Planet Haskell

November 07, 2024

Donnacha Oisín Kidney

POPL Paper—Algebraic Effects Meet Hoare Logic in Cubical Agda

Posted on November 7, 2024
Tags:

New paper: “Algebraic Effects Meet Hoare Logic in Cubical Agda”, by myself, Zhixuan Yang, and Nicolas Wu, will be published at POPL 2024.

Zhixuan has a nice summary of it here.

The preprint is available here.

by Donnacha Oisín Kidney at November 07, 2024 12:00 AM

March 18, 2024

Haskell Interlude

45: András Kovács

In this episode, András Kovács is being interviewed by Andres Löh and Matthias Pall Gissurarson. We learn how to go from economics to functional programming, how GHC's runtime system is superior to Rust's, the importance of looking at GHC's Core for spotting stray closures, and why staging might be the answer to all your optimisation problems.

March 18, 2024 08:00 AM

Michael Snoyman

How I Stay Organized

When I describe the Yesod web framework, one of the terms I use is the boundary issue. Internally, I view Yesod as an organized, structured, strongly typed ecosystem. But externally, it's dealing with all the chaos of network traffic. For example, within Yesod, we have clear typing delineations between normal strings, HTML, and raw binary data. But the network layer simply throws around bytes for all three. The boundary issue in Yesod is the idea that, before chaotic, untyped, unorganized data enters the system, it has to be cleaned, sanitized, typed, and then ingested.

This represents my overall organizational system too. I've taken a lot of inspiration from existing approaches, notably Getting Things Done and Inbox Zero. But I don't follow any such philosophy dogmatically. If your goal in reading this blog post is to get organized, I'd recommend reading this, searching for articles on organization, and then determining how you'd like to organize your life.

The process

I like to think of chaotic versus ordered systems. Chaotic systems are sources of stuff: ideas, work items, etc. There are some obvious chaotic sources:

  • Mobile app notifications

  • Incoming emails

  • Phone calls

  • Signal/WhatsApp messages

I think most of us consider these kinds of external interruptions to be chaotic. It doesn't matter what you're in the middle of, the interruption happens and you have to choose how to deal with it. (Note: that may include ignoring it, or putting notifications on silent.)

However, there's another source of chaos, arguably more important than the above: yourself. When I'm sitting working on some code and a thought comes up, it's an internally-driven interruption, and often harder to shake than something external.

Taking heavy inspiration from Getting Things Done, my process is simple for this: record the idea and move on. There are of course caveats to that. If I think of something that demands urgent attention (e.g., "oh shoot I left the food on the stove") chaos will reign. But most of the time, I'm either working on something else, taking a shower, or kicking back reading a book when one of these ideas comes up. The goal is to get the idea into one of the ordered systems so I can let go of it and get back to what I was doing.

For me, my ordered systems are basically my calendar, my todo list, and various reminders from the tools that I use. I'll get into the details of that below.

Other people

How do you treat other people in a system like this? While I think in reality there's a spectrum, we can talk about the extremes:

  • Chaotic people: these are people who don't follow your rules for organization, and will end up randomizing you. This could be a demanding boss, a petulant child, or a telemarketer trying to sell you chaos insurance (I'm sure that's a thing). In these cases, I treat the incoming messages with chaos mode: jot down all work items/ideas, or simply handle them immediately.

  • Ordered people: these are people you can rely on to participate in your system. In an ideal world, this would include your coworkers, close friends and family, etc. With these people, you can trust that "they have the ball" is equivalent to writing down the reminders in your ordered systems.

That's a bit abstract, so let's get concrete. Imagine I'm on a call with a few other developers and we're dividing up the work on the next feature we're implementing. Alice takes work item A, Bob takes work item B, etc. Alice is highly organized, so I rely on her to record the work somewhere (personal todo list, team tracker, Jira... somewhere). But suppose Bob is... less organized. I'd probably either create the Jira issue for Bob and assign it to him, or put a reminder in my own personal systems to follow up and confirm that Bob actually recorded this.

You may think that this kind of redundancy is going overboard. However, I've had to use this technique often to keep projects moving forward. I try as much as possible to encourage others to follow these kinds of organized systems. Project management is, to a large extent, trying to achieve the same goal. But it's important to be honest about other people's capabilities and not rely on them being more organized than they're capable of.

As mentioned, no one is 100% on either the order or chaos side. Even the most chaotic person will often remember to follow up on the most important actions, and even the most ordered will lose track of things from time to time.

Tooling

Once you have the basic system in mind for organizing things, you need to choose appropriate tooling to make it happen. "Tooling" here could be as simple as a paper-and-pen you carry around and write everything down. However, given how bad my handwriting is and the fact that I'm perpetually connected to an electronic device of some kind, I prefer the digital approach.

My tooling choices for organization come down to the following:

Todoist

I use Todoist as my primary todo list application. I've been very happy with it, and the ability to have shared projects has been invaluable. My wife (Miriam, aka LambdaMom) and I use a shared Todoist project for managing topics like purchases for the house, picking up medicines at the pharmacy, filing taxes, etc. And yes, having my spouse be part of the "ordered world" is a wonderful thing. We've given the advice of shared todo lists to many of our friends.

One recommendation if you have a large number of tasks scheduled each day: leverage your todo app's mechanisms for setting priorities and times of day for performing a task. When you have 30 items to cover in a day, including things like "take allergy medicine in the afternoon" and similar, it's easy to miss urgent items. In Todoist, I regularly use the priority feature to push work items to the top.

Calendars

While todo lists track work items and deliverables, calendars track specific times when actions need to be taken: show up to a meeting, go to the doctor, etc. I don't think anyone's too surprised by the idea of using a calendar to stay organized.

Email

Email is another classic organization method. Email is actually a much better ordered system than many other forms of communication, since it has:

  • Unread: things that need to be processed and organized

  • Read in inbox: things that have gone through initial processing but require more work

  • Snooze: for me a killer feature. Plenty of emails do not require immediate attention. In the past I used to create Todoist items for following up on emails that needed more work. But snoozing email is now a common feature in almost every mail system I use, and I rely on it heavily.

Other chat apps

But most communication these days is not happening in email. We have work-oriented chat (like Slack) and personal chat applications (Signal, WhatsApp, etc). My approach to these is:

  • If the app provides a "remind me later" feature, I use it to follow up on things later.

  • If the app doesn't provide such a feature, I add a reminder to Todoist.

Technically I could use "mark as unread" in many cases too. However, I prefer not doing that. You may have noticed that, with the approaches above, you'll very quickly get to 0 active notifications in your apps: no emails waiting to be processed, no messages waiting for a response. You'll have snoozed emails pop up in the future, "remind me later" messages that pop up, and an organized todo list with all the things you need to follow up on.

Notifications and interruptions

This is an area I personally struggle in. Notifications from apps are interruptions, and with the methods above I'm generally able to minimize the impact of an interruption. However, minimizing isn't eliminating: there's still a context switch. Overall, there are two main approaches you can take:

  • Receive all notifications and interruptions and always process them. This makes sure you aren't missing something important and aren't blocking others.

  • Disable notifications while you're in "deep work" and check in occasionally. This allows better work time, but may end up dropping the ball on something important.

For myself, which mode I operate in depends largely on my role. When I'm working as an individual contributor on a codebase, it's less vital to respond immediately, and I may temporarily disable notifications. When I'm leading a project, I try to stay available to answer things immediately to avoid blocking people.

My recommendation here is:

  • Establish some guidelines with the rest of your team about different signaling mechanisms to distinguish between "please answer at some point when you have a chance" and "urgent top priority please answer right now." This can be separate groups/channels with different notification settings, a rule that urgent topics require a phone call, or anything else.

  • Try to use tools that are optimized for avoiding distractions. I've been particularly enamored with Twist recently, which I think nails a sweet spot for this. I'm hoping to follow up with a blog post on team communication tools. (That's actually what originally inspired me to write this post.)

Work organization

I've focused here on personal organization, and the tools I use for that. Organizing things at work falls into similar paradigms. Instead of an individual todo list, at work we'll use project management systems. Instead of tracking messages in WhatsApp, at work it might be Teams. For the most part, the same techniques transfer over directly to the work tools.

One small recommendation: don't overthink the combining/separating of items between work and personal. I went through a period trying to keep the two completely separate, and I've gone through periods of trying to combine it all together. At this point, I simply use whatever tool seems best at the time. That could be a Jira issue, or a Todoist item, or even "remind me later" on a Slack message.

As long as the item is saved and will come up later in a reasonable timeframe, consider the item handled for now, and rely on the fact that it will pop back up (in sprint planning, your daily todo list review, or a notification from Slack) when you need to work on it.

Emotions

A bit of a word of warning for people who really get into organization. It's possible to take things too far, and relate to all impediments to your beautifully organized life as interruptions/distractions/bad things. Sometimes it's completely legitimate to respond with frustration: getting an email from your boss telling you that requirements on a project changed is difficult to deal with, regardless of your organizational system. Having a telemarketer call in the middle of dinner is always unwanted.

But taken too far, a system like this can lead you to interpreting all external interruptions as negative. And it can allow you to get overly upset by people who are disrupting your system by introducing more chaos. Try to avoid letting defense of the system become a new source of stress.

Also, remember that ultimately you are the arbiter of what you will do. Just because someone has sent you an email asking for something doesn't mean you're obligated to create a todo item and follow up. You're free to say no, or (to whatever extent it's appropriate, polite, and professional) simply ignore such requests. You control your life, not your todo program, your inbox, or anyone who knows how to ask for something.

My recommendation: try to remember that this system isn't a goal unto itself. You're trying to make your life better by organizing things. You expect that you won't hit 100%, and that others will not be following the same model. Avoiding the fixation on perfection can make all the difference.

Further reading

For now, I'm just including one "further reading" link. Overall, I really like Todoist as an app, but appreciate even more the thought they put into how the app would tie into a real organizational system. This guide is a good example:

Beyond that, I'd recommend looking up getting things done and inbox zero as search terms. And as I find other articles (or people put them in the comments), I'll consider expanding the list.

March 18, 2024 12:00 AM

March 17, 2024

Oleg Grenrus

ST with an early exit

Posted on 2024-03-17 by Oleg Grenrus

Implementation

I wish there were an early exit functionality in the ST monad. This need comes time to time when writing imperative algorithms in Haskell.

It's very likely there is a functional version of an algorithm, but it might be that ST-version is just simply faster, e.g. by avoiding allocations (as allocating even short lived garbage is not free).

But there are no early exit in the ST monad.

