity in URLs
• Leaky abstraction by omission
• Structural equality for better tests
• Simplifying code with Decorated Commands
• Unit testing private helper methods
• Keep IDs internal with REST
• The Equivalence contravariant functor
• Waiting to never happen
• Test Double clocks
• An applicative reservation validation example in C#
• ASP.NET validation revisited
• Coalescing DTOs
• Stubs and mocks break encapsulation
• An initial proof of concept of applicative assertions in C#
• Warnings-as-errors friction
• A restaurant example of refactoring from example-based to property-based testing
• Decomposing CTFiYH's sample code base
• Do ORMs reduce the need for mapping?
• The case of the mysterious comparison

That the above list represents the outtakes from the book's example code base should give you an idea of the richness of it.

I may add to the list in the future if I write more articles that use the book's example code base.

Conclusion #

A decade after my first book, I've finally written a completely new book. Over the years, I had a few false starts. This book wasn't the book I thought that I'd be writing if you'd asked me five years ago, but when it finally dawned on me that the topic ought to be Code That Fits in Your Head: Heuristics for Software Engineering, the book almost wrote itself.

This one is for all the software developers out there who aspire to improve their practical skills.

Comments