Enumerate Wordle combinations with an applicative functor

Monday, 17 January 2022 16:39:00 UTC

An example of ad-hoc programming.

Like so many others, I recently started solving the daily Wordle puzzle. As is normal when one is a beginner, I initially struggled a bit. One day, I couldn't find a good word to proceed.

To be clear, this article isn't really about Wordle strategies or tools. Rather, it's an example of ad-hoc programming. Particularly, it's an example of how the applicative nature of lists can be useful when you need to enumerate combinations. While I've previously shown a similar example, I think one more is warranted.

Inured from tears #

Last Monday, I'd started with the word TEARS. (I've subsequently learned that better starting words exist.) The puzzle responded with a yellow E and a green R:

Wordle after entering TEARS.

In case you haven't played the game, this means that the fourth letter of the hidden word is an R, and that the word also contains an E. The second letter, however, is not an E. Also, the hidden word contains neither T, A, nor S.

While the green and yellow letters may be repeated, you only have six guesses, so it's a good idea to attempt to exhaust as many letters as possible. The first compatible word with five distinct letters that I could think of was INURE.

Wordle after entering TEARS and INURE.

This gave me a bit of new information. The hidden word also contains a U, but not as the third letter. Furthermore, the E isn't the last letter either. Keep in mind that from TEARS we also know that E isn't the second letter.

While I believe myself to have a decent English vocabulary, at this point I was stuck. While I knew that the E had to be in either first or third position, I couldn't think of a single word that fit all observations.

After trying to come up with a word for ten minutes, I decided that, instead of giving up, I'd use the applicative nature of lists to generate all possible combinations. I was hoping that with the observations already in place, there wouldn't be too many to sift through.

Combinations #

While you could do this in other languages (such as F# or C#), it's natural to use Haskell because it natively understands applicative functors. Thus, I launched GHCi (Glasgow Haskell Compiler interactive - the Haskell REPL).

Wordle is kind enough to show a keyboard with colour-coded keys:

Wordle keyboard.

All letters except the dark ones remain valid, so I first defined a list of all available letters:


The variable avail is a String, but in Haskell, a String is a type synonym for a (linked) list of Char values (characters). Since lists form an applicative functor, that'll work.

Most of the letters are still in play - only five letters are already out of the picture: T, I, A, S, and N. Thus, availstill spells out most of the alphabet.

Next, I wrote an expression that enumerates all five-letter combinations of these letters, with one constraint: The fourth letter must be an R:

> candidates = (\x y z æ ø -> [x,y,z,æ,ø]) <$> avail <*> avail <*> avail <*> "R" <*> avail

The leftmost expression ((\x y z æ ø -> [x,y,z,æ,ø])) is a lambda expression that takes five values (one from each list of available letters) and combines them to a single list. Each value (x, y, and so on) is a Char value, and since String in Haskell is the same as [Char], the expression [x,y,z,æ,ø] is a String. In Danish we have three more letters after z, so I after I ran out of the the usual Latin letters, I just continued with the Danish æ and ø.

Notice that between each of the <*> operators (apply) I've supplied the list of available letters. In the fourth position there's no choice, so there the list contains only a single letter. Recall that a String is a list of characters, so "R" is the same as ['R'].

How many combinations are there? Let's ask GHCi:

> length candidates

Almost 200,000. That's a bit much to look through, but we can peek at the first ten as a sanity check:

> take 10 candidates

There are no promising words in this little list, but I never expected that.

I needed to narrow down the options.

Filtering #

How do you make a collection smaller? You could filter it.

candidates contains illegal values. For example, the third value in the above list (of the ten first candidates) is "QQQRE". Yet, we know (from the INURE attempt) that the last letter isn't E. We can filter out all strings that end with E:

> take 10 $ filter (\s -> s!!4 /= 'E') candidates

In Haskell, !! is the indexing operator, so s!!4 means the (zero-based) fourth element of the string s. /= is the inequality operator, so the lambda expression (\s -> s!!4 /= 'E') identifies all strings where the fifth element (or fourth element, when starting from zero) is different from E.

We know more than this, though. We also know that the second element can't be E, and that the third element isn't U, so add those predicates to the filter:

> take 10 $ filter (\s -> s!!1 /= 'E' && s!!2 /= 'U' && s!!4 /= 'E') candidates

How many are left?

> length $ filter (\s -> s!!1 /= 'E' && s!!2 /= 'U' && s!!4 /= 'E') candidates

Still too many, but we aren't done yet.

Notice that all of the first ten values shown above are invalid. Why? Because the word must contain at least one E, and none of them do. Let's add that predicate:

> take 10 $ filter (\s -> s!!1 /= 'E' && s!!2 /= 'U' && s!!4 /= 'E' && 'E' `elem` s) candidates

The Boolean expression 'E' `elem` s means that the character 'E' must be an element of the string (list) s.

The same rule applies for U:

> take 10 $ filter (\s -> s!!1 /= 'E' && s!!2 /= 'U' && s!!4 /= 'E' && 'E' `elem` s && 'U' `elem` s)

There's a great suggestion already! The eighth entry spells QUERY! Let's try it:

Wordle after entering TEARS, INURE and QUERY.

QUERY was the word of the day!

A bit embarrassing that I couldn't think of query, given that I often discuss Command Query Separation.

Was that just pure luck? How many suggestions are left in the filtered list?

> length $ filter (\s -> s!!1 /= 'E' && s!!2 /= 'U' && s!!4 /= 'E' && 'E' `elem` s && 'U' `elem` s)

Okay, a bit lucky. If I ask GHCi to display the filtered list in its entirety, no other word jumps out at me, looking like a real word.

Conclusion #

While I admit that I was a bit lucky that QUERY was among the first ten of 1,921 possible combinations, I still find that applicative combinations are handy as an ad-hoc tool. I'm sure there are more elegant ways to solve a problem like this one, but for me, this approach had low ceremony. It was a few lines of code in a terminal. Once I had the answer, I could just close the terminal and no further clean-up was required.

I'm sure other people have other tool preferences, and perhaps you'd like to leave a comment to the effect that you have a better way with Bash, Python, or APL. That's OK, and I don't mind learning new tricks.

I do find this capability of applicative functors to do combinatorics occasionally useful, though.


It's not "better", but here's a similar approach in Python.

from operator import contains

candidates = ((x,y,z,æ,ø) for x in avail for y in avail for z in avail for æ in "R" for ø in avail)
filtered = [s for s in candidates if s[1] != "E" and s[2] != "U" and s[4] != "E" and contains(s, "E") and contains(s, "U")]

for candidate in filtered[:10]:
    print(*candidate, sep="")

2022-01-21 19:08 UTC

Type-safe DI composition

Monday, 10 January 2022 06:41:00 UTC

DI Containers aren't type-safe. What are the alternatives?

In April 2020 I published an article called Unit bias against collections. My goal with the article was to point out a common cognitive bias. I just happened to use .NET's built-in DI Container as an example because I'd recently encountered a piece of documentation that exhibited the kind of bias I wanted to write about.

This lead to a discussion about the mental model of the DI Container:

"Yeah, I wasn't explicit and clarified in a second tweet. I didn't mean in the services example, but in general where it helps if the reader's mental model of the code has 1 item from the collection, because that's how it is in real life. SingleOrDefault would enforce this."

The point made by Bogdan Galiceanu highlights the incongruity between the container's API and the mental model required to work with it.

IServiceCollection recap #

The API in case belongs to IServiceCollection, which is little more than a collection of ServiceDescriptor objects. Each ServiceDescriptor describes a service, as the name implies.

Given an IServiceCollection you can, for example, register an IReservationsRepository instance:

var connStr = Configuration.GetConnectionString("Restaurant");
services.AddSingleton<IReservationsRepository>(sp =>
    var logger =
    var postOffice = sp.GetService<IPostOffice>();
    return new EmailingReservationsRepository(
        new LoggingReservationsRepository(
            new SqlReservationsRepository(connStr)));

This adds a ServiceDescriptor entry to the collection. (Code examples are from Code That Fits in Your Head.)

Later, you can remove and replace the service for test purposes:

internal sealed class SelfHostedApi : WebApplicationFactory<Startup>
    protected override void ConfigureWebHost(IWebHostBuilder builder)
        builder.ConfigureServices(services =>
                new FakeDatabase());

Here I use RemoveAll, even though I 'know' there's only one service of that type. Bogdan Galiceanu's argument, if I understand it correctly, is that it'd be more honest to use SingleOrDefault, since we 'know' that there's only one such service.

I don't bring this up to bash on either Bogdan Galiceanu or the IServiceCollection API, but this exchange of ideas provided another example that DI Containers aren't as helpful as you'd think. While they do provide some services, they require significant mental overhead. You have to 'know' that this service has only one instance, while another service may have two implementations, and so on. As the size of both code base and team grows, keeping all such knowledge in your head becomes increasingly difficult.

The promise of object-orientation was always that you shouldn't have to remember implementation details.

Particularly with statically typed programming languages you should be able to surface such knowledge as static type information. What would a more honest, statically typed DI Container look like?

Statically typed containers #

Over a series of articles I'll explore how a statically typed DI Container might look:

The first of these articles show a Haskell prototype, while the rest of the articles use C#. If you don't care about Haskell, you can skip the first article.

As the title of the last article implies, this exploration only concludes that type-safe DI Containers are isomorphic to Pure DI. I consider Pure DI the simplest of these approaches, suggesting that there's no point in type-safe DI Containers of the kinds shown here.