Recent GHC added delimited continuations. The TL;DR is that delimited continuations is somewhat like goto:

  • newPromptTag# creates a label (tag)
  • prompt# brackets the computation
  • control# kind of jumps (goes to) the end of enclosing prompt bracket, and continues from there.

So let's use this functionality to implement a version of ST which has an early exit. It turns out to be quite simple.

The ST monad is define like:

newtype ST s a = ST (State# s -> (# State# s, a #)

and we change it by adding an additional prompt tag argument:

newtype EST e s a = EST
    { unEST :: forall r. PromptTag# (Either e r)
            -> State# s -> (# State# s, a #) 
    }

(Why forall r.? We'll see soon).

It's easy to lift normal ST computations into EST ones:

liftST :: ST s a -> EST e s a
liftST (ST f) = EST (\_ -> f)

so EST is a generalisation of ST, good.

Now we need a way to run EST computations, and also a way to early exit in them.

The early exit is the simpler one. Given that tag prompt brackets the whole computation, we simply jump to the end with Left e. We ignore the captured continuation, we have no use for it.

earlyExitEST :: e -> EST e s any
earlyExitEST e = EST (\tag -> control0## tag (\_k s -> (# s, Left e #)))

Now, the job for runEST is to create the tag and prompt the computation:

runEST :: forall e a. (forall s. EST e s a) -> Either e a
runEST (EST f) = runRW#
    -- create tag
    (\s0 -> case newPromptTag# s0 of {
    -- prompt
    (# s1, tag #) -> case prompt# tag
         -- run the `f` inside prompt,
         -- and once we get to the end return `Right` value
         (\s2 -> case f tag s2 of (# s3, a #) -> (# s3, Right a #)) s1 of {
    (# _, a #) -> a }})

runRW# and forgetting the state at the end is the same as in runST, for comparison:

runST :: (forall s. ST s a) -> a
runST (ST st_rep) = case runRW# st_rep of (# _, a #) -> a
-- See Note [runRW magic] in GHC.CoreToStg.Prep

With all the pieces in place, we can run few simple examples:

-- | >>> ex1
-- Left 'x'
ex1 :: Either Char Bool
ex1 = runEST $ earlyExitEST 'x'

-- | >>> ex2
-- Right True
ex2 :: Either Char Bool
ex2 = runEST (return True)

Comments & wrinkles

Early exit is one of the simplest "effect" you can implement with delimited continuations. This is the throwing part of the exceptions, with only top-level exception handler. It's a nice exercise (and a brain twister) to implement catch blocks.

One wrinkle in this implementation is the control0## (not control0#) function I used. The delimited continuations primops are made to work only with RealWorld, not arbitrary State# tokens.

I think this is unnecessary specialization GHC issue #24165, I was advice to simply use unsafeIOToST, so I did:

control0##
    :: PromptTag# a
    -> (((State# s -> (# State# s, b #)) -> State# s -> (# State# s, a #))
                                         -> State# s -> (# State# s, a #))
    -> State# s -> (# State# s, b #)
control0## = unsafeCoerce# control0#

This still feels silly, especially realizing that the (only) example in the delimited continuations proposal goes like

type role CC nominal representational
newtype CC ans a = CC (State# RealWorld -> (# State# RealWorld, a #))
  deriving (Functor, Applicative, Monad) via IO

runCC :: (forall ans. CC ans a) -> a
runCC (CC m) = case runRW# m of (# _, a #) -> a

but if you look at that, it's just a ST monad done weirdly:

newtype ST s a = ST (State# RealWorld -> (# State# RealWorld, a #))
-- not using `s` argument !?

There might be a good reason why CC should be done like that (other than than primops are RealWorld specific), but the proposal doesn't explain that difference. To me having phantom ans instead of using nominally it as in ST is suspicious.

Conclusion

Delimited continutations are fun and could be very useful.

But surprisingly, at the moment of writing I cannot find any package on Hackage using them for anything! Search for newPromptTag returns only false positives (ghc-lib etc) right now. I wonder why they are unused?

Please try them out!

March 17, 2024 12:00 AM

March 14, 2024

Magnus Therning

Hackage revisions in Nix

Today I got very confused when using callHackageDirect to add the openapi3 package gave me errors like this

> Using Parsec parser
> Configuring openapi3-3.2.3...
> CallStack (from HasCallStack):
>   withMetadata, called at libraries/Cabal/Cabal/src/Distribution/Simple/Ut...
> Error: Setup: Encountered missing or private dependencies:
> base >=4.11.1.0 && <4.18,
> base-compat-batteries >=0.11.1 && <0.13,
> template-haskell >=2.13.0.0 && <2.20

When looking at its entry on Hackage those weren't the version ranges for the dependencies. Also, running ghc-pkg list told me that I already had all required packages at versions matching what Hackage said. So, what's actually happening here?

It took me a while before remembering about revisions but once I did it was clear that callHackageDirect always fetches the initial revision of a package (i.e. it fetches the original tar-ball uploaded by the author). After realising this it makes perfect sense – it's the only revision that's guaranteed to be there and won't change. However, it would be very useful to be able to pick a revision that actually builds.

I'm not the first one to find this, of course. It's been noted and written about on the discource several years ago. What I didn't find though was a way to influence what revision that's picked. It took a bit of rummaging around in the nixpkgs code but finally I found two variables that's used in the Hackage derivation to control this

  • revision - a string with the number of the revision, and
  • editedCabalFile - the SHA256 of the modified Cabal file.

Setting them is done using the overrideCabal function. This is a piece of my setup for a modified set of Haskell packages:

hl = nixpkgs.haskell.lib.compose;

hsPkgs = nixpkgs.haskell.packages.ghc963.override {
  overrides = newpkgs: oldpkgs: {
    openapi3 = hl.overrideCabal (drv: {
      revision = "4";
      editedCabalFile =
        "sha256-a5C58iYrL7eAEHCzinICiJpbNTGwiOFFAYik28et7fI=";
    }) (oldpkgs.callHackageDirect {
      pkg = "openapi3";
      ver = "3.2.3";
      sha256 = "sha256-0F16o3oqOB5ri6KBdPFEFHB4dv1z+Pw6E5f1rwkqwi8=";
    } { });

It's not very ergonomic, and I think an extended version of callHackageDirect would make sense.

March 14, 2024 09:31 PM

March 13, 2024

GHC Developer Blog

GHC 9.10.1-alpha1 is now available

GHC 9.10.1-alpha1 is now available

bgamari - 2024-03-13

The GHC developers are very pleased to announce the availability of the first alpha release of GHC 9.10.1. Binary distributions, source distributions, and documentation are available at downloads.haskell.org.

We hope to have this release available via ghcup shortly.

GHC 9.10 will bring a number of new features and improvements, including:

  • The introduction of the GHC2024 language edition, building upon GHC2021 with the addition of a number of widely-used extensions.

  • Partial implementation of the GHC Proposal #281, allowing visible quantification to be used in the types of terms.

  • Extension of LinearTypes to allow linear let and where bindings

  • The implementation of the exception backtrace proposal, allowing the annotation of exceptions with backtraces, as well as other user-defined context

  • Further improvements in the info table provenance mechanism, reducing code size to allow IPE information to be enabled more widely

  • Javascript FFI support in the WebAssembly backend

  • Improvements in the fragmentation characteristics of the low-latency non-moving garbage collector.

  • … and many more

A full accounting of changes can be found in the release notes. As always, GHC’s release status, including planned future releases, can be found on the GHC Wiki status.

Many will notice that this release comes a fair bit later than the previously-announced schedule. While this delay has been attributable to a variety factors, the most recent cause is a set of issues with GHC 9.10’s binary distributions on Windows (#24542). Instead of continuing to hold up the release process while we sort out this situation, we have instead provided this alpha without the usual assortment of Windows binary distributions. We expect to have this resolved by alpha 2; apologies to eager Windows testers for this delay.

We would like to thank GitHub, IOG, the Zw3rk stake pool, Well-Typed, Tweag I/O, Serokell, Equinix, SimSpace, the Haskell Foundation, and other anonymous contributors whose on-going financial and in-kind support has facilitated GHC maintenance and release management over the years. Finally, this release would not have been possible without the hundreds of open-source contributors whose work comprise this release.

As always, do give this release a try and open a [ticket][] if you see anything amiss.

by ghc-devs at March 13, 2024 12:00 AM

March 12, 2024

Tweag I/O

Software Identifiers through the eyes of Nix

This is an answer to a recent request for comments issued by CISA, the United States “Cybersecurity and Infrastructure Security Agency”, about software identifiers. Unfortunately I wasn’t aware of this request for comments early enough and thus too late to comment officially. But CISA encouraged me to publish the answer as a separate blog post. The Guix team similarly published their own answer


Dear CISA team,

I appreciate your effort to gather comments about your recently released “Software Identification Ecosystem Option Analysis” white paper. As you say in the Executive Summary, “Organizations of all sizes must track what software they own and operate to perform user support, inventory administration, and vulnerability management”. I would go further and claim that for any software system that will be modified and reassembled — which is basically always — precise knowledge and control over the components and how they should be put together is crucial. Precise naming, identifiers, are the basis of that. In that light, I would like to bring to your attention another noteworthy technology that hasn’t been mentioned in this study, Nix and its sister project Guix, that achieves exactly this.

Nix is a powerful package manager, offering a very distinctive approach to deploying software. It achieves very high levels of reproducibility and provenance tracking of software artifacts by design, utilizing a functional and declarative language to describe software builds and their dependencies.

In fact, the levels of reproducibility it achieves are so high that Nix can robustly rely on an “input-addressed” storage, an identification model that names software artifacts by hashing everything required to build them, as opposed to hashing their content once assembled. This unique input-addressed approach is very powerful because it allows computing the identifier of a software asset without assembling it.

“Software artifacts” in Nix can be anything from sources and data assets to executable binary packages. And, importantly, “everything required to build” is not limited to source code and data assets, as in most intrinsic identification models, but comprises all commands, configuration, and recursively identified dependencies that are required to assemble the asset in a very strict sandbox. This controlled environment ensures that the description and the identifier of a software asset (called “package closure” in Nix) are complete, capturing all ingredients that went into the final output.

The reproducibility of core packages of the Nix distribution NixOS is very high, automatically tested, and can in principle be used for the independent, and decentralized verification of the content of software artifacts, as demonstrated by this implementation developed within the European Commission’s Next Generation Internet program.

In addition, these advantageous properties allowed the Nix community to construct Nixpkgs, the largest, and most up to date, open software library available. As required by the Nix model, this enormous repository of software assets comprises not only a description of the components that have been used to assemble them but also everything else necessary to produce the final packages, including, besides build instructions and configuration, a global dependency graph of all these assets. And, besides the guarantees that Nix furnishes by design for completeness and reproducibility, the packages go through automatic tests executed by an associated CI, and the hands of tens of thousands of regular users.

The screenshot below shows the CI build example of the open source computer game EmptyEpsilon. Omitting some details in the screenshot below, rja769qkxhiha7mbhq5bjmkjd0d5l1v0-empty-epsilon-2023.06.17 in the Derivation store path field, is the unique identifier of the particular version of this software package realized in the context of all the dependencies and configuration that are defined in a specific version (commit) of the Nixpkgs library that this CI build is attached to. “release.nix” is the entrypoint into the Nixpkgs library of software assets where EmptyEpsilon is defined using the Nix language; the mentioned .drv file contains an intermediate, raw build recipe that was generated from the Nix expressions. Software assets can come with attached metadata such as license information or short descriptions, and some data such as the closure size comprising the software and all its dependencies (build- and run-time) can easily be computed. More details on this can be found here or here.

CI build details of EmptyEpsilon

The exceptional completeness, robustness and precision of this approach uniquely positions Nix (and Guix) as highly valuable tool for automatically creating and managing accurate, reliable and reproducible software bills of materials (SBOMs) that can be employed to address the challenges outlined in the CISA notice. In fact, several projects exist (e.g. 1,2,3) that aim to automatically generate SBOMs from Nix expressions, or connect Nix packages to NIST’s national vulnerability database. The Nix model for identifiers works very well together with SoftWare Hash IDentifiers (SWHID) for the full state of version control system repositories that are developed by Software Heritage.

Finally, and certainly most importantly, I would like to emphasize that tens of thousands of users are demonstrating every day that applying this model comes without overhead. In fact, the precision and robustness of these software identifiers comes with a multitude of additional benefits. All this is not magic but enabled by a tool that follows a rigorous deployment model, the output of decades of academic experimentation and research. This is why the user bases of Nix and Guix, still emerging technologies, are rapidly growing.

March 12, 2024 12:00 AM

March 11, 2024

Joachim Breitner

Convenient sandboxed development environment

I like using one machine and setup for everything, from serious development work to hobby projects to managing my finances. This is very convenient, as often the lines between these are blurred. But it is also scary if I think of the large number of people who I have to trust to not want to extract all my personal data. Whenever I run a cabal install, or a fun VSCode extension gets updated, or anything like that, I am running code that could be malicious or buggy.

In a way it is surprising and reassuring that, as far as I can tell, this commonly does not happen. Most open source developers out there seem to be nice and well-meaning, after all.

Convenient or it won’t happen

Nevertheless I thought I should do something about this. The safest option would probably to use dedicated virtual machines for the development work, with very little interaction with my main system. But knowing me, that did not seem likely to happen, as it sounded like a fair amount of hassle. So I aimed for a viable compromise between security and convenient, and one that does not get too much in the way of my current habits.

For instance, it seems desirable to have the project files accessible from my unconstrained environment. This way, I could perform certain actions that need access to secret keys or tokens, but are (unlikely) to run code (e.g. git push, git pull from private repositories, gh pr create) from “the outside”, and the actual build environment can do without access to these secrets.

The user experience I thus want is a quick way to enter a “development environment” where I can do most of the things I need to do while programming (network access, running command line and GUI programs), with access to the current project, but without access to my actual /home directory.

I initially followed the blog post “Application Isolation using NixOS Containers” by Marcin Sucharski and got something working that mostly did what I wanted, but then a colleague pointed out that tools like firejail can achieve roughly the same with a less “global” setup. I tried to use firejail, but found it to be a bit too inflexible for my particular whims, so I ended up writing a small wrapper around the lower level sandboxing tool https://github.com/containers/bubblewrap.

Selective bubblewrapping

This script, called dev and included below, builds a new filesystem namespace with minimal /proc and /dev directories, it’s own /tmp directories. It then binds-mound some directories to make the host’s NixOS system available inside the container (/bin, /usr, the nix store including domain socket, stuff for OpenGL applications). My user’s home directory is taken from ~/.dev-home and some configuration files are bind-mounted for convenient sharing. I intentionally don’t share most of the configuration – for example, a direnv enable in the dev environment should not affect the main environment. The X11 socket for graphical applications and the corresponding .Xauthority file is made available. And finally, if I run dev in a project directory, this project directory is bind mounted writable, and the current working directory is preserved.

The effect is that I can type dev on the command line to enter “dev mode” rather conveniently. I can run development tools, including graphical ones like VSCode, and especially the latter with its extensions is part of the sandbox. To do a git push I either exit the development environment (Ctrl-D) or open a separate terminal. Overall, the inconvenience of switching back and forth seems worth the extra protection.

Clearly, isn’t going to hold against a determined and maybe targeted attacker (e.g. access to the X11 and the nix daemon socket can probably be used to escape easily). But I hope it will help against a compromised dev dependency that just deletes or exfiltrates data, like keys or passwords, from the usual places in $HOME.

Rough corners

There is more polishing that could be done.

  • In particular, clicking on a link inside VSCode in the container will currently open Firefox inside the container, without access to my settings and cookies etc. Ideally, links would be opened in the Firefox running outside. This is a problem that has a solution in the world of applications that are sandboxed with Flatpak, and involves a bunch of moving parts (a xdg-desktop-portal user service, a filtering dbus proxy, exposing access to that proxy in the container). I experimented with that for a bit longer than I should have, but could not get it to work to satisfaction (even without a container involved, I could not get xdg-desktop-portal to heed my default browser settings…). For now I will live with manually copying and pasting URLs, we’ll see how long this lasts.

  • With this setup (and unlike the NixOS container setup I tried first), the same applications are installed inside and outside. It might be useful to separate the set of installed programs: There is simply no point in running evolution or firefox inside the container, and if I do not even have VSCode or cabal available outside, so that it’s less likely that I forget to enter dev before using these tools.

    It shouldn’t be too hard to cargo-cult some of the NixOS Containers infrastructure to be able to have a separate system configuration that I can manage as part of my normal system configuration and make available to bubblewrap here.

So likely I will refine this some more over time. Or get tired of typing dev and going back to what I did before…

The script

The dev script (at the time of writing)

by Joachim Breitner (mail@joachim-breitner.de) at March 11, 2024 08:39 PM

March 08, 2024

Mark Jason Dominus

Werewolf ammunition

This week I read on Tumblr somewhere this intriguing observation:

how come whenever someone gets a silver bullet to kill a werewolf or whatever the shell is silver too. Do they know that part gets ejected or is it some kind of scam

Quite so! Unless you're hunting werewolves with a muzzle-loaded rifle or a blunderbuss or something like that. Which sounds like a very bad idea.

Once you have the silver bullets, presumably you would then make them into cartidge ammunition using a standard ammunition press. And I'd think you would use standard brass casings. Silver would be expensive and pointless, and where would you get them? The silver bullets themselves are much easier. You can make them with an ordinary bullet mold, also available at Wal-Mart.

Anyway it seems to me that a much better approach, if you had enough silver, would be to use a shotgun and manufacture your own shotgun shells with silver shot. When you're attacked by a werewolf you don't want to be fussing around trying to aim for the head. You'd need more silver, but not too much more.

I think people who make their own shotgun shells usually buy their shot in bags instead of making it themselves. A while back I mentioned a low-tech way of making shot:

But why build a tower? … You melt up a cauldron of lead at the top, then dump it through a copper sieve and let it fall into a tub of water at the bottom. On the way down, the molten lead turns into round shot.

That's for 18th-century round bullets or maybe small cannonballs. For shotgun shot it seems very feasible. You wouldn't need a tower, you could do it in your garage. (Pause while I do some Internet research…) It seems the current technique is a little different: you let the molten lead drip through a die with a small hole.

Wikipedia has an article on silver bullets but no mention of silver shotgun pellets.

Addendum

I googled the original Tumblr post and found that it goes on very amusingly:

catch me in the woods the next morning with a metal detector gathering up casings to melt down and sell to more dumb fuck city shits next month

by Mark Dominus (mjd@plover.com) at March 08, 2024 08:55 AM

Well-Typed.Com

GHC activities report: December 2023–February 2024

This is the twenty-second edition of our GHC activities report, which describes the work on GHC, Cabal and related projects that we are doing at Well-Typed. The current edition covers roughly the months of December 2023 to February 2024. You can find the previous editions collected under the ghc-activities-report tag.

Many thanks to our sponsors who make this work possible: Anduril, Hasura and Juspay. In addition, we are grateful to Mercury for funding specific work on improved performance for developer tools on large codebases, and to the Sovereign Tech Fund for funding work on Cabal.

However, we need more sponsorship to sustain the team! If your company might be able to contribute funding to sustain this work, please read about how you can help or get in touch.

Of course, Haskell tooling is a large community effort, and Well-Typed’s contributions are just a small part of this. This report does not aim to give an exhaustive picture of all GHC work that is ongoing, and there are many fantastic features currently being worked on that are omitted here simply because none of us are currently involved in them. Furthermore, the aspects we do mention are still the work of many people. In many cases, we have just been helping with the last few steps of integration. We are immensely grateful to everyone contributing to GHC!

Team

The GHC team at Well-Typed currently consists of Ben Gamari, Andreas Klebinger, Matthew Pickering, Zubin Duggal, Sam Derbyshire and Rodrigo Mesquita, with Hannes Siebenhandl joining the team in January and Finley McIlwaine moving to another client project. In addition, many others within Well-Typed are contributing to GHC more occasionally.

Releases

Zubin released GHC 9.6.4 in January and GHC 9.8.2 in February. We are now working towards the release of GHC 9.10 later in the year. Check out the GHC status page for more information on release plans.

Eras profiling

Matthew and Zubin recently implemented a new profiling mode, eras profiling, that can give insight into when particular objects are allocated. This can be a great boon in diagnosing memory leaks in long-running programs.

Check out our blog post introducing eras profiling for more information about this new feature, and an exploration of how we used this new profiling mode to diagnose a memory leak in GHCi. Matthew also used eras profiling to diagnose a space leak in GHC’s simplifier (!11914).

The combination of eras profiling and ghc-debug works particularly well for analysing memory leaks, so Zubin has been making various improvements to ghc-debug (MR 32), including improving how it handles profiled executables (MR 35, MR 36).