Conclusion #

Some people like DI Containers. I don't, because they take away type-safety without providing much benefit to warrant the trade-off. A commonly suggested benefit of DI Containers is lifetime management, but you can trivially implement type-safe lifetime management with Pure DI. I don't find that argument compelling.

This article series examines if it's possible to create a 'better' DI Container by making it more type-safe, but I ultimately conclude that there's no point in doing so.

Next: Type-level DI Container prototype.

To ID or not to ID

Monday, 03 January 2022 08:57:00 UTC

How to model an ID that sometimes must be present, and sometimes not.

I'm currently writing a client library for Criipto that partially implements the actions available on the Fusebit API. This article, however, isn't about the Fusebit API, so even if you don't know what that is, read on. The Fusebit API is just an example.

This article, rather, is about how to model the absence or presence of an Id property.

User example #

The Fusebit API is an HTTP API that, as these things usually do, enables you to create, read, update, and delete resources. One of these is a user. When you create a user, you supply such data as firstName, lastName, and primaryEmail:

POST /v1/account/acc-123/user HTTP/2
authorization: Bearer 938[...]
content-type: application/json

HTTP/2 200
content-type: application/json; charset=utf-8

Notice that you're supposed to POST the user representation without an ID. The response, however, contains an updated representation of the resource that now includes an id. The id (in this example usr-8babf0cb95d94e6f) was created by the service.

To summarise: when you create a new user, you can't supply an ID, but once the user is created, it does have an ID.

I wanted to capture this rule with the F# type system.

Inheritance #

Before we get to the F# code, let's take a detour around some typical C# code.

At times, I've seen people address this kind of problem by having two types: UserForCreation and CreatedUser, or something like that. The only difference would be that CreatedUser would have an Id property, whereas UserForCreation wouldn't. While, at this time, the rule of three doesn't apply yet, such duplication still seems frivolous.

How does an object-oriented programmer address such a problem? By deriving CreatedUser from UserForCreation, of course!

public class CreatedUser : UserForCreation
    public string Id { getset; }

I'm not too fond of inheritance, and such a design also comes with a built-in flaw: Imagine a method with the signature public CreatedUser Create(UserForCreation user). While such an API design clearly indicates that you don't have to supply an ID, you still can. You can call such a Create method with a CreatedUser object, since CreatedUser derives from UserForCreation.

CreatedUser user = resource.Create(new CreatedUser
    Id = "123",
    FirstName = "Sue",
    LastName = "Flay",
    Email = "suoffle@example.org"

Since CreatedUser contains an ID, this seems to suggest that you can call the Create method with a user with an ID. What would you expected from such a possibility? In the above code example, what would you expect the value of user.Id to be?

It'd be reasonable to expect user.Id to be "123". This seems to indicate that it'd be possible to supply a client-generated user ID, which would then be used instead of a server-generated user ID. The HTTP API, however, doesn't allow that.

Such a design is misleading. It suggests that CreatedUser can be used where UserForCreation is required. This isn't true.

Generic user #

I was aware of the above problem, so I didn't even attempt to go there. Besides, I was writing the library in F#, not C#, and while F# enables inheritance as well, it's not the first modelling option you'd typically reach for.

Instead, my first attempt was to define user data as a generic record type:

type UserData<'a> =
        Id : 'a
        FirstName : string option
        LastName : string option
        Email : MailAddress option
        Identities : Identity list
        Permissions : Permission list

(The Fusebit API also enables you to supply Identities and Permissions when creating a user. I omitted them from the above C# example code because this detail is irrelevant to the example.)

This enabled me to define an impure action to create a user:

// ApiClient -> UserData<unit> -> Task<Result<UserData<string>, HttpResponseMessage>>
let create client (userData : UserData<unit>) = task {
    let jobj = JObject ()
    |> Option.iter (fun fn -> jobj.["firstName"<- JValue fn)
    |> Option.iter (fun ln -> jobj.["lastName"<- JValue ln)
    |> Option.iter
        (fun email -> jobj.["primaryEmail"<- email |> string |> JValue)
        |> List.map Identity.toJToken
        |> List.toArray
        |> JArray
        let aobj = JObject ()
            |> List.map Permission.toJToken
            |> List.toArray
            |> JArray
    let json = jobj.ToString Formatting.None
    let relativeUrl = Uri ("user", UriKind.Relative)
    let! resp = Api.post client relativeUrl json
    if resp.IsSuccessStatusCode
        let! content = resp.Content.ReadAsStringAsync ()
        let jtok = JToken.Parse content
        let createdUser = parseUser jtok
        return Ok createdUser
    else return Error resp }

Where parseUser is defined like this:

// JToken -> UserData<string>
let private parseUser (jobj : JToken) =
    let uid = jobj.["id"] |> string
    let fn = jobj.["firstName"] |> Option.ofObj |> Option.map string
    let ln = jobj.["lastName"] |> Option.ofObj |> Option.map string
    let email =
        |> Option.ofObj
        |> Option.map (string >> MailAddress)
        Id = uid
        FirstName = fn
        LastName = ln
        Email = email
        Identities = []
        Permissions = []

Notice that, if we strip away all the noise from the User.create action, it takes a UserData<unit> as input and returns a UserData<string> as output.

Creating a value of a type like UserData<unit> seems a little odd:

let user =
        Id = ()
        FirstName = Some "Helen"
        LastName = Some "Back"
        Email = Some (MailAddress "hellnback@example.net")
        Identities = []
        Permissions = []

It gets the point across, though. In order to call User.create you must supply a UserData<unit>, and the only way you can do that is by setting Id to ().

Not quite there... #

In the Fusebit API, however, the user resource isn't the only resource that exhibits the pattern of requiring no ID on creation, but having an ID after creation. Another example is a resource called a client. Adopting the above design as a template, I also defined ClientData as a generic record type:

type ClientData<'a> =
        Id : 'a
        DisplayName : string option
        Identities : Identity list
        Permissions : Permission list

In both cases, I also realised that the record types gave rise to functors. A map function turned out to be useful in certain unit tests, so I added such functions as well:

module Client =
    let map f c =
            Id = f c.Id
            DisplayName = c.DisplayName
            Identities = c.Identities
            Permissions = c.Permissions

The corresponding User.map function was similar, so I began to realise that I had some boilerplate on my hands.

Besides, a type like UserData<'a> seems to indicate that the type argument 'a could be anything. The map function implies that as well. In reality, though, the only constructed types you'd be likely to run into are UserData<unit> and UserData<string>.

I wasn't quite happy with this design, after all...

Identifiable #

After thinking about this, I decided to move the generics around. Instead of making the ID generic, I instead made the payload generic by introducing this container type:

type Identifiable<'a> = { Id : string; Item : 'a }

The User.create action now looks like this:

// ApiClient -> UserData -> Task<Result<Identifiable<UserData>, HttpResponseMessage>>
let create client userData = task {
    let jobj = JObject ()
    |> Option.iter (fun fn -> jobj.["firstName"<- JValue fn)
    |> Option.iter (fun ln -> jobj.["lastName"<- JValue ln)
    |> Option.iter
        (fun email -> jobj.["primaryEmail"<- email |> string |> JValue)
        |> List.map Identity.toJToken
        |> List.toArray
        |> JArray
        let aobj = JObject ()
            |> List.map Permission.toJToken
            |> List.toArray
            |> JArray
    let json = jobj.ToString Formatting.None
    let relativeUrl = Uri ("user", UriKind.Relative)
    let! resp = Api.post client relativeUrl json
    if resp.IsSuccessStatusCode
        let! content = resp.Content.ReadAsStringAsync ()
        let jtok = JToken.Parse content
        let createdUser = parseUser jtok
        return Ok createdUser
    else return Error resp }

Where parseUser is defined as:

// JToken -> Identifiable<UserData>
let private parseUser (jtok : JToken) =
    let uid = jtok.["id"] |> string
    let fn = jtok.["firstName"] |> Option.ofObj |> Option.map string
    let ln = jtok.["lastName"] |> Option.ofObj |> Option.map string
    let email =
        |> Option.ofObj
        |> Option.map (string >> MailAddress)
    let ids =
        match jtok.["identities"with
        | null -> []
        | x -> x :?> JArray |> Seq.map Identity.parse |> Seq.toList
    let perms =
        |> Option.ofObj
        |> Option.toList
        |> List.collect (fun j ->
            j.["allow"] :?> JArray
            |> Seq.choose Permission.tryParse
            |> Seq.toList)
        Id = uid
        Item =
                FirstName = fn
                LastName = ln
                Email = email
                Identities = ids
                Permissions = perms

The required input to User.create is now a simple, non-generic UserData value, and the successful return value an Identifiable<UserData>. There's no more arbitrary ID data types. The ID is either present as a string or it's absent.

You could also turn the Identifiable container into a functor if you need it, but I've found no need for it so far. Wrapping and unwrapping the payload from the container is easy without supporting machinery like that.

This design is still reusable. The equivalent Client.create action takes a non-generic ClientData value as input and returns an Identifiable<ClientData> value when successful.

C# translation #

There's nothing F#-specific about the above solution. You can easily define Identifiable in C#:

public sealed class Identifiable<T>
    public Identifiable(string id, T item)
        Id = id;
        Item = item;
    public string Id { get; }
    public T Item { get; }
    public override bool Equals(object obj)
        return obj is Identifiable<T> identifiable &&
               Id == identifiable.Id &&
               EqualityComparer<T>.Default.Equals(Item, identifiable.Item);
    public override int GetHashCode()
        return HashCode.Combine(Id, Item);

I've here used the explicit class-based syntax to define an immutable class. In C# 9 and later, you can simplify this quite a bit using record syntax instead (which gets you closer to the F# example), but I chose to use the more verbose syntax for the benefit of readers who may encounter this example and wish to understand how it relates to a less specific C-based language.

Conclusion #

When you need to model interactions where you must not supply an ID on create, but representations have IDs when you query the resources, don't reach for inheritance. Wrap the data in a generic container that contains the ID and a generic payload. You can do this in languages that support parametric polymorphism (AKA generics).

Label persistent test data with deletion dates

Monday, 27 December 2021 06:34:00 UTC

If you don't clean up after yourself, at least enable others to do so.

I'm currently developing a software library that interacts with a third-party HTTP API to automate creation of various resources at that third party. While I use automated testing to verify that my code works, I still need to run my automation code against the real service once in while. After all, I'd like to verify that I've correctly interpreted the third party's documentation.

I run my tests against a staging environment. The entire purpose of the library is to create resources, so all successful tests leave behind new 'things' in that staging environment.

I'm not the only person who's testing against that environment, so all sorts of test entries accumulate.

Test data accretion #

More than one developer is working with the third-party staging environment. They create various records in the system for test purposes. Often, they forget about these items once the test is complete.

After a few weeks, you have various entries like these:

  • Foo test Permit Client
  • Fo Permit test client
  • Paul Fo client from ..id
  • Paul Verify Bar Test Client
  • Pauls test
  • SomeClient
  • michael-template-client

Some of these may be used for sustained testing. Others look suspiciously like abandoned objects.

Does it matter that stuff like this builds up?

Perhaps not, but it bothers my sense of order. If enough stuff builds up, it may also make it harder to find the item you actually need, and rarely, there may be real storage costs associated with the jetsam. But realistically, it just offends my proclivity for tidiness.

Label ephemeral objects explicitly #

While I was testing my library's ability to create new resources, it dawned on me that I could use the records' display names to explicitly label them as temporary.

At first, I named the objects like this:

Test by Mark Seemann. Delete if older than 10 minutes.

While browsing the objects via a web UI (instead of the HTTP API), however, I realised that the creation date wasn't visible in the UI. That makes it hard to identify the actual age.

So, instead, I began labelling the items with a absolute time of safe deletion:

Test by Mark Seemann. Delete after 2021-11-23T13:13:00Z.

I chose to use ISO 8601 Zulu time because it's unambiguous.

Author name #

As you can tell from the above examples, I explicitly named the object Test by Mark Seemann. The word Test indicates to everyone else that this is a test resource. The reason I decided to include my own name was to make it clear to other readers who to contact in case of doubt.

While I find a message like Delete after 2021-11-23T13:13:00Z quite clear, you can never predict how other readers will interpret a text. Thus, I left my name in the title to give other people an opportunity to contact me if they have questions about the record.

Conclusion #

This is just a little pleasantry you can use to make life for a development team a little more agreeable.

You may not always be able to explicitly label a test item. Some records don't have display names, or the name field is too short to allow a detailed, explicit label.

You may also feel that this isn't worth the trouble, and perhaps it isn't.

I usually clean up after having added test data, but sometimes one forgets. When working in a shared environment, I find it considerate to clearly label test data to indicate to others when it's safe to delete it.

Changing your organisation

Monday, 20 December 2021 06:41:00 UTC

Or: How do I convince X to adopt Y in software development?

In January 2012 a customer asked for my help with software architecture. The CTO needed to formulate a strategy to deal with increasing demand for bring-your-own-device (BYOD) access to internal systems. Executives brought their iPhones and iPads to work and expected to be able to access and interact with custom-developed and bespoke internal line-of-business applications.

Quite a few of these were running on the Microsoft stack, which was the reason Microsoft Denmark had recommended me.

I had several meetings with the developers responsible for enabling BYOD. One guy, in particular, kept suggesting a Silverlight solution. I pointed out that Silverlight wouldn't run on Apple devices, but he wouldn't be dissuaded. He was certain that this was the correct solution, and I could tell that he became increasingly frustrated with me because he couldn't convince me.

We'll return to this story later in this essay.

Please convince my manager #

Sometimes people ask me questions like these:

  • How do I convince my manager to let me use F#?
  • How do I convince my team mates to adopt test-driven development?
  • How do I convince the entire team that code quality matters?

Sometimes I receive such questions via email. Sometimes people ask me at conferences or user groups.

To quote Eric Lippert: Why are you even asking me?

To be fair, I do understand why people ask me, but I have few good answers.

Why ask me? #

I suppose people ask me because I've been writing about software improvement for decades. This blog dates back to January 2009, and my previous MSDN blog goes back to January 2006. These two resources alone contain more than 700 posts, the majority of which are about some kind of suggested improvement. Granted, there's also the occasional rant, by I think it's fair to say that my main motivation for writing is to share with readers what I think is right and good.

I do write a lot about potential improvements to software development.

As most readers have discovered, my book about Dependency Injection (with Steven van Deursen) is more than just a manual to a few Dependency Injection Containers, and my new book Code That Fits in Your Head is full of suggested improvements.

Add to that my Pluralsight courses, my Clean Coders videos, and my conference talks, and there's a clear pattern to most of my content.

Maven #

In The Tipping Point Malcolm Gladwell presents a model for information dissemination, containing three types of people required for an idea to take hold: Connectors, mavens, and salesmen.

In that model, a maven is an 'information specialist' - someone who accumulates knowledge and shares it with others. If I resemble any of these three types of people, I'm a maven. I like learning and discovering new ways of doing things, and obviously I like sharing that information. I wouldn't have blogged consistently for sixteen years if I didn't feel compelled to do so.

My role, as I see it, is to find and discover better ways of writing software. Much of what I write about is something that I've picked up somewhere else, but I try to present it in my own way, and I'm diligent about citing sources.

When people ask me concrete questions, like how do I refactor this piece of code? or how do write this in a more functional style?, I present an answer (if I can).

The suggestions I make are just that: It's a buffet of possible solutions to certain problems. If you encounter one of my articles and find the proposed technique useful, then that makes me happy and proud.

Notice the order of events: Someone has a problem, finds my content, and decides that it looks like a solution. Perhaps (s)he remembers my name. Perhaps (s)he feels that I helped solve a problem.

Audience members that come to my conference talks, or readers who buy my books, may not have a concrete problem to solve, but they still voluntarily seeks me out - perhaps because of previous exposure to my content.

To reiterate:

  • You have a problem (concrete or vague)
  • Perhaps you come across my content
  • Perhaps you find it useful

Perhaps you think that I convinced you that 'my way' is best. I didn't. You were already looking for a solution. You were ready for a change. You were open.

Salesman #

When you ask me about how to convince your manager, or your team mates, you're no longer asking me a technical question. Now you're asking about how to win friends and influence people. You don't need a maven for that; you need a salesman. That's not me.

I've had some success applying changes to software organisations, but in all cases, the reason I was there in the first place was because the organisation itself (or members thereof) had asked me to come and help them. When people want your help changing things, convincing them to try something new isn't a hard sell.

When I consult development teams, I'm not there to sell them new processes; I'm there to help them use appropriate solutions at opportune moments.

I'm not a salesman. Just because I convinced you that, say, property-based testing is a good idea doesn't mean I can convince anyone else. Keep in mind that I probably convinced you because you were ready for a change.

Your manager or your team mates may not be.

Bide your time #

While I'm no salesman, I've managed to turn people around from time to time. The best strategy, I've found, is to wait for an opportunity.

As long as everything is going well, people aren't ready to change.

"Only a crisis - actual or perceived - produces real change. When that crisis occurs, the actions that are taken depend on the ideas that are lying around. That, I believe, is our basic function: to develop alternatives to existing policies, to keep them alive and available until the politically impossible becomes the politically inevitable"

If you have solutions at the ready, you may be able to convince your manager or your colleagues to try something new if a crisis occurs. Don't gloat - just present your suggestion: What if we did this instead?

Vocabulary #

Indeed, I'm not a salesman, and while I can't tell you how to sell an idea to an unwilling audience, I can tell you how you can weaken your position: Make it all about you.

Notice how questions like the above are often phrased: my manager will not let me... or how do I convince my colleagues?

I actually didn't much like How to Win Friends and Influence People, but it does present the insight that in order to sway people, you have to consider what's in it for them.

I had to be explicitly told this before I learned that lesson.

In the second half of the 2000s I was attached to a software development project at a large Danish company. After a code review, I realised that the architecture of the code was all wrong!

In order to make the project manager aware of the issue, I wrote an eight-page internal 'white paper' and emailed it to the project manager (let's call him Henk).

Nothing happened. No one addressed the problem.

A few weeks later, I had a scheduled one-on-one meeting with my own manager in Microsoft, and when asked if I had any issues, I started to complain about this problem I'd identified.

My manager looked at me for a moment and asked me: How do you think receiving that email made Henk feel?

It had never crossed my mind to think about that. It was my problem! I discovered it! I viewed myself as a great programmer and architect because I had perceived such a complex issue and was able to describe it clearly. It was all about me, me, me.

When we programmers ask how to convince our managers to 'let us' use TDD, or FP, or some other 'cool' practice, we're still focused on us. What's in it for the manager?

When we talk about code quality and lack thereof, 'ugly code', refactoring, and other such language, a non-coding manager is likely to see us as primadonnas out of touch with reality: We have features to ship, but the programmers only care about making the code 'pretty'.

I offer no silver bullet to convince other people that certain techniques are superior, but do consider introducing suggestions by describing the benefits they bring: Wouldn't it be cool if we could decrease our testing phase from three weeks to three days?, Wouldn't it be nice if we could deploy every week?, Wouldn't it be nice if we could decrease the number of errors our users encounter?