A new home for GHC’s internals

GHC’s base library has long served a dual purpose: on one hand it is the user-facing standard library interface, but at the same time it contains many internal details used to implement the standard library. This dual purpose lead to problems for both implementors and users alike, as internal interfaces are freely interspersed with long-stable interfaces intended for general consumption. Even worse, the documentation of base often provided little guidance to users regarding which interfaces fell into which category.

Earlier this year, the Core Libraries Committee and GHC Team agreed a path to improve this situation by splitting base into three libraries: base, ghc-internal, and ghc-experimental. Our hope is that this approach will allow us to solve several problems at once:

  • base gives users a clearly-demarcated set of stable interfaces, overseen by the Core Libraries Commiteee.
  • ghc-experimental gives developers of new language and library features a dedicated place to iterate on their designs while still allowing usage to users willing to accept a slightly lower degree of stability.
  • ghc-internal provides a home for internal implementation details that are not intended for consumption by users, and potentially change from release to release.

Ben has been working on implementing this split by separating out definitions that belong in the ghc-internal package (!11400). This split has lead to a number of improvements across the ecosystem, ranging from Haddock improvements (see Haddock issues 1629, 1630) to compiler bug-fixes (#24436) and implementation cleanups (#24472).

Exception backtraces

Ben has been working to land his long-running and long-awaited Exception Backtrace Proposal (!8869) following extensive discussions with the Core Libraries Committee. This is expected to form part of GHC 9.10 and will be a major step towards making exception diagnosis easier for users.

GHC Steering Committee and GHC2024

Adam has now taken on the role of Secretary to the GHC Steering Committee, following Joachim Breitner stepping down after many years of dedicated service in the role. His first major task as secretary has been seeking new volunteers to serve on the commitee. If you would be interested, please read more and get in touch.

The committee has updated the collection of recommended language extensions by introducing GHC2024. GHC 9.10 will ship with GHC2024 available (!12084), but it is unclear when it will become the default (see ghc-proposals MR 632).

STM correctness and performance

Andreas has been diagnosing progress and performance issues with STM prompted by a user reporting STM starvation problems (#24142). In particular:

  • STM transaction performance scales badly with the number of TVars involved (#24410), because the current implementation uses a linked list to keep track of all TVars used by a transaction. Ben explored one approach for improving this situation, using a hashmap for these lookups (!12030).

  • Transactions with a large number of TVars may perform badly (#24427) due to a check performed by the RTS each time Haskell threads return to the scheduler. This check identifies potentially non-terminating STM transactions by validating the transaction’s view of the STM memory against the memory’s current state. While very useful, this check is somewhat costly to perform, and under the current implementation can also lead to false negatives when multiple validations happen in parallel. It is likely that the best solution for this issue is to perform validations less frequently, especially on long running transactions.

  • In pathological cases, two transactions run in parallel may be unable to make progress (#24446), even if all transactions are read only. This should be solvable with a rework of how TVars are locked during validation.

Unfortunately, fixing these issues will require further work.

Specialisation and late plugins

Finley has been exploring techniques to make it easier to diagnose issues with specialisation in large applications, such as poor runtime performance due to overloaded calls not being specialised. One workaround for such problems is exposing all unfoldings and using aggresive specialisation, but this tends to lead to poor compile-time performance instead.

Motivated by these investigations he added “late plugins,” which are plugins that are run at the very end of the Core pipeline, after the addition of late cost centres (!11765). This allows plugins to analyse and modify the Core that is compiled down to STG, without the changes ending up in interface files.

Cabal

Matthew, Rodrigo and Sam have been working to address longstanding architectural and maintenance issues in the Cabal library and the cabal-install build tool. This work is being supported by the Sovereign Tech Fund as discussed in our previous blog post.

Some of the changes have included:

  • Designing and implementing a new build-type: Hooks feature to provide a path towards deprecating build-type: Custom. Based on community feedback, Sam iterated on the design, with a particular focus on pre-build rules, arriving at a design inspired by Cloud Haskell, using static pointers. See the detailed HF Tech Proposal for an in-depth explanation of the design and its benefits. The implementation is now being prepared for review (PR 9551).

  • Disentangling implicit global state from the Cabal library, allowing it to take a working directory as an argument instead of using the working directory of the current process (PR 9718). This is intended to allow directly calling the Cabal library to build packages in a concurrent setting.

  • Working on a design and prototype implementation for private dependencies (issue 4035), allowing packages to express the fact that they do not expose any types from a dependency in their API. This gives greater flexibility to construct build plans, potentially making library version upgrades easier, and allows tests and benchmarks to compare different versions of the same library.

  • Making the testsuite more robust, including refactoring it to run tests in a separate temporary directory so they are not influenced by the external configuration of the user’s system (PR 9717).

  • Allowing per-component builds with Haskell Program Coverage (HPC) information (PR 9464).

  • Refactoring to eliminate long-standing code duplication that was a regular source of bugs in the logic for building components (PR 9602) and in glob support (PR 9673).

  • Fixing several longstanding bugs with the install command often ignoring CLI flags (PR 9697).

  • Robustly handling the same GHC version having been compiled from source multiple times (PR 9618), as the GHC version number is not enough to ensure ABI-compatibility.

  • Many more bug fixes and refactorings to improve maintainability and robustness of the codebase (e.g. PR 9524 PR 9554).

GHC bug fixes

  • Ben investigated memory-ordering issues using ThreadSanitizer and fixed numerous data races (!9372, !11795, !11768).

  • Ben fixed a thread-safety issue due to GHC’s use of the C strerror utility (#24344).

  • Sam fixed a 9.8 regression in shadowing error messages involving record fields with no field selectors (!11981).

  • Hannes fixed a 9.8 regression in how Haddock resolves qualified references (!11920).

  • Zubin fixed a regression in which GHC reported a poor error message in the presence of module cycles including hs-boot files (!11718, !11792).

  • Zubin fixed cross-module module breakpoints using incorrect cost centres (!11892).

  • Sam and Andreas fixed a variety of bugs in the handling of fused-multiply-add primops that were added in GHC 9.8.1 (!11587, !11893, !11902, !11987).

  • Ben fixed a subtle bug in the implementation of unique generation on 32-bit platforms (!11802).

  • Andreas fixed a bug in the C foreign-function interface that was introduced by using sub-word-sized arguments (!11989).

  • Zubin set -DPROFILING when compiling C++ sources with profiling (!11871).

  • Matthew fixed an off-by-one error when handling info-table provenance entries (!11873).

  • Zubin fixed a bug with ghcup-metadata generation (!11791).

  • Zubin updated the users’ guide to take into account the unrestricted overloaded labels GHC proposal, which landed in GHC 9.6 (!11774).

  • Hannes fixed a bug arising from GHC being installed at a filepath that includes spaces on Windows (!11938).

Build system, CI and distribution improvements

  • Ben carried out a number of submodule bumps in preparation for the GHC 9.10 release.

  • Rodrigo allowed the configure script to use autoconf 2.72 (!11942).

  • Matthew fixed a bug in the configuration of hsc2hs when building GHC, which was the source of linker errors (#24050, !11384).

  • Matthew updated the CI images, with a particular focus on improving the testing of the LLVM backend on CI (#24369, !11976).

  • Matthew ensured that documentation is built on more configuration in CI (e.g. on alpine, rocky8, Windows, Darwin) (!12134).

  • Ben adapted GHC to LLVM’s new pass manager CLI (!8999).

by adam, andreask, ben, finley, hannes, matthew, rodrigo, sam, zubin at March 08, 2024 12:00 AM

March 07, 2024

Tweag I/O

Extending destination-passing style programming to arbitrary data types in Linear Haskell

Three years ago, a blog post introduced destination-passing style (DPS) programming in Haskell, focusing on array processing, for which the API was made safe thanks to Linear Haskell. Today, I’ll present a slightly different API to manipulate arbitrary data types in a DPS fashion, and show why it can be useful for some parts of your programs.

The present blog post is mostly based on my recent paper Destination-passing style programming: a Haskell implementation, published at JFLA 2024. It assumes basic knowledge of Linear Haskell and intermediate fluency in Haskell.

Tail Modulo Cons

Haskell is a lazy language by default, but a lot of algorithms are in fact more efficient in a strict setting. That’s one reason why Haskell has been extending support for opt-in strictness, via strict field annotation for example.

Non-tail recursive functions such as map are decently efficient in a lazy setting. On strict data structures, however, non-tail recursive consume stack space. That’s why the quest for tail-recursive implementations is even more central in strict languages such as OCaml than in Haskell.

If any function can be made tail-recursive using a CPS transformation, this transformation trades stack space for heap space (where the built continuations are allocated), which is rarely a win performance-wise. We actually want to focus on tail-recursive implementations which don’t resort to continuations, and unfortunately, some functions don’t have one in a purely functional setting.

For example, some functions are almost tail-recursive, in the sense that the recursive call is the penultimate computation in the returned value, and the last one is just a constructor application. This is actually the case for map:

map :: (a -> b) -> [a] -> [b]
map f [] = []
map f (x : xs) = (f x) : map f xs

One could argue that a tail-recursive version of map could be written using an accumulator to store the result list, and then reverse it at the end. But that incurs an extra linear operation (reversing the accumulator) that isn’t present in the naive version.

Actually, Bour et al. found in 2021 that whenever a function is of this particular shape — with just a constructor application over the recursive call — named tail-recursive modulo constructor, this function can be easily and automatically converted into an equivalent tail-recursive one in destination-passing style (DPS).

In OCaml1, this transformation happens entirely in the compiler. In this blog post, on the other hand, I’ll show how to do it in user-land in Haskell thanks to linear types, which make the API safe.

For map, here’s the translation to DPS Haskell, although we will come back to it later:

mapDPS :: (a -> b) -> [a] -> Dest [b] %1 -> ()
mapDPS f [] d = fill @'[] d
mapDPS f (x : xs) = let !(dh :: Dest b, dt :: Dest [b]) = fill @'(:) d
                     in fillLeaf (f x) dh `lseq` mapDPS f xs dt

It’s time to see what DPS means and what it offers.

What is Destination-passing style programming?

Destination-passing style (DPS) is a programming idiom in which functions don’t return a result, but rather write their result directly to a memory location they receive as a parameter. This gives more control over memory to the caller of a function, instead of that control lying exclusively in the hands of the callee. In non-GC languages, or for array processing (as in the aforementioned blog post), it allows the allocation of a big chunk of memory at once, and then gives each piece of the program the responsibility to fill a small part of that chunk (represented by a glorified pointer, aka. destination), giving mostly alloc-free code. In early imperative languages such as C, this is actually quite common: memcpy and strcpy both receive a destination as a parameter.

In the context of a functional, immutable, GC-based language, we cannot circumvent the allocation of heap objects to obtain alloc-free code. Instead, we get one interesting feature: being able to build functional structures in the reverse order compared to the regular constructor-based approach. This goes hand-in-hand with the ability to create and manipulate incomplete data structures (containing unspecified fields, aka. holes) safely. This is exactly what we will focus on in this blog post.

Incomplete structures, you say?

An incomplete structure can be seen as a tree of constructor objects, much like a regular data structure. However, some of the constructor’s fields might not be specified, leaving holes in the structure.

Having incomplete structures is very different from having optional fields in a structure represented by the Maybe a type. For so-called incomplete structures, we do not indicate the absence of value (or presence of a hole) through a different type for the leaf itself, but instead we forbid any read on the whole structure as long as (at least) one hole exists somewhere. That way, the field’s value can (in fact, must) be updated later without allocating the whole structure a second time.

To update a yet-unfilled field of an incomplete structure, we use a destination. A destination is a unique pointer to a hole inside an incomplete structure, that is no longer usable as soon as the hole has been filled. Those pointers are carried alongside the structure until they get consumed. As a result, destinations are also a way to know whether or not a structure has any remaining holes. When an incomplete structure no longer has any accompanying destinations, it can be read safely.

At this point, incomplete structures could be seen as the nemesis of Haskell, as they bring a form of mutability and a range of memory errors if not handled properly. However, with a proper linear API, which is the real novelty in this blog post, they are both powerful and safe to use. In particular, a linear discipline on destinations guarantees that:

  1. when a structure no longer has accompanying destinations, it’s a complete structure (that is, it has no holes remaining);
  2. once a hole has been filled with a value, that value cannot be changed anymore (i.e. holes are write-once).

Implementation of Incomplete structures

As teased before, I introduce an opaque data Incomplete a b to represent incomplete objects. The a part is the structure being built that may contain holes, and the b part carries the destinations pointing to these holes. Destinations are raw pointers under the hood, put in a pretty box: data Dest a represents a pointer to a hole of type a.

What can we do with an Incomplete a b? We cannot read the structure on the a side (yet) as long as the b side still contains destinations (as they indicate the presence of holes in the a side). The b side is what must be linearly consumed to make the structure readable. What we can do is map over the b side, to consume the destinations little-by-little until there are none left. This is exposed through a (linear) Functor instance:

instance Control.Functor (Incomplete a) where
  fmap :: (b %1 -> c) %1 -> Incomplete a b %1 -> Incomplete a c
  (<&>) :: Incomplete a b %1 -> (b %1 -> c) %1 -> Incomplete a c  -- flipped arguments

This Functor instance lets us access destinations of an incomplete object through a linear continuation of type b %1 -> c.

Let’s take a step back and look at our previous example. mapDPS has signature (a -> b) -> [a] -> Dest [b] %1 -> (). This means that mapDPS f list is in fact a linear continuation of type Dest [b] %1 -> ().

In other terms, given an incomplete structure having a hole of type [b], i.e. i :: Incomplete u (Dest [b]), we can write the result of f mapped to list to this hole, using i <&> mapDPS f list. The resulting structure will have type Incomplete u () (no more destinations), and can be made readable as we’ll see in a moment.

Here we can see the essence of DPS: functions get less responsibility as they don’t get to choose where they write their result; instead, the output location is now passed as an explicit parameter to the function. Moreover, inside a function such as mapDPS, we can, and in fact we have no choice but to forget about the global structure we are building — it becomes implicit — and only focus on the handling of destinations. The Functor instance is thus the glue that lets us assign a location to a producer of data like mapDPS for it to write its output.

Operating on Dests

Let’s take a closer look at the mapDPS implementation:

mapDPS f [] d = fill @'[] d
mapDPS f (x : xs) = let !(dh :: Dest b, dt :: Dest [b]) = fill @'(:) d
                     in fillLeaf (f x) dh `lseq` mapDPS f xs dt

In the base case, there is no element left in the input list, but we still receive a destination d :: Dest [b] that needs to be dealt with linearly. The only meaningful operation here is to write the empty list to the hole represented by d, which is what fill @'[] d does.

The recursive case is more interesting:

  • one cons cell should be added to the list, carrying the value f x :: b;
  • we somehow need to create another destination of type Dest [b] to pass to the recursive call.

All of that is done in two steps, using fill @'(:) and then fillLeaf.

fill @'(:) d is first used to add a new hollow cons cell (:) _h _t :: [b] at the end of the linked list, that is to say, a cons cell with unspecified fields (both the head _h and tail _t are holes). Under the hood, it allocates the new hollow cons cell, writes its address into the destination d :: Dest [b], and returns one destination dh :: Dest b pointing to the hole _h, and one destination dt :: Dest [b] pointing to the hole _t. This gives the signature fill @'(:) :: Dest [b] %1 -> (Dest b, Dest [b]).

Then, fillLeaf is used to fill the destination dh :: Dest b (representing the “value part” of the newly added cons cell) with the result of f x :: b. fillLeaf :: a -> Dest a %1 -> () is in fact pretty simple. It takes a value, a destination, and writes the value address to the hole represented by the destination. The destination is linearly consumed in the process.

After doing that, only one destination remains unconsumed: dt :: Dest [b]. This is exactly the destination that will be passed to the recursive call! It corresponds to the new “end” of the linked list.

We directly see here how fill @'(:) extends an (incomplete) list by adding one new “slot” at the end; whereas cons (:) is usually used to extend a normal linked list from the front. This is what I meant in the introduction by building functional structures in the reverse order.

What I just presented is not, in fact, restricted to lists. It can be used to build any kind of structure, as long as it implements Generic. This is mostly the only constraint fill has; it can be used for all sorts of constructors. For example, we can build a binary tree in a similar way, starting with the root, and extending it progressively in a top-down fashion, using fill @'Leaf and fill @'Node (assuming data Tree a = Leaf | Node a (Tree a) (Tree a) deriving Generic).

Creating and disposing of Incompletes

One can create a new, empty Incomplete using alloc :: Linearly %1 -> Incomplete a (Dest a). This function exchanges a Linearly token (see below) for an Incomplete of the chosen type a. The resulting Incomplete has a single destination that points to its root of type a. In other terms, even the root of the new structure is a hole at the moment, that will be specified later with the first use of fill or fillLeaf.

Conversely, as soon as we have an Incomplete with only unit () on the b side, the absence of destination indicates that the structure on the a side is complete. So we can make it readable by getting out of the Incomplete wrapper using fromIncomplete :: Incomplete a () %1 -> Ur a.

It is valid to use the built structure in a non-linear fashion (justifying why it is wrapped in Ur in return position of fromIncomplete) because it has been made of non-linear elements only: fillLeaf is non-linear in its first argument, and the spine of the structure can be duplicated without breaking linearity.

The last missing piece of this API is linearly :: (Linearly %1 -> Ur b) %1 -> Ur b, whose definition is shared with the one from a previous blog post about linear scopes. linearly delimits a scope in which linear objects can be used. Only non-linear objects can escape this scope (because of the Ur restriction on the return type as before), such as complete structures finalized with a call to fromIncomplete.

The Linearly type, of which an instance is supplied by linearly, is a linear token which can be duplicated to give birth to any number of Incompletes, but each of them will still have to be managed linearly.

With these final ingredients, we can complete our definition of a tail-recursive map:

map :: (a -> b) -> [a] -> [b]
map f l =
  unur $ linearly $ \token ->
    fromIncomplete $ alloc token <$> \d ->
      mapDPS f l d

Performance

The current implementation behind the API is based on compact regions as they make it easy to operate on memory without too much tension with the garbage collector. However, they incur extra copying in some contexts, which makes it hard sometimes to compete with optimized lazy Haskell code.

At the moment, the mapDPS implementation is slightly more efficient memory-wise than the optimized lazy one for large lists (and less efficient for smaller lists). The same kind of results are obtained for the different use-cases we benchmarked in Section 6 of the associated paper. I expect a next implementation without compact regions, taking place directly in the GC heap, to have better performance.

In addition, the DPS techniques detailed here are proven to be really efficient in strict languages. This work might thus inspire performance and expressiveness improvements in languages other than Haskell.

Conclusion

The API presented in this blog post defines a small set of tooling sufficient to create and operate safely on incomplete data structures in Haskell through destination-passing style programming. It is more general than the constructor-based building approach usually used in functional programming languages, and also more general than DPS tooling introduced by Tail Modulo Cons in OCaml. It is also a nice example of how linear types can be used to enforce a write-once memory model in Haskell.

The full prototype API is available here. It currently requires a custom GHC version to work, but I hope I will be able to merge the few primops required for DPS programming into GHC in the future.

I would like to thank Arnaud Spiwack for his solid support and feedback on all pieces of this work.


  1. OCaml has had experimental support for Tail-recursion Modulo Constructor since version 4.14.0.

March 07, 2024 12:00 AM

March 06, 2024

Mark Jason Dominus

Optimal boxes with and without lids

Sometime around 1986 or so I considered the question of the dimensions that a closed cuboidal box must have to enclose a given volume but use as little material as possible. (That is, if its surface area should be minimized.) It is an elementary calculus exercise and it is unsurprising that the optimal shape is a cube.

Then I wondered: what if the box is open at the top, so that it has only five faces instead of six? What are the optimal dimensions then?

I did the calculus, and it turned out that the optimal lidless box has a square base like the cube, but it should be exactly half as tall.

For example the optimal box-with-lid enclosing a cubic meter is a 1×1×1 cube with a surface area of .

Obviously if you just cut off the lid of the cubical box and throw it away you have a one-cubic-meter lidless box with a surface area of . But the optimal box-without-lid enclosing a cubic meter is shorter, with a larger base. It has dimensions $$2^{1/3} \cdot 2^{1/3} \cdot \frac{2^{1/3}}2$$

and a total surface area of only . It is what you would get if you took an optimal complete box, a cube, that enclosed two cubic meters, cut it in half, and threw the top half away.

I found it striking that the optimal lidless box was the same proportions as the optimal complete box, except half as tall. I asked Joe Keane if he could think of any reason why that should be obviously true, without requiring any calculus or computation. “Yes,” he said. I left it at that, imagining that at some point I would consider it at greater length and find the quick argument myself.

Then I forgot about it for a while.

Last week I remembered again and decided it was time to consider it at greater length and find the quick argument myself. Here's the explanation.