You could be wrong #

Have you ever felt frustrated that you couldn't convince other people to do it your way, despite knowing that you were right?

Recall the developer from the introduction, the one who kept insisting that Silverlight was the right solution to the organisation's BYOD problem. He was clearly convinced that he had the solution, and he was frustrated that I kept rejecting it.

We may scoff at such obvious ignorance of facts, but he had clearly drunk the Microsoft Kool-Aid. I could tell, because I'd been there myself. When you're a young programmer, you may easily buy into a compelling narrative. Microsoft evangelists were quite good at their game back then, as I suppose their Apple and Linux counterparts were and are. As an inexperienced developer, you can easily be convinced that a particular technology will solve all problems.

When you're young and inexperienced, you can easily feel that you're absolutely right and still be unequivocally wrong.

Consider this the next time you push your agenda. Perhaps your manager and colleagues reject your ideas because they're actually bad. A bit of metacognition is often appropriate.

Conclusion #

How do you convince your manager or team mates to do things better?

I don't know; I'm not a salesman, but in this essay, I've nonetheless tried to reflect on the question. I think it helps to consider what the other party gains from accepting a change, but I also think it's a waiting game. You have to be patient.

As an economist, I could also say much about incentives, but this essay is already long enough as it is. Still, when you consider how your counterparts react to your suggestions, reflect on how they are incentivised.

Even if you do everything right, make the best suggestions at the most opportune times, you may find yourself in a situation systemically rigged against doing the right thing. Ultimately, as Martin Fowler quipped, either change your organisation, or change your organisation.

Backwards compatibility as a profunctor

Monday, 13 December 2021 07:01:00 UTC

In order to keep backwards compatibility, you can weaken preconditions or strengthen postconditions.

Like the previous articles on Postel's law as a profunctor and the Liskov Substitution Principle as a profunctor, this article is part of a series titled Some design patterns as universal abstractions. And like the previous articles, it's a bit of a stretch including the present article in that series, since backwards compatibility isn't a design pattern, but rather a software design principle or heuristic. I still think, however, that the article fits the spirit of the article series, if not the letter.

Backwards compatibility is often (but not always) a desirable property of a system. Even in Zoo software, it pays to explicitly consider versioning. In order to support Continuous Delivery, you must be able to evolve a system in such a way that it's always in a working state.

When other systems depend on a system, it's important to maintain backwards compatibility. Evolve the system, but support legacy dependents as well.

Adding new test code #

In Code That Fits in Your Head, chapter 11 contains a subsection on adding new test code. Despite admonitions to the contrary, I often experience that programmers treat test code as a second-class citizen. They casually change test code in a more undisciplined way than they'd edit production code. Sometimes, that may be in order, but I wanted to show that you can approach the task of editing test code in a more disciplined way. After all, the more you edit tests, the less you can trust them.

After a preliminary discussion about adding entirely new test code, the book goes on to say:

You can also append test cases to a parametrised test. If, for example, you have the test cases shown in listing 11.1, you can add another line of code, as shown in listing 11.2. That’s hardly dangerous.

Listing 11.1 A parametrised test method with three test cases. Listing 11.2 shows the updated code after I added a new test case. (Restaurant/b789ef1/Restaurant.RestApi.Tests/ReservationsTests.cs)

[InlineData(null"j@example.net""Jay Xerxes", 1)]
[InlineData("not a date""w@example.edu""Wk Hd", 8)]
[InlineData("2023-11-30 20:01"null"Thora", 19)]
public async Task PostInvalidReservation(

Listing 11.2 A test method with a new test case appended, compared to listing 11.1. The line added is highlighted. (Restaurant/745dbf5/Restaurant.RestApi.Tests/ReservationsTests.cs)

[InlineData(null"j@example.net""Jay Xerxes", 1)]
[InlineData("not a date""w@example.edu""Wk Hd", 8)]
[InlineData("2023-11-30 20:01"null"Thora", 19)]
[InlineData("2022-01-02 12:10""3@example.org""3 Beard", 0)]
public async Task PostInvalidReservation(

You can also add assertions to existing tests. Listing 11.3 shows a single assertion in a unit test, while listing 11.4 shows the same test after I added two more assertions.

Listing 11.3 A single assertion in a test method. Listing 11.4 shows the updated code after I added more assertions. (Restaurant/36f8e0f/Restaurant.RestApi.Tests/ReservationsTests.cs)


Listing 11.4 Verification phase after I added two more assertions, compared to listing 11.3. The lines added are highlighted. (Restaurant/0ab2792/Restaurant.RestApi.Tests/ReservationsTests.cs)

var content = await response.Content.ReadAsStringAsync();

These two examples are taken from a test case that verifies what happens if you try to overbook the restaurant. In listing 11.3, the test only verifies that the HTTP response is 500 Internal Server Error. The two new assertions verify that the HTTP response includes a clue to what might be wrong, such as the message No tables available.

I often run into programmers who’ve learned that a test method may only contain a single assertion; that having multiple assertions is called Assertion Roulette. I find that too simplistic. You can view appending new assertions as a strengthening of postconditions. With the assertion in listing 11.3 any 500 Internal Server Error response would pass the test. That would include a 'real' error, such as a missing connection string. This could lead to false negatives, since a general error could go unnoticed.

Adding more assertions strengthens the postconditions. Any old 500 Internal Server Error will no longer do. The HTTP response must also come with content, and that content must, at least, contain the string "tables".

This strikes me as reminiscent of the Liskov Substitution Principle. There are many ways to express it, but in one variation, we say that subtypes may weaken preconditions and strengthen postconditions, but not the other way around. You can think of of subtyping as an ordering, and you can think of time in the same way, as illustrated by figure 11.1. Just like a subtype depends on its supertype, a point in time 'depends' on previous points in time. Going forward in time, you’re allowed to strengthen the postconditions of a system, just like a subtype is allowed to strengthen the postcondition of a supertype.

A type hierarchy forms a directed graph, as indicated by the arrow from subtype to supertype. Time, too, forms a directed graph, as indicated by the arrow from t2 to t1. Both present a way to order elements.

Figure 11.1 A type hierarchy forms a directed graph, as indicated by the arrow from subtype to supertype. Time, too, forms a directed graph, as indicated by the arrow from t2 to t1. Both present a way to order elements.

Think of it another way, adding new tests or assertions is fine; deleting tests or assertions would weaken the guarantees of the system. You probably don’t want that; herein lie regression bugs and breaking changes.

The book leaves it there, but I find it worthwhile to expand on that thought.

Function evolution over time #

As in the previous articles about x as a profunctor, let's first view 'a system' as a function. As I've repeatedly suggested, with sufficient imagination, every operation looks like a function. Even an HTTP POST request, as suggested in the above test snippets, can be considered a function, albeit one with the IO effect.

You can envision a function as a pipe. In previous articles, I've drawn horizontal pipes, with data flowing from left to right, but we can also rotate them 90° and place them on a timeline:

Function pipes on a timeline.

As usually depicted in Western culture, time moves from left to right. In a stable system, functions don't change: The function at t1 is equal to the function at t2.

The function in the illustration takes values belonging to the set a as input and returns values belonging to the set b as output. A bit more formally, we can denote the function as having the type a -> b.

We can view the passing of time as a translation of the function a -> b at t1 to a -> b at t2. If we just leave the function alone (as implied by the above figure), it corresponds to mapping the function with the identity function.

Clients that rely on the function are calling it by supplying input values from the set a. In return, they receive values from the set b. As already discussed in the article about Postel's law as a profunctor, we can illustrate such a fit between client and function as snugly fitting pipes:

Function pipes and clients on a timeline.

As long as the clients keep supplying elements from a and expecting elements from b in return, the function remains compatible.

If we have to change the function, which kind of change will preserve compatibility?

We can make the function accept a wider set of input, and let it return narrower set of output:

Function pipe, flanged pipe, and clients on a timeline.

This will not break any existing clients, because they'll keep calling the function with a input and expecting b output values. The drawing is similar to the drawings from the articles on Postel's law as a profunctor and The Liskov Substitution Principle as a profunctor. It seems reasonable to consider backwards compatibility in the same light.

Profunctor #

Consider backwards compatible function evolution as a mapping of a function a -> b at t1 to a' -> b' at t2.

What rules should we institute for this mapping?

In order for this translation to be backwards compatible, we must be able to translate the larger input set a' to a; that is: a' -> a. That's the top flange in the above figure.

Likewise, we must be able to translate the original output set b to the smaller b': b -> b'. That's the bottom nozzle in the above figure.

Thus, armed with the two functions a' -> a and b -> b', we can translate a -> b at t1 to a' -> b' at t2 in a way that preserves backwards compatibility. More formally:

(a' -> a) -> (b -> b') -> (a -> b) -> (a' -> b')

This is exactly the definition of dimap for the Reader profunctor!

Arrow directions #

That's why I wrote as I did in Code That Fits in Your Head. The direction of the arrows in the book's figure 11.1 may seem counter-intuitive, but I had them point in that direction because that's how most readers are used to see supertypes and subtypes depicted.

When thinking of concepts such as Postel's law, it may be more intuitive to think of the profunctor as a mapping from a formal specification a -> b to the more robust implementation a' -> b'. That is, the arrow would point in the other direction.

Likewise, when we think of the Liskov Substitution Principle as rule about how to lawfully derive subtypes from supertypes, again we have a mapping from the supertype a -> b to the subtype a' -> b'. Again, the arrow direction goes from supertype to subtype - that is, in the opposite direction from the book's figure 11.1.

This now also better matches how we intuitively think about time, as flowing from left to right. The arrow, again, goes from t1 to t2.

Most of the time, the function doesn't change as time goes by. This corresponds to the mapping dimap id id - that is, applying the identity function to the mapping.

Implications for tests #

Consider the test snippets shown at the start of the article. When you add test cases to an existing test, you increase the size of the input set. Granted, unit test inputs are only samples of the entire input set, but it's still clear that adding a test case increases the input set. Thus, we can view such an edit as a mapping a -> a', where a ⊂ a'.

Likewise, when you add more assertions to an existing set of assertions, you add extra constraints. Adding an assertion implies that the test must pass all of the previous assertions, as well as the new one. That's a Boolean and, which implies a narrowing of the allowed result set (unless the new assertion is a tautological assertion). Thus, we can view adding an assertion as a mapping b -> b', where b' ⊂ b.

This is why it's okay to add more test cases, and more assertions, to an existing test, whereas you should be weary of the opposite: It may imply (or at least allow) a breaking change.

Conclusion #

As Michael Feathers observed, Postel's law seems universal. That's one way to put it.

Another way to view it is that Postel's law is a formulation of a kind of profunctor. And it certainly seems as though profunctors pop up here, there, and everywhere, once you start looking for that idea.

We can think of the Liskov Substitution Principle as a profunctor, and backwards compatibility as well. It seems reasonable enough: In order to stay backwards compatible, a function can become more tolerant of input, or more conservative in what it returns. Put another way: Contravariant in input, and covariant in output.

The Liskov Substitution Principle as a profunctor

Monday, 06 December 2021 06:52:00 UTC

With a realistic example in C#.

Like the previous article on Postel's law as a profunctor, this article is part of a series titled Some design patterns as universal abstractions. And like the previous article, it's a bit of a stretch to include the present article in that series, since the Liskov Substitution Principle (LSP) isn't a design pattern, but rather a software design principle or heuristic. I still think, however, that the article fits the spirit of the article series, if not the letter.

As was also the case for the previous article, I don't claim that any of this is new. Michael Feathers and Rúnar Bjarnason blazed that trail long before me.

LSP distilled #

In more or less natural language, the LSP states that subtypes must preserve correctness. A subtype isn't allowed to change behaviour in such a way that client code breaks.

Note that subtypes are allowed to change behaviour. That's often the point of subtyping. By providing a subtype, you can change the behaviour of a system. You can, for example, override how messages are sent, so that an SMS becomes a Slack notification or a Tweet.

If client code 'originally' supplied correct input for sending an SMS, this input should also be valid for posting a Tweet.

Specifically (paraphrasing the Wikipedia entry as of early November 2021):

  • Subtypes must be contravariant in input
  • Subtypes must be covariant in output
  • Preconditions must not be strengthened in the subtype
  • Postconditions must not be weakened in the subtype

There's a bit more, but in this article, I'll focus on those rules. The first two we already know. Since any function is already a profunctor, we know that functions are contravariant in input and covariant in output.

The LSP, however, isn't a rule about a single function. Rather, it's a rule about a family of functions. Think about a function a -> b as a pipe. You can replace the pipe segment with another pipe segment that has exactly the same shape, but replacing it with a flanged pipe also works, as long as the input flange is wider, and the nozzle narrower than the original pipe shape.

The Liskov Substitution Principle illustrated with flanged pipes.

On the other hand, if you narrow the pipe at the input and widen it at the output, spillage will happen. That's essentially what the LSP states: The upper, green, flanged pipe is a good replacement for the supertype (the blue pipe in the middle), while the lower, orange, flanged pipe is not useful.

The previous article already described that visual metaphor when it comes to co- and contravariance, so this article will focus on pre- and postconditions. My conjecture is that this is just another take on co- and contravariance.

Supertype example #

When encountering statements about subtypes and supertypes, most people tend to think about object inheritance, but that's just one example. As I've previously outlined, anything that can 'act as' something else is a subtype of that something else. Specifically, an interface implementation is a subtype of the interface, and the interface itself the supertype.

Consider, as a supertype example, this interface from my book Code That Fits in Your Head:

public interface IReservationsRepository
    Task Create(int restaurantId, Reservation reservation);
    Task<IReadOnlyCollection<Reservation>> ReadReservations(
        int restaurantId, DateTime min, DateTime max);
    Task<Reservation?> ReadReservation(int restaurantId, Guid id);
    Task Update(int restaurantId, Reservation reservation);
    Task Delete(int restaurantId, Guid id);

Specifically, in this article, I'll focus exclusively on the ReadReservations method. You can imagine that there's an interface with only that method, or that when subtyping the interface in the following, we vary only that method and keep everything else fixed.

What might be the pre- and postconditions of the ReadReservations method?

ReadReservations preconditions #

The most basic kind of precondition is captured by the parameter types. In order to be able to call the method, you must supply an int and two DateTime instances. You can't omit any of them or replace one of the DateTime values with a Guid. In a statically typed language, this is obvious, and the compiler will take care of that.

Both int and DateTime are value types (structs), so they can't be null. Had one of the parameters been a reference type, it'd be appropriate to consider whether or not null constitutes valid input.

So far, we've only discussed static types. Of course a subtype must satisfy the compiler, but what other pre-conditions might be implied by ReadReservations?

The purpose of the method is to enable client code to query a data store and retrieve the reservations for a particular restaurant and in a particular time interval.

Is any restaurantId acceptable? 1? 0? -235?

It's probably a distraction that the restaurant ID is even an int. After all, you don't add IDs together, or multiply one with another one. That an ID is an integer is really just a leaky implementation detail - databases like it best when IDs are integers. I should actually have defined the restaurant ID as an opaque object with Value Object semantics, but I didn't (like other humans, I'm imperfect and lazy). The bottom line is that any number is as good as any other number. No precondition there.

What about the two DateTime parameters? Are DateTime.MinValue or DateTime.MaxValue valid values? Probably: If you'd like to retrieve all reservations in the past, you could ask for DateTime.MinValue as min and DateTime.Now as max. On the other hand, it'd be reasonable to require that min should be less than (or equal to?) max. That sounds like a proper precondition, and one that's difficult to express as a type.

We may also consider it a precondition that the object that implements the ReadReservations method is properly initialised, but I'll take that as a given. Making sure of that is the responsibility of the constructor, not the ReadReservations method.

To summarise, apart from the types and other things handled by the compiler, there's only one additional pre-condition: that min must be less than max.

Are there any postconditions?

ReadReservations postconditions #

The code base for the book obeys Command-Query Separation. Since ReadReservations returns data, it must be a Query. Thus, we can assume that calling it isn't supposed to change state. Thus, postconditions can only be statements about the return value.

Again, static typing takes care of the most fundamental postconditions. An implementation can't return a double or an UriBuilder. It must be a Task, and that task must compute an IReadOnlyCollection<Reservation>.

Why IReadOnlyCollection, by the way? That's my attempt at describing a particular postcondition as a type.

The IReadOnlyCollection<T> interface is a restriction of IEnumerable<T> that adds a Count. By adding the Count property, the interface strongly suggests that the collection is finite.

IEnumerable<T> implementations can be infinite sequences. These can be useful as functional alternatives to infinite loops, but are clearly not appropriate when retrieving reservations from a database.

The use of IReadOnlyCollection tells us about a postcondition: The collection of reservations is finite. This is, however, captured by the type. Any valid implementation of the interface ought to make that guarantee.

Is there anything else? Is it okay if the collection is empty? Yes, that could easily happen, if you have no reservations in the requested interval.

What else? Not much comes to mind, only that we'd expect the collection to be 'stable'. Technically, you could implement the GetEnumerator method so that it generates Count random Reservation objects every time you enumerate it, but none of the built-in implementations do that; that's quite a low bar, as postconditions go.

To summarise the postconditions: None, apart from a well-behaved implementation of IReadOnlyCollection<Reservation>.

SQL implementation #

According to the LSP, a subtype should be allowed to weaken preconditions. Keep in mind that I consider an interface implementation a subtype, so every implementation of ReadReservations constitutes a subtype. Consider the SQL Server-based implementation:

public async Task<IReadOnlyCollection<Reservation>> ReadReservations(
    int restaurantId,
    DateTime min,
    DateTime max)
    var result = new List<Reservation>();
    using var conn = new SqlConnection(ConnectionString);
    using var cmd = new SqlCommand(readByRangeSql, conn);
    cmd.Parameters.AddWithValue("@RestaurantId", restaurantId);
    cmd.Parameters.AddWithValue("@Min", min);
    cmd.Parameters.AddWithValue("@Max", max);
    await conn.OpenAsync().ConfigureAwait(false);
    using var rdr =
        await cmd.ExecuteReaderAsync().ConfigureAwait(false);
    while (await rdr.ReadAsync().ConfigureAwait(false))
    return result.AsReadOnly();
private const string readByRangeSql = @"
    SELECT [PublicId], [At], [Name], [Email], [Quantity]
    FROM [dbo].[Reservations]
    WHERE [RestaurantId] = @RestaurantId AND
          @Min <= [At] AND [At] <= @Max";

This implementation actually doesn't enforce the precondition that min ought to be less than max. It doesn't have to, since the code will run even if that's not the case - the result set, if min is greater than max, will always be empty.

While perhaps not useful, weakening this precondition doesn't adversely affect well-behaved clients, and buggy clients are always going to receive empty results. If this implementation also fulfils all postconditions, it's already LSP-compliant.

Still, could it weaken preconditions even more, or in a different way?

Weakening of preconditions #

As Postel's law suggests, a method should be liberal in what it accepts. If it understands 'what the caller meant', it should perform the desired operation instead of insisting on the letter of the law.

Imagine that you receive a call where min is midnight June 6 and max is midnight June 5. While wrong, what do you think that the caller 'meant'?

The caller probably wanted to retrieve the reservations for June 5.

You could weaken that precondition by swapping back min and max if you detect that they've been swapped.

Let's assume, for the sake of argument, that we make the above ReadReservations implementation virtual. This enables us to inherit from SqlReservationsRepository and override the method:

public override Task<IReadOnlyCollection<Reservation>> ReadReservations(
    int restaurantId,
    DateTime min,
    DateTime max)
    if (max < min)
        return base.ReadReservations(restaurantId, max, min);
    return base.ReadReservations(restaurantId, min, max);

While this weakens preconditions, it breaks no existing clients because they all 'know' that they must pass the lesser value before the greater value.

Strengthening of postconditions #

Postel's law also suggests that a method should be conservative in what it sends. What could that mean?

In the case of ReadReservations, you may notice that the result set isn't explicitly sorted. Perhaps we'd like to also sort it on the date and time:

public override async Task<IReadOnlyCollection<Reservation>> ReadReservations(
    int restaurantId,
    DateTime min,
    DateTime max)
    var query =
        min < max ?
            base.ReadReservations(restaurantId, min, max) :
            base.ReadReservations(restaurantId, max, min);
    var reservations = await query.ConfigureAwait(false);
    return reservations.OrderBy(r => r.At).ToList();

This implementation retains the weakened precondition from before, but now it also explicitly sorts the reservations on At.

Since no client code relies on sorting, this breaks no existing clients.

While the behaviour changes, it does so in a way that doesn't violate the original contract.

Profunctor #

While we've used terms such as weaken preconditions and strengthen postconditions, doesn't this look an awful lot like co- and contravariance?

I think it does, so let's rewrite the above implementation using the Reader profunctor.

First, we'll need to express the original method in the shape of a function like a -> b - that is: a function that takes a single input and returns a single output. While ReadReservations return a single value (a Task), it takes three input arguments. To make it fit the a -> b mould, we have to convert those three parameters to a Parameter Object.

This enables us to write the original implementation as a function:

Func<QueryParameters, Task<IReadOnlyCollection<Reservation>>> imp =
    q => base.ReadReservations(q.RestaurantId, q.Min, q.Max);

If we didn't want to weaken any preconditions or strengthen any postconditions, we could simply call imp and return its output.

The above weakened precondition can be expressed like this:

Func<QueryParameters, QueryParameters> pre =
    q => q.Min < q.Max ?
        new QueryParameters(q.RestaurantId, q.Min, q.Max) :
        new QueryParameters(q.RestaurantId, q.Max, q.Min);

Notice that this is a function from QueryParameters to QueryParameters. As above, it simply swaps Min and Max if required.

Likewise, we can express the strengthened postcondition as a function:

Func<Task<IReadOnlyCollection<Reservation>>, Task<IReadOnlyCollection<Reservation>>> post =
    t => t.Select(rs => rs.OrderBy(r => r.At).ToReadOnly());

The Select method exists because Task forms an asynchronous functor.

It's now possible to compose imp with pre and post using DiMap:

Func<QueryParameters, Task<IReadOnlyCollection<Reservation>>> composition = imp.DiMap(pre, post);

You can now call composition with the original arguments:

composition(new QueryParameters(restaurantId, min, max))

The output of such a function call is entirely equivalent to the above, subtyped ReadReservations implementation.

In case you've forgotten, the presence of a lawful DiMap function is what makes something a profunctor. We already knew that functions are profunctors, but now we've also seen that we can use this knowledge to weaken preconditions and strengthen postconditions.

It seems reasonable to conjecture that the LSP actually describes a profunctor.

It seems to me that the profunctor composition involved with the LSP always takes the specialised form where, for a function a -> b, the preprocessor (the contravariant mapping) always takes the form a -> a, and the postprocessor (the covariant mapping) always takes the form b -> b. This is because polymorphism must preserve the shape of the original function (a -> b).

Conclusion #

We already know that something contravariant in input and covariant in output is a profunctor candidate, but the Liskov Substitution Principle is usually expressed in terms of pre- and postconditions. Subtypes may weaken preconditions and strengthen postconditions, but not the other way around.

Evidence suggests that you can phrase these rules as a profunctor specialisation.

Next: Backwards compatibility as a profunctor.


It seems to me that the profunctor composition involved with the LSP always takes the specialised form where, for a function a -> b, the preprocessor (the contravariant mapping) always takes the form a -> a, and the postprocessor (the covariant mapping) always takes the form b -> b. This is because polymorphism must preserve the shape of the original function (a -> b).

I think this depends on the language and your perspective on subtype polymorphism. In Java, a subclass Y of a superclass X can override a method m on X that has return type b with a method on Y that has return type c provided c is a subtype of b. To be clear, I am not saying that the instance returned from Y::m can have runtime-type c, though that is also true. I am saying that the compile-time return type of Y::m can be c.

With this in mind, I am willing to argue that the postprocessor can take the form b -> c provided c is a subtype of b. As the programmer, that is how I think of Y::m. And yet, if someone has a instance of Y with compile-time type X, then calling m returns something with compile-time type b by composing my b -> c function with the natural c -> b function from subtype polymorphism to get the b -> b function that you suggested is required.

2021-12-10 05:28 UTC

Tyson, thank you for writing. Yes, true, that nuance may exist. This implies that the LSP is recursive, which really isn't surprising.

A function a -> b may be defined for types a and b, where both are, themselves, polymorphic. So, if a is HttpStyleUriParser and b is MemberInfo, we'd have a function Func<HttpStyleUriParser, MemberInfo> (or HttpStyleUriParser -> MemberInfo if we want to stick with an ML-style type signature - we can imagine that it's an F# function).

If Func<HttpStyleUriParser, MemberInfo> is our polymorphic target signature, according to C# co- and contravariance, we're allowed to vary both input and output. We can, for example, widen the input type to UriParser and restrict the output type to Type. Thus, a specific function of the type Func<UriParser, Type> is compatible with a general function of the type Func<HttpStyleUriParser, MemberInfo>:

Func<HttpStyleUriParser, MemberInfo> general = specific;

Thus, you're correct that both pre- and postprocessor may take the form a' -> a and b -> b', where a' is a supertype of a, and b is a supertype of b'. This is true for C# function delegates, since they're defined with the in and out keywords. You can put in and out on your own interface definitions as well, but most people don't bother (I rarely do, either).

As with all practical recursion you need base cases to stop the recursion. While you can define polymorphic functions (or classes) where input parameters and return types are themselves polymorphic, etc., these should still obey the LSP.

2021-12-14 07:12 UTC

Postel's law as a profunctor

Monday, 29 November 2021 07:10:00 UTC

When viewing inputs and outputs as sets, Postel's law looks like a profunctor.

This article is part of a series titled Some design patterns as universal abstractions. Including the present article in that series is a bit of a stretch, since Postel's law isn't really a design pattern, but rather a software design principle or heuristic. I still think, however, that the article fits the spirit of the article series, if not the letter.

This article is heavily inspired by Michael Feathers' article The Universality of Postel's Law, in which he writes:

[Postel's law] has been paraphrased over the years as “Be liberal in what you accept, and conservative in what you send” and for people who are mathematically inclined: “be contravariant in your inputs and covariant in your outputs.”

A thing contravariant in input and covariant in output sounds like a profunctor, but why does Michael Feathers write that about Postel's law?

In this article, I'll try to explain.

Perfect fit #

Postel's law is a statement about functions, methods, procedures, or whatever else you'd like to call them. As I've previously outlined, with sufficient squinting, we can think about methods and other operations as functions, so in this article I'll focus on functions.

Functions don't stand alone. Functions have callers. Some other entity, usually client code, passes some input data to the function, which then performs its work and returns output data. When viewed with a set-based perspective, we can depict a function as a pipe:

Horizontal pipe.

Client code often use the output of one function as input for another:

Int3 isEven = EncodeEven(number);
Int3 decremented = Decrement(isEven);

Even though this code example uses an explicit intermediary variable (isEven), it's equivalent to function composition:

var composition = EncodeEven.Compose(Decrement);

where Compose can be implemented as:

public static Func<A, C> Compose<ABC>(this Func<A, B> f, Func<B, C> g)
    return x => g(f(x));

Such a composition we can depict by appending one pipe after another:

Two horizontal pipes composed one after the other.

This works, but is brittle. It's a close fit. The output set of the first function has to exactly fit the input set of the second function. What happens if the pipes don't perfectly align?

Misalignment #

Functions compose when they fit perfectly, but in the real world, that's rarely the case. For example, it may turn out that Decrement is defined like this:

static Int3 Decrement(Int3 i)
    if (i == 0)
        throw new ArgumentOutOfRangeException(
            "Can't decrement 0.");
    return i - (Int3)1;

This function is undefined for 0. If we wanted to peek at the set diagram 'inside' the pipe, we might depict the function like this:

Decrement function set diagram.

In a sense, it's still a mapping from our hypothetical 3-bit integer to 3-bit integer, but it's a partial function.

Another way to depict the mapping, however, is to constrain the domain to [1..7], and narrow the codomain to the function's image, producing a bijection:

Total decrement function set diagram.

Such sets are a little harder to express in code, because how do you represent a set with seven elements? Often, you'd stick with an implementation like the above Decrement function.

This turns out to be unfortunate, however, because EncodeEven is defined like this:

static Int3 EncodeEven(Int3 i)
    return i.IsEven ? (Int3)1 : (Int3)0;

As a set diagram, we might depict it like this:

Set diagram for the EncodeEven function.

It turns out that half the inputs into the above composition don't work! It's almost as though the pipes are misaligned:

Misaligned pipes.

This can easily happen, also in the real world:

Misaligned drain pipes.

This is also why Michael Feathers writes:

We can see Postel in the physical world too. Every time you see a PVC pipe with a flanged end, you’re seeing something that serves as a decent visual metaphor for Postel’s Law. Those pipes fit well together because one end is more accepting.

In other words, there's nothing new in any of the above. I've just been supplying the illustrations.

Flanges #

How should we interpret the idea of flanges? How do we illustrate them? Here's a way:

Flanged pipe.

Given our set-based interpretation of things, how should we interpret a flange? Let's isolate one of them. It doesn't matter which one, but lets consider the left flange. If we attempt to make it transparent, we could also draw it like this:

Transparent flange.

What does that look like? It looks like a mapping from one set to another.

The left-hand set is slightly larger than the right-hand set, but the illustration includes neither the elements of each set nor the arrows that connect them.

If we think of the 'original' function as a function from the set A to the set B we can also write it in pseudo-code as A -> B. In Haskell you'd exactly write A -> B if A and B were two concrete types. Polymorphically, though, you'd write any function as a -> b, or in C# as Func<A, B>.

Let's think of any function a -> b as the 'perfect fit' case. While such a function composes with, say, a function b -> c, the composition is brittle. It can easily become misaligned.

How do we add flanges to the function a -> b?

As the above illustration of the flange implies, we can think of the flange as another function. Perhaps we should call the slightly larger set to the left a+ (since it's 'like' a, just larger - that is, more liberal). With that nomenclature, the flange would be a function a+ -> a.

Likewise, the right flange would be a function b -> b-. Here, I've called the narrower set of the flange b- because it's smaller (more conservative) than b.

Thus, the flanged pipe is just the composition of these three functions: a+ -> a, a -> b, and b -> b-:

Flange composition.

That's exactly how dimap is defined in Haskell:

dimap ab cd bc = cd . bc . ab

The implementation code uses other letters, and recall that Haskell is typically read from right to left. As its name implies, ab is a function a -> b, bc is a function b -> c, and cd is a function c -> d.

In other words, Postel's law is a description of the Reader profunctor, or, as Michael Feathers put it: Be contravariant in your inputs and covariant in your outputs.

Conclusion #

Postel's law is a useful design principle to keep in mind. Intuitively, it makes sense to think of it as making sure that pipes are flanged. The bigger the receiving flange is, and the smaller the nozzle is, the easier it is to compose the flanged pipe with other (flanged) pipes.

Using mostly visual metaphor, this article demonstrates that this is equivalent with being contravariant in input and covariant in output, and thus that the principle describes a profunctor.

Postel's law, however, isn't the only design principle describing a profunctor.

Next: The Liskov Substitution Principle as a profunctor.

Functions as pipes

Monday, 22 November 2021 06:30:00 UTC

A visual metaphor.

A recent article on types as sets briefly touched on functions as sets. For example, you can think of Boolean negation as a set of two arrows:

Boolean negation set diagram.

Here the arrows stay within the set, because the function is a function from the set of Boolean values to the set of Boolean values.

In general, however, functions aren't always endomorphisms. They often map one set of values to another set. How can we think of this as sets?

As was also the case with the article on types as sets, I'd like to point out that this article isn't necessarily mathematically rigorous. I'm neither a computer scientist nor a mathematician, and while I try to be as correct as possible, some hand-waving may occur. My purpose with this article isn't to prove a mathematical theorem, but rather to suggest what I, myself, find to be a useful metaphor.

Boolean negation visualised with domain and codomain #

Instead of visualising Boolean negation within a single set, we can also visualise it as a mapping of elements from an input set (the domain) to an output set (the codomain):

Boolean negation illustrated as arrows from the set on the left to the set on the right.

In this figure, the domain is to the left and the codomain is on the right.

How do we visualise more complex functions? What if the domain isn't the same as the codomain?

Boolean and #

Let's proceed slowly. Let's consider Boolean and next. While this function still involves only Boolean values, it combines two Boolean values into a single Boolean value. It'd seem, then, that in order to visualise this mapping, we'd need two sets to the left, and one to the right. But perhaps a better option is to visualise the domain as pairs of Boolean values:

Boolean and visualised as arrows between sets.

To the left, we have four pairs that each map to the Boolean values on the right.

Even numbers #

Perhaps using only Boolean values is misleading. Even when dealing with pairs, the above example may fail to illustrate that we can think of any function as a mapping from a domain to a codomain.

Imagine that there's such a thing as a 3-bit number. Such a data structure would be able to represent eight different numbers. To be clear, I'm only 'inventing' such a thing as a 3-bit number because it's still manageable to draw a set of eight elements.

How would we illustrate a function that returns true if the input is an even 3-bit number, and false otherwise? We could make a diagram like this:

Set diagram of a function to determine whether a 3-bit number is even.

On the left-hand side, I've only labelled two of the numbers, but I'm sure you can imagine that the rest of the elements are ordered from top to bottom: 0, 1, 2, 3, 4, 5, 6, 7. To the right, the two elements are true (top) and false (bottom).

Given this illustration, I'm sure you can extrapolate to 32-bit integers and so on. It's impractical to draw, but the concept is the same.

Encoding #

So far, we've only looked at functions where the codomain is the set of Boolean values. How does it look with other codomains?

We could, for example, imagine a function that 'encodes' Boolean values as 3-bit numbers, false corresponding (arbitrarily) to 5 and true to 2. The diagram for that function looks like this:

Set diagram of encoding Boolean values as 3-bit numbers.

Now the set of Boolean values is on the left, with true on top.

A function as a pipe #

In all examples, we have the domain to the left and the codomain on the right, connected with arrows. Sometimes, however, we may want to think about a function as just a single thing, without all the details. After all, the diagrams in this article are simple only because the examples are toy examples. Even a simple function like this one would require a huge diagram:

public static bool IsPrime(int candidate)

An input type of int corresponds to a set with 4,294,967,296 elements. That's a big set to draw, and a lot of arrows, too!

So perhaps, instead, we'd like to have a way to illustrate a function without all the details, yet still retaining this set-based way of thinking.

Let's return to one of the earlier examples, but add a shape around it all:

Set diagram of even function enclosed in pipe.

This seems like a fair addition to make. It starts to look like the function is enclosed in a pipe. Let's make the pipe opaque:

Horizontal pipe.

In architecture diagrams, a horizontal pipe is a common way to illustrate that some sort of data processing takes place, so this figure should hardly be surprising.

Composition #

You may say that I've been cheating. After all, in a figure like the one that illustrates an isEven function, the domain is larger than the codomain, yet I've kept both ovals of the same size. Wouldn't the following be a fairer depiction?

Set diagram where the codomain is drawn smaller.

If we try to enclose this diagram in an opaque pipe, it'd look like this:

Horizontal cone.

This mostly looks like a (bad) perspective drawing of a pipe, but it does start to suggest how functions fit together. For example, the output of this isEven function is the Boolean set, which is also the input of, for example the Boolean negation function (! or not). This means that if the shapes fit together, we can compose the pipes:

Horizontal cone composed with horizontal pipe.

Continuing this line of thinking, we can keep on composing the shapes as long as the output fits the input of the next function. For example, the output of Boolean negation is still the Boolean set, which is also the domain of the above 'encoding' function:

Narrowing cone composed with pipe and widening cone.

We can even, if we'd like to peek into the composition, make the pipes transparent again, to illustrate what's going on:

Transparent narrowing cone composed with transparent pipe and transparent widening cone.

As long as the right-hand side of one pipe fits the left-hand side of another pipe, it indicates that you can compose these two functions.

Haskell translation #

For completeness' sake, let's try to express these three functions, as well as their composition, in a programming language. Since Haskell already comes with a composition operator (.), it fits nicely. It also already comes with two of the three functions we'll need:

even :: Integral a => a -> Bool
not :: Bool -> Bool

Thanks to Haskell's type-class feature, even works for any Integral instance, so if we imagine that our hypothetical 3-bit number is an Integral instance, it'll work for that type of input as well.

The remaining function is trivial to implement:

encode :: Num a => Bool -> a
encode  True = 2
encode False = 5

Since Integral is a supertype of Num, if our 3-bit number is an Integral instance, it's also a Num instance.

The composition implied by the above figure is this:

composition :: Integral a => a -> a
composition = encode . not . even

Haskell is typically read from right to left, so this composition starts with even, continues with not, and concludes with encode.

Let's call it:

> composition 3
> composition 4

composition 3 first passes through even, which returns False. False then passes through not, which returns True. Finally, True passes through encode, which returns 2.

I'll leave the exegesis of composition 4 as an exercise for the reader.

C# translation #

In C#, imagine that an Int3 data type exists. You can now define the three functions like this:

Func<Int3, boolisEven = number => number.IsEven;
Func<boolboolnot = b => !b;
Func<bool, Int3> encode = b => b ? (Int3)2 : (Int3)5;

Given a Compose extension method on Func<A, B>, you can now compose the functions like this:

Func<Int3, Int3> composition = isEven.Compose(not).Compose(encode);

This composition works just like the above Haskell example, and produces the same results.

Conclusion #

A function takes input and returns output. Even if the function takes multiple arguments, we can think of an argument as a single object: a tuple or Parameter Object.

Thus, we can think of a function as a mapping from one set to another. While we can illustrate a specific mapping (such as even, not, and encode), it's often useful to think of a function as a single abstract thing. When we think of functions as mappings from sets to sets, it makes intuitive sense to visualise the abstraction as a pipe.

This visual metaphor works for object-oriented programming as well. With sufficient mental gymnastics, functions are isomorphic to methods, so the pipe metaphor works beyond pure functions.

Types as sets

Monday, 15 November 2021 06:37:00 UTC

To a certain degree, you can think of static types as descriptions of sets.

If you've ever worked with C#, Java, F#, Haskell, or other compiled languages, you've encountered static types in programming. In school, you've probably encountered basic set theory. The two relate to each other in illuminating ways.

To be clear, I'm neither a mathematician nor a computer scientist, so I'm only basing this article on my layman's understanding of these topics. Still, I find some of the correspondences to be useful when thinking about certain programming topics.

Two elements #

What's the simplest possible set? For various definitions of simple, probably the empty set. After that? The singleton set. We'll skip these, and instead start with a set with two elements:

Set with two elements.

If you had to represent such a set in code, how would you do it?

First, you'd have to figure out how to distinguish the two elements from each other. Giving each a label seems appropriate. What do you call them? Yin and yang? Church and state? Alice and Bob?

And then, how do you represent the labels in code, keeping in mind that they somehow must 'belong to the same set'? Perhaps as an enum?

public enum Dualism

As a data definition, though, an enum is a poor choise because the underlying data type is an integer (int by default). Thus, a method like this compiles and executes just fine:

public Dualism YinOrYang()
    return (Dualism)42;

The Dualism returned by the function is neither Yin nor Yang.

So, how do you represent a set with two elements in code? One option would be to Church encode it (try it! It's a good exercise), but perhaps you find something like the following simpler:

public sealed class Dualism
    public static readonly Dualism  Yin = new Dualism(false);
    public static readonly Dualism Yang = new Dualism( true);
    public Dualism(bool isYang)
        IsYang = isYang;
    public bool IsYang { get; }
    public bool IsYin => !IsYang;

With this design, a method like YinOrYang can't cheat, but must return either Yin or Yang:

public Dualism YinOrYang()
    return Dualism.Yin;

Notice that this variation is based entirely off a single member: IsYang, which is a Boolean value.

In fact, this implementation is isomorphic to bool. You can create a Dualism instance from a bool, and you can convert that instance back to a Boolean value via IsYang, without loss of information.

This holds for any two-element set: it's isomorphic to bool. You could say that the data type bool is equivalent to 'the' two-element set.

More, but still few, elements #

Is there a data type that corresponds to a three-element set? Again, you can always use Church encoding to describe a data type with three cases, but in C#, the easiest backing type would probably be bool? (Nullable<bool>). When viewed as a set, it's a set inhabited by the three values false, true, and null.

How about a set with four elements? A pair of Boolean values seems appropriate:

public sealed class Direction
    public readonly static Direction North = new Direction(falsefalse);
    public readonly static Direction South = new Direction(false,  true);
    public readonly static Direction  East = new Direction( truefalse);
    public readonly static Direction  West = new Direction( true,  true);
    public Direction(bool isEastOrWestbool isLatter)
        IsEastOrWest = isEastOrWest;
        IsLatter = isLatter;
    public bool IsEastOrWest { get; }
    public bool IsLatter { get; }

The Direction class is backed by two Boolean values. We can say that a four-element set is isomorphic to a pair of Boolean values.

How about a five-element set? If a four-element set corresponds to a pair of Boolean values, then perhaps a pair of one Boolean value and one Nullable<bool>?

Alas, that doesn't work. When combining types, the number of possible combinations is the product, not the sum, of individual types. So, a pair of bool and bool? would support 2 × 3 = 6 combinations: (false, null), (false, false), (false, true), (true, null), (true, false), and (true, true).

Again, a Church encoding (try it!) is an option, but you could also do something like this:

public sealed class Spice
    public readonly static Spice   Posh = new Spice(0);
    public readonly static Spice  Scary = new Spice(1);
    public readonly static Spice   Baby = new Spice(2);
    public readonly static Spice Sporty = new Spice(3);
    public readonly static Spice Ginger = new Spice(4);
    private readonly int id;
    private Spice(int id)
        this.id = id;
    public override bool Equals(object obj)
        return obj is Spice spice &&
               id == spice.id;
    public override int GetHashCode()
        return HashCode.Combine(id);

This seems like cheating, since it uses a much larger underlying data type (int), but it still captures the essence of a set with five elements. You can't add more elements, or initialise the class with an out-of-range id since the constructor is private.

The point isn't so much how to implement particular classes, but rather that if you can enumerate all possible values of a type, you can map a type to a set, and vice versa.

More elements #

Which type corresponds to this set?

A 256-element set.

I hope that by now, it's clear that this set could correspond to infinitely many types. It's really only a matter of what we call the elements.

If we call the first element 0, the next one 1, and then 2, 3, and so on up to 255, the set corresponds to an unsigned byte. If we call the elements -128, -127, -126 up to 127, the set corresponds to a signed byte. They are, however, isomorphic.

Likewise, a set with 65,536 elements corresponds to a 16-bit integer, and so on. This also holds for 32-bit integers, 64-bit integers, and even floating point types like float and double. These sets are just too large to draw.

Infinite sets #

While most languages have built-in number types based on a fixed number of bytes, some languages also come with number types that support arbitrarily large numbers. .NET has BigInteger, Haskell comes with Integer, and so on. These numbers aren't truly infinite, but are limited by machine capacity rather than the data structure used to represent them in memory.

Another example of a type with arbitrary size is the ubiquitous string. There's no strict upper limit to how large strings you can create, although, again, your machine will ultimately run out of memory or disk space.

Theoretically, there are infinitely many strings, so, like BigInteger, string corresponds to an infinite set. This also implies that string and BigInteger are isomorphic, but that shouldn't really be that surprising, since everything that's happening on a computer is already encoded as (binary) numbers - including strings.

Any class that contains a string field is therefore also isomorphic to an (or the?) infinite set. Two string fields also correspond to infinity, as does a string field paired with a bool field, and so on. As soon as you have just one 'infinite type', the corresponding set is infinite.

Constrained types #

How about static types (classes) that use one or more built-in types as backing fields, but on top of that impose 'business rules'?

This one, for example, uses a byte as a backing field, but prohibits some values:

public sealed class DesByte
    private readonly byte b;
    public DesByte(byte b)
        if (12 <= b && b <= 19)
            throw new ArgumentOutOfRangeException(nameof(b), "[12, 19] not allowed.");
        this.b = b;

While this class doesn't correspond to the above 256-element set, you can still enumerate all possible values:

A 248-element set.

But then what about a class like this one?

public sealed class Gadsby
    private readonly string manuscript;
    public Gadsby(string manuscript)
        if (manuscript.Contains('e', StringComparison.OrdinalIgnoreCase))
            throw new ArgumentException(
                "The manuscript may not contain the letter 'e'.",
        this.manuscript = manuscript;

While the constructor prohibits any string that contains the letter e, you can still create infinitely many string values even with that constraint.

You can, however, still conceivably enumerate all possible Gadsby values, although the corresponding set would be infinitely large.

Obviously, this isn't practical, but the point isn't one of practicality. The point is that you can think of types as sets.

Function types #

So far we've only covered 'values', even though it's trivial to create types that correspond to infinitely large sets.

In type systems, functions also have types. In Haskell, for example, the type Int -> Bool indicates a function that takes an Int as input and return a Bool. This might for example be a function that checks whether the number is even.

Likewise, we can write the type of not (Boolean negation) as Bool -> Bool. In a set diagram, we can illustrate it like this:

Boolean negation set diagram.

Each element points to the other one: true points to false, and vice versa. Together, the two arrows completely describe the not function (the ! operator in C#).

The two arrows describing Boolean negation themselves form a set.

If we remove the original elements (true and false) of the set, it becomes clearer that these two arrows also form a set.

In general, we can think of functions as sets of arrows, but still sets. Many of these sets are infinite.

Conclusion #

Set theory is a branch of mathematics, and so is type theory. Having no formal education in either, I don't claim that types and sets are the same. A quick web search implies that while there are many similarities between types and sets, there are also differences. In this article, I've highlighted some similarities.

Thinking about types as sets can be helpful in everyday programming. In test-driven development, for example, equivalence partitioning provides insights into which test inputs to use. Being able to consider a system under test's inputs as sets (rather than types) makes this easier.

As future articles will cover, it also becomes easier to think about Postel's law and the Liskov substitution principle.


Any class that contains a string field is therefore also isomorphic to an (or the?) infinite set.

It is isomorphic to an infinite set. More generally, any class that contains a field of type T is isomorphic to a set with cardinality at least at big as the cardinality of the inhabitants of T. There are infinite sets with different cardinalities, and string is isomorphic to the smallest of these, which is known as countably infinite or Aleph-nought. Any class that contains a field of type string -> bool is isomorphic to a set with cardinality at least Aleph-one since that function type is isomorphic to a set with that cardinality.

2021-11-15 14:39 UTC

Tyson, thank you for writing. I wasn't kidding when I wrote that I'm not a mathematician. Given, however, that I've read and more or less understood The Annotated Turing, I should have known better. That book begins with a lucid explanation of Cantor's theorem.

Does this practically impact the substance of the present article?

2021-11-16 06:51 UTC
Does this practically impact the substance of the present article?

Nope. I think the article is good. I think everything you said is correct. I just wanted to elaborate a bit at the one point where you conveyed some hesitation.

2021-11-20 15:52 UTC

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