Take the cube and saw it into two equal halves. Each of these is a lidless five-sided box like the one we are trying to construct. The original cube enclosed a certain volume with the minimum possible material. The two half-cubes each enclose half the volume with half the material.

If there were a way to do better than that, you would be able to make a lidless box enclose half the volume with less than half the material. Then you could take two of those and glue them back together to get a complete box that enclosed the original volume with less than the original amount of material. But we already knew that the cube was optimal, so that is impossible.

by Mark Dominus (mjd@plover.com) at March 06, 2024 12:00 PM

Well-Typed.Com

The Haskell Unfolder Episode 21: testing without a reference

Today, 2024-03-06, at 1930 UTC (11:30 am PST, 2:30 pm EST, 7:30 pm GMT, 20:30 CET, …) we are streaming the 21th episode of the Haskell Unfolder live on YouTube.

The Haskell Unfolder Episode 21: testing without a reference

The best case scenario when testing a piece of software is when we have a reference implementation to compare against. Often however such a reference is not available, begging the question how to test a function if we cannot verify what that function computes exactly. In this episode we will consider how to define properties to verify the implementation of Dijkstra’s shortest path algorithm we discussed in Episode 20; you may wish to watch that episode first, but it’s not required: we will mostly treat the algorithm as a black box for the sake of testing it.

About the Haskell Unfolder

The Haskell Unfolder is a YouTube series about all things Haskell hosted by Edsko de Vries and Andres Löh, with episodes appearing approximately every two weeks. All episodes are live-streamed, and we try to respond to audience questions. All episodes are also available as recordings afterwards.

We have a GitHub repository with code samples from the episodes.

And we have a public Google calendar (also available as ICal) listing the planned schedule.

by andres, edsko at March 06, 2024 12:00 AM

March 05, 2024

Mark Jason Dominus

Children and adults see in very different ways

I was often struck with this thought when my kids were smaller. We would be looking at some object, let's say a bollard.

The kid sees the actual bollard, as it actually appears, and in detail! She sees its shape and texture, how the paint is chipped and mildewed, whether it is straight or crooked.

I don't usually see any of those things. I see the bollard abstractly, more as an idea of a “bollard” than as an actual physical object. But instead I see what it is for, and what it is made of, and how it was made and why, and by whom, all sorts of things that are completely invisible to the child.

The kid might mention that someone was standing by the crooked bollard, and I'd be mystified. I wouldn't have realized there was a crooked bollard. If I imagined the bollards in my head, I would have imagined them all straight and identical. But kids notice stuff like that.

Instead, I might have mentioned that someone was standing by the new bollard, because I remembered a couple of years back when one of them was falling apart and Rich demolished it and put in a new one. The kid can't see any of that stuff.

by Mark Dominus (mjd@plover.com) at March 05, 2024 02:18 AM

March 04, 2024

Mark Jason Dominus

Even without an alien invasion, February 22 on Talos I would have been a shitshow

One of my favorite videogames of the last few years, maybe my most favorite, is Prey. It was published in 2017, and developed by Arkane, the group that also created Dishonored. The publisher (Bethesda) sabotaged Prey by naming it after a beloved 2006 game also called Prey, with which it had no connection. Every fan of Prey (2006) who was hoping for a sequel was disappointed and savaged it. But it is a great, great game.

(I saw a video about the making of the 2017 Prey in which Raphael Colantonio talked about an earlier game of theirs, Dark Messiah of Might and Magic, which was not related to the Might and Magic series. But the publisher owned the Might and Magic IP, and thought the game would sell better if it was part of their established series. They stuck “Might and Magic” in the title, which disappointed all the Might and Magic fans, who savaged it. Then when Bethesda wanted to name Prey (2017) after their earlier game Prey (2006), Colantonio told them what had gone wrong the previous time they tried that strategy. His little shrug after he told that story broke my heart a little.)

This article contains a great many spoilers for the game, and also assumes you are familiar with the plot. It is unlikely to be of interest to anyone who is not familiar with Prey. You have been warned.

(If you're willing to check it out on my say-so, here's a link. I suggest you don't read the description, which contains spoilers. Just buy it and dive in.)


A recent question on Reddit's r/prey forum asked what would have happened if the Typhon organisms had not broken out when they did. The early plot of Prey is all there, but it is a little confusing, because several things were happening at once. The short answer to the question though, is that February 22, 2035 would have been the worst day of Alex Yu's life even if his magnificent space station hadn't been overrun by terrifying black aliens.

Morgan escapes the sim lab anyway

January had contingency plans for at least two situations. One was a Typhon escape, which we know all about.

But there was another plan for another situation. Morgan was having her memory erased before each round of testing. January explains that there was a procedure that was supposed to bring Morgan back up to speed after the tests were over. We know this procedure was followed for some time: Morgan's office has been used. Her assistant Jason Chang still hasn't gotten over his delight at working for such a hot boss. There are puzzled emails around asking why she never remembers her office combination. There's painful email from Mikhaila asking why Morgan is snubbing her. Clearly, at some point in the recent past, Morgan was still walking around the station in between tests, working and talking to people.

January's second contingency plan was in case Alex stopped bringing Morgan back up to speed after each round of tests, and just kept her in the simulation day after day — perhaps even more than once per day. (No wonder her eye is red!) And crucially, that plan was already in motion on February 22, the day the Typhon escaped.

The first thing that happens to the player in Prey is that Morgan fails all the tests. (“Is she…” “Yes, she's… hiding behind the chair.”) Why? We find out later Morgan was supposed to receive neuromods that would give her Typhon powers such as mimicry. They didn't. Marco Simmons says he installed exactly what Patricia brought down. There's email in the sim lab that asks Neuromod to check that something isn't wrong with the production process. But nothing is wrong with the production process. What really went wrong is that January had secretly replaced the neuromods with fakes, so that when they were removed from Morgan's brain, her memory wouldn't be affected. The next time Morgan woke up in her apartment and it was still March 15, she would realize what was happening.

It's hard to guess just what would have happened next, but I'm sure it would have been rather dramatic.

But that's not all

There are at least five other situations that would have blown up that same day. February 22 2035 on Talos I was always going to be an incredible shitshow.

  1. Emmanuel Mendez

    Because Frank Jones is a fuckup, Emmanuel Mendez has become aware that the escape pods don't work. He has decided to alert the crew by reprogramming the giant floating billboards to display “ESCAPE PODS ARE FAKE”. He completes this task on February 22 but dies without activating the program that will change the display.

    Those billboards are visible from everywhere on the station, including the cafeteria.

  2. Halden Graves

    Halden Graves, head of the Neuromod Division, has just figured out that the neuromods, even the non-Typhon ones, are made with exotic material from the Typhon, and he completely loses his shit, to the point of chopping open his own head to get them out. That might attract some attention.

  3. Josh Dalton

    On February 22, Josh Dalton murdered Lane Carpenter with the BFG 9000 and then fled with it into the GUTS.

  4. Alton Weber

    Weber is on the Life Support security team. He has had a paranoid breakdown and stolen a shotgun. There's probably going to be a firefight outside the Life Support restrooms.

  5. Mikhaila Ilyushin

    Mikhaila is about to be arrested. Alex already suspected that something about her was fishy. Mikhaila has sent Divya Naaz to install snooping devices in the doors in Psychotronics, and Divya has been caught. Alex isn't going to wait any longer to stop Ilyushin.

Coming soon

These are starting to fall apart but the shit won't really hit the fan until sometime after February 22.

  1. Annelise Gallegos and Quinten Purvis

    Annelise Gallegos has been overcome by her conscience and is blowing the whistle on the experiments in Psychotronics and the murder of the “Volunteers”. On February 22, Alex has ordered Sarah Elazar to arrest her, as soon as her shift is over. The Typhon escape prevents that. What would have happened to Gallegos? I suspect she would would have died in an ⸢unfortunate accident⸣.

    But it's too late for the Yus. Gallegos has already prepared her thumb drive with all the damning evidence, and Quinten Purvis has hidden with it in a cargo container. If the Typhon hadn't gotten loose that day, he would have been on his way to Earth with it.

  2. Hunter Hale

    Shuttles are supposed to take the Volunteers back to Earth when their service is complete. The Shuttle Bay flight control staff have recently noticed that the shuttles are not going straight back to Earth, but are stopping somewhere else just after leaving, and then proceeding to Earth on a slightly altered course.

    What's really happening is that the shuttle pilot, Hunter Hale, makes a stop at the Psychotronics airlock and drops off some or all of the Volunteers so they can be turned into Neuromods.

    Alex is paying Hale five times the normal salary to keep his mouth shut about this, but HR has noticed and is asking questions about it. Between the flight control staff and HR, the truth is going to come out.

  3. Sarah Elazar

    Elazar suspects that the Yus are up to something dirty. She doesn't know what yet, but she's going to find out.

  4. Disappearing neuromods

    Everyone seems to be pilfering neuromods. Emmanuella Da Silva has some stashed in the drop ceiling of the Shuttle Bay locker room. Yuri Kimura has four under her desk, and Elias Black is blackmailing her. Lorenzo Calvino has some in both of his secret safes. Lily Morris has them hidden in the fire alarms in half a dozen places around the station. That dumbass Grant Lockwood has tried to walk back to Earth with his stolen neuromods.

    I probably missed a few, they're all over.

    (I said none of this would come to light until after February 22, but it won't be long before someone wonders what became of Lockwood. It's also possible Alex will find out about the Lily Morris conspiracy that day, from Eddie Voss. I almost feel sorry for Alex.)

Minor shit

Not giant disasters, but troublesome nevertheless.

  1. Lorenzo Calvino

    It won't be long before someone, probably Miyu Okabe, figures out that Lorenzo Calvino has a severe, progressive mental impairment.

  2. Price Broadway

    Broadway, the alcoholic in Waste Processing, is endangering everyone's lives by leaving empty vodka bottles in the eel tanks. His supervisor knows and has reported him to HR. She says HR will help, but I imagine they'll just fire him.

    Maybe he'll end up on Hunter Hale's shuttle home.

  3. Volunteers

    What's up with the Volunteers in the dormitory in Neuromod Division? Some of them are stealing and selling supplies. Other are stealing dangerous equipment and weapons. What for?

Drama drama drama

Even without the Typhon, Prey could have been a great game!

You play Morgan, of course. The first fifteen minutes are the same, right up until Bellamy would have died.

When you wake up for the second time on March 15, you figure out what is happening, and confront the Sim Lab staff. You escape, go rogue, and make your way to the Arboretum to confront Alex. Meanwhile all sorts of stuff is going down. Alton Weber is on a rampage in Life Support. Josh Dalton is loose in the GUTS. You'll have to deal with him to get to the Arboretum. (What, did you think you were going to take the elevator?) Somewhere along the line you find out about Purvis in the cargo container and have to decide how to handle that. And then Mendez changes the billboards and there's a panic…

What else?

A lot is happening on Talos I. I probably left something out.

(The shortage of escape pods doesn't count. Someone would have noticed long ago that there aren't nearly enough. I think we have to assume that there are more escape pods than we see in the game. Perhaps Morgan's simulation omitted them.)

(And in my headcanon, that poor schmuck Kevin Hague never does find out his wife has cheated on him with the asshole football star.)

Let me know what I missed.

by Mark Dominus (mjd@plover.com) at March 04, 2024 03:49 AM

March 01, 2024

Haskell Interlude

44: José Manuel Calderón Trilla

Wouter and Niki interview Jose Calderon, the new Executive Director of the Haskell Foundation. Jose tells why he applied for the job, how he sees the foundation developing over the coming years, and how you can get involved in the Haskell community.

by Haskell Podcast at March 01, 2024 02:00 PM

February 29, 2024

Gabriella Gonzalez

The siren song of domain-specific languages

The siren song of domain-specific languages

I’ve seen a lot of engineering teams mistakenly believe that they can author domain-specific languages for less technical users on a budget. In particular they seem to believe that if they create this domain-specific language then the less technical users will be able to thoughtlessly churn out a bunch of code in that language and there won’t be any problem and they can then move onto the next project. This rarely works out in the way that people hope it will.

In the best case scenario, your less technical users will churn out a large amount of code using your domain-specific language (which is exactly the outcome you hoped for!) and that corpus of code will push the boundaries of what your language is capable of (like performance, compilation speed, features, or supporting integrations). The larger your userbase the greater the demand will be to improve your language in a myriad of ways.

In the worst case scenario your users will find increasingly inane ways to do things wrong with your language despite your best efforts and you will be expected to clean up their mess because you sold the project on the premise of “our users are not going to have to think”.

… and in either case this process will never end; the project will never be in a “done state” and require permanent staffing. Hell, even if you staff an entire team to support this language it’s still often a struggle to keep up with the needs of less technical users.

This tradeoff can still tempt businesses because it’s appealing to replace skilled labor with unskilled labor. The reasoning goes that a small investment of more skilled labor (the authors of the domain-specific language) can enable a larger pool of less skilled labor (the less technical users) to do most of the work. However, what you will often find in practice is that this larger group of less technical users is frequently blocked without continuous assistance from the engineers who created the language.

So in practice you’re not actually replacing skilled labor with unskilled labor. Rather, you’re merely “laundering” skilled labor as unskilled labor and creating more work for your engineers to make them seem more replaceable than they actually are.

I do think there are situations where domain-specific languages make sense, but typically not on the scale of a software engineering organization or even a small product. I personally think this sort of division of labor tends to only work on the scale of an open source ecosystem where you get a large enough economy of scale.

by Gabriella Gonzalez (noreply@blogger.com) at February 29, 2024 12:47 PM

Tweag I/O

Bazel remote execution with rules_nixpkgs

Tweag developed rules_nixpkgs to empower Bazel users with the ability to leverage Nix’s reproducible builds and its extensive package registry. That ruleset has proven to be especially advantageous in endeavors demanding intricate dependency administration and the maintenance of uniform build environments.

However, rules_nixpkgs is incompatible with remote execution. This is a major limitation given that remote execution is possibly the main reason why people switch to Bazel. And that rules_nixpkgs provides a great way to configure hermetic toolchains, which are an important ingredient for reliable remote execution. There is no trivial fix as can be seen in the related, longstanding open issue. At Tweag we investigated a promising solution presented at Bazel eXchange 2022 (recording), but these ideas were never implemented in a public proof of concept.

In this post, we will present our new remote execution infrastructure repo and walk you through the required steps to comprehend and replicate how it achieves remote execution with rules_nixpkgs.

The remote execution limitation

When we make use of rules_nixpkgs, we instruct Bazel to use packages from nixpkgs rather than those from the host system. This means that when we try to build a C++ project, Bazel won’t use the gcc compiler, which is typically found under /usr/bin, but instead will use the compiler specified by rules_nixpkgs and provided by Nix, typically stored under some /nix/store/<unique_hash>-gcc/bin directory.

Bazel distinguishes actions to import external dependencies from regular build actions. The former are always executed locally1, while the latter can be distributed using remote execution. rules_nixpkgs falls into the former category and invokes Nix to download and install the required /nix/store/<unique_hash>-gcc path locally on your machine.

This scenario works fine when we’re building locally. However, when we enable remote execution, rules_nixpkgs still installs dependencies locally, while the build happens on another machine, which will not have those paths available, so it will inevitably fail.

Initial setup with remote execution

For our proof of concept, we decided to use Buildbarn to provide the remote execution endpoint and infrastructure. Buildbarn provides Kubernetes manifests that we can use to deploy all the necessary Buildbarn components for remote execution to work. We’ll be using the examples from the bb-deployments repository to test our setup, but also modifying it to make use of rules_nixpkgs.

To replicate our implementation you’ll need a working Buildbarn infrastructure, which in this case would be a Kubernetes cluster. You can use our guide to set up a cluster on AWS.

Test remote execution without rules_nixpkgs

To make sure that everything is working as expected, we’ll use the @abseil-hello Bazel target which is available in the Buildbarn deployments repo. This example does not use rules_nixpkgs, yet. You can clone the bb-deployments repository, if you want to follow along.

  • Get the service endpoint of the Buildbarn executor service (frontend). If you’re deploying on a cloud provider this would be a load-balancer.
$ kubectl get services -n buildbarn
NAME        TYPE           CLUSTER-IP      EXTERNAL-IP                         PORT(S)                      AGE
browser     ClusterIP      172.20.22.171   <none>                              7984/TCP                     8d
frontend    LoadBalancer   172.20.126.97   xxxxx.us-east-1.elb.amazonaws.com   8980:31657/TCP               8d
scheduler   ClusterIP      172.20.83.110   <none>                              8982/TCP,8983/TCP,7982/TCP   8d
storage     ClusterIP      None            <none>                              8981/TCP                     8d
  • Update .bazelrc to use the remote executor endpoint of our environment
...
build:remote-exec --remote_executor=grpc://[endpoint-from-previous-step]
...

Now we can try building the @abseil-hello target using the remote execution infrastructure. Note that we’ll be using a custom toolchain specific to the default executors created by Buildbarn.

bazel build --config=remote-ubuntu-22-04 @abseil-hello//:hello_main

Test remote execution with rules_nixpkgs

Once we have validated that our setup works we can create a new target that uses rules_nixpkgs.

Update .bazelversion to use 6.4 which is a version supported by rules_nixpkgs (any other version on the 6.x should work as well).

Update the WORKSPACE file with the following:

http_archive(
    name = "io_tweag_rules_nixpkgs",
    strip_prefix = "rules_nixpkgs-244ae504d3f25534f6d3877ede4ee50e744a5234",
    urls = ["https://github.com/tweag/rules_nixpkgs/archive/244ae504d3f25534f6d3877ede4ee50e744a5234.tar.gz"],
)

load("@io_tweag_rules_nixpkgs//nixpkgs:repositories.bzl", "rules_nixpkgs_dependencies")
rules_nixpkgs_dependencies()

load("@io_tweag_rules_nixpkgs//nixpkgs:nixpkgs.bzl", "nixpkgs_git_repository", "nixpkgs_package", "nixpkgs_cc_configure")

load("@io_tweag_rules_nixpkgs//nixpkgs:toolchains/go.bzl", "nixpkgs_go_configure") # optional

nixpkgs_git_repository(
    name = "nixpkgs",
    revision = "23.11",
)

nixpkgs_cc_configure(
  repository = "@nixpkgs",
  name = "nixpkgs_config_cc",
  attribute_path = "clang",
)

This is the standard boilerplate to install rules_nixpkgs on our Bazel workspace. We’re also creating a reference to the nixpkgs repository, and a C++ toolchain using clang.

Next, we create a new cc_binary target in BUILD.bazel with a simple hello-world program.

$ cat BUILD.bazel
...
cc_binary(
    name = "hello-world",
    srcs = ["hello-world.cc"],
)

$ cat hello-world.cc
#include <iostream>

int main(int argc, char** argv) {
  std::cout << "Hello world!" << std::endl;
  return 0;
}

Now we need to update the custom Buildbarn toolchain used by the executors to reference @nixpkgs_config_cc. Update the file tools/remote-toolchains/BUILD.bazel and replace the instances of @remote_config_cc with @nixpkgs_config_cc.

We can try building the application using the C++ toolchain we defined with rules_nixpkgs. We expect this to fail because the executors are not Nix-aware yet.

$ bazel build --config=remote-ubuntu-22-04 @abseil-hello//:hello_main

...
ERROR: /home/user/.cache/bazel/_bazel_user/5ce2ca33a49034ed7557e24d70204ce5/external/com_google_absl/absl/base/BUILD.bazel:324:11: Compiling absl/base/internal/throw_delegate.cc failed: (Exit 34): Remote Execution Failure:
Invalid Argument: Failed to run command: Failed to start process: fork/exec /nix/store/n37gxbg343hxin3wdryx092mz2dkafy8-clang-wrapper-16.0.6/bin/cc: no such file or directory
...

Because the executors don’t have the /nix/store available, they cannot resolve the compiler path which is generated locally on our machine when we invoke bazel build.

Now let’s see how we can solve this problem by configuring the executors to access a shared /nix/store via NFS.

NFS-based solution

Our solution involves a Nix server that bridges this gap. This server manages and synchronizes the Nix dependencies across the Bazel build environment.

Here’s how it works:

  1. During bazel build the rules_nixpkgs repository rules will build and copy any Nix derivation to the remote Nix server.

  2. The Nix server will export the /nix/store directory tree via a read-only NFS mount share to the executors.

  3. When a build is triggered, all necessary dependencies are already available on the executors, allowing for the build process to continue.

Workflow overview

Implementation-wise, we’ll need to make the following changes to the Buildbarn infrastructure:

  • A Nix server. This could be a VM with Nix installed that is exporting the /nix/store directory as a read-only NFS share over the private network. We’ll need SSH access on that server from the machine that invokes bazel build.

  • Kubernetes executors with the exported NFS share mounted.

For a detailed setup guide and implementation specifics, refer to our infrastructure repository.

To instruct rules_nixpkgs to copy the nix derivations to the server we’ll need to create an entry in our SSH config (typically found under ~/.ssh/config) with the remote server and then set the environment variable BAZEL_NIX_REMOTE with the name of that entry.

# SSH Configuration
$ cat ~/.ssh/config
Host nix-server
  Hostname [public-ip]
  IdentityFile [ssh-private-key]
  Port [ssh-port]
  User [ssh-user]

Testing out remote execution again

With the new setup, we can try building the project again.

$ export BAZEL_NIX_REMOTE=nix-server
$ bazel clean --expunge # To refetch the Nix derivations
$ bazel build --config=remote-ubuntu-22-04 @abseil-hello//:hello_main

You should now see lines like the following, confirming communication with the Nix server

...
Analyzing: target @abseil-hello//:hello_main (0 packages loaded, 0 targets configured)
    Fetching repository @nixpkgs_config_cc_info; Remote-building Nix derivation 9s
...

And the build should be successful.

Conclusion

In this post, we explored the challenges and our solution for integrating rules_nixpkgs with remote execution in Bazel. Of course this solution is not perfect and it comes with some shortcomings that end user should be aware of.

  • The first issue is about cache eviction. Caching all the Nix paths over the long term is not practical from a storage standpoint. That’s why we need a way to mark the required paths, and garbage collect the others. A Nix path should be available as long as a client may trigger a remote build that uses it. However, there’s no way to determine when a client no longer needs a specific path. A simple solution will be to invalidate the least used paths. That will require a tighter integration with the Bazel APIs in order to track the Nix path usage.

  • The second issue relates to NFS performance. This depends on the infrastructure and workloads in operation. At least we want to tune the NFS synchronization to the point that the paths are available before any build begins. Slow synchronization between the NFS server and client can lead to failed builds.


  1. Bazel has an experimental feature that enables remotable repository rule actions. However, their capabilities are too limited to support the rules_nixpkgs use-case.

February 29, 2024 12:00 AM

February 27, 2024

Oleg Grenrus

More QualifiedDo examples

Posted on 2024-02-27 by Oleg Grenrus

Qualified do-notation, QualifiedDo, is a nice syntactical extension in GHC. Probably the best its property is that it changes semantics only locally, by using explicit "annotation": by qualifying the do keyword1. This means that enabling the extension doesn't change meaning of other & existing code.

I'll give two examples of QualifiedDo applications.

First example: COMPLETE pattern synonyms

GHC had long had PatternSynonyms. One use case for pattern synonyms is to provide backward compatibility when data type constructors change: preserving old constructor names and arguments as a compatibility pattern synonym.

For example, we used to have data Solo = Solo a. Recently the constructor was renamed to MkSolo to avoid name punning. To not break all the code using Solo constructor there compatibility pattern synonym was added:

pattern Solo :: a -> Solo
pattern Solo x = MkSolo x
{-# COMPLETE Solo #-}

The COMPLETE pragma says that a pattern match using Solo pattern synonym is complete, so we wouldn't get incomplete pattern match warnings2.

But COMPLETE support is (ironically) incomplete. If we have a do block like

broken :: Monad m => m (Solo a) -> m a
broken s = do
    Solo x <- s
    return x

the GHC will error because we don't have MonadFail instance (to desugar incomplete pattern match: Could not deduce (MonadFail m), that is GHC issue #15681). There are various workarounds, but I don't remember anyone mentioning QualifiedDo.

If we write a small helper module

module M ((>>=), (>>), fail) where

import Prelude ((>>=), (>>), Monad, String, error)
import GHC.Stack

fail :: (Monad m, HasCallStack) => String -> m a
fail = error

we can change broken into something which works:

import qualified M

works :: Monad m => m (Solo a) -> m a
works s = M.do
    Solo x <- s
    return x

Now if GHC needs to fail, it will simply error.

I hope that it's obvious that this is a band-aid: if you are relying on fail doing something useful (e.g. in Maybe), this will obviously break your program. But as QualifiedDo usage is explicitly annotated it's not a spooky action at the distance. And HasCallStack annotation should help you find the mistakes if any happen.

Second example: zero-overhead effects

At work I have been (adjacently) working with the code building on top of io-sim. TL;DR you write your code using (a lot of) type-classes, and then can either run your code in real IO (production) or in a simulator IOSim (for tests). But I'm getting slightly anxious thinking about having all I/O code being abstracted using type-classes making the true IO case potentially go slow. (This is mtl-like take on effect handling, but even effectful or something based on delimited continuations aren't zero-overhead: the overhead is there, just smaller).

What we truly want is a complete specialisation of effect-related type-classes, so there aren't any abstraction bits left when the use case is concrete (in mtl approach we can theoretically get there, but not in practice. In effectful or delimited-continuations a small cost is always there, but it doesn't rely that much on compiler optimising well).

Most likely, if your code isn't pushing both the I/O and CPU utilization at the same time, either approach will work ok. Compare that to data science done in Python: Python is a quite slow glue language, but it's combining bigger fast running "primitive" blocks. So if there is very little glue code, and the most work is done inside the abstracted primitives, the glue being tacky doesn't matter.

But can we do better?

In GHC we can do better using staging i.e. Typed Template Haskell (TTH). At first I was worried that TTH syntactic overhead will be off-putting until I remembered that QualifiedDo extension exists!

We can write code like:

import qualified SIO

example :: SIO.SIO i m => i FilePath -> m ()
example fn = SIO.do
  contents <- SIO.readFile fn
  SIO.putStr contents

that looks like normal Haskell. If we were forced to use >>= like operator explicitly, e.g. writing

example' :: SIO.SIO i m => i FilePath -> m ()
example' fn =
  SIO.readFile fn >>>= \contents ->
  SIO.putStr contents

it wouldn't be as nice.

The SIO type class has the part which looks almost like Monad, but not exactly:

class SIO i m | m -> i where
  (>>=)    :: m a -> (i a -> m b) -> m b

The "pure" values are wrapped inside type constructor i (for identity).

The readFile and putStr are also in the same type-class (could be different, doesn't really matter):

  readFile :: i FilePath -> m ByteString
  putStr   :: i ByteString -> m ()

We can have concrete instances, like IO (or actually IOSim) for tests:

instance SIO Identity IO where
  (>>=) :: forall a b. IO a -> (Identity a -> IO b) -> IO b
  (>>=) = coerce (bindIO @a @b)

  readFile = coerce BS.readFile
  putStr = coerce BS.putStr

But because we are liberated from the restricting shape of the Monad type class, we can have instance for CodeQ from template-haskell:

newtype CodeIO a = CodeIO { unCodeIO :: CodeQ (IO a) }

instance SIO CodeQ CodeIO where
  m >>= k     = CodeIO
    [|| bindIO $$(unCodeIO m) (\x -> $$(unCodeIO (k [|| x ||]))) ||]
  readFile fn = CodeIO [|| BS.readFile $$fn ||]
  putStr bs   = CodeIO [|| BS.putStr $$bs ||]

Then in our main production module we can splice the example in like

spliced :: FilePath -> IO ()
spliced fn = $$(SIO.unCodeIO $ SIO.do
    example [|| fn ||]
    example [|| fn ||])

and the generated code has no effect handling abstractions; in fact not even a Monad, as we used thenIO and bindIO building blocks:

spliced fn_a3kY =
    (GHC.Base.thenIO
       ((GHC.Base.bindIO (Data.ByteString.readFile fn_a3kY))
          (\ x_a3m2 -> Data.ByteString.putStr x_a3m2)))
      ((GHC.Base.bindIO (Data.ByteString.readFile fn_a3kY))
         (\ x_a3m3 -> Data.ByteString.putStr x_a3m3))

We have a precise control (but also a responsibility) to control the inlining of building blocks (i.e. if we want example let-bound first and then called twice, we must do that manually: power comes with responsibility). This is either a pro or con, depending on your POV. I think this is a pro if you go this far caring about the performance. If GHC Haskell had a type-class like mechanism with full monomorphisation guarantee, we'd would still like to to control inlining.

You may also worry that "wont staging generate a lot of code". Yes it will, but so would full monomorphisation (of templates in C++ or traits in Rust). It's a behaviour we arguably want, but it's GHC which may be worried and don't do too good job. With staging we could also do modular code-generation too, making layered type-class hierarchy, generating i.e. "pre-splicing" intermediate layers (layers like in three layer cake).

Conclusion

QualifiedDo is a neat GHC extension. We saw two more examples of its usage, where we want something like regular Monad desugaring, but which doesn't fit the Monad type-class. I also think we could have more of Qualified* syntactic extensions.


  1. In comparison ApplicativeDo applies globally. These design choices are probably not-so-intentional. For QualifiedDo it would require some additional setting to change all do statements in the source module (like -fplugin takes a module name). OTOH ApplicativeDo main motivation (using it with haxl) was to use it globally. But if you want to use it only in some do statement, you can't. Similarly OverloadedStrings applies to all string literals, and in the same way for all of them. Compare to Python which has kind of "QualifiedStrings" with string literals very differently: imagine writing T."this is text" but still having "this is string" :: String, without any type-class resolution.↩︎

  2. GHC doesn't try to reason about completeness through pattern synonyms: you may want to keep a pattern synonym group intentionally incomplete (so extending an otherwise abstract type with new ones isn't a breaking change), or to tell that something is complete (due to invariant you maintain, but GHC has no chance figuring out).↩︎

February 27, 2024 12:00 AM

February 23, 2024

GHC Developer Blog

GHC 9.8.2 is now available

GHC 9.8.2 is now available

Zubin Duggal - 2024-02-23

The GHC developers are happy to announce the availability of GHC 9.8.2. Binary distributions, source distributions, and documentation are available on the release page.

This release is primarily a bugfix release addressing many issues found in the 9.8 series. These include:

A full accounting of changes can be found in the release notes. As some of the fixed issues do affect correctness users are encouraged to upgrade promptly.

We would like to thank Microsoft Azure, GitHub, IOG, the Zw3rk stake pool, Well-Typed, Tweag I/O, Serokell, Equinix, SimSpace, Haskell Foundation, and other anonymous contributors whose on-going financial and in-kind support has facilitated GHC maintenance and release management over the years. Finally, this release would not have been possible without the hundreds of open-source contributors whose work comprise this release.

As always, do give this release a try and open a ticket if you see anything amiss.

Enjoy!

-Zubin

by ghc-devs at February 23, 2024 12:00 AM

February 22, 2024

Gabriella Gonzalez

Unification-free ("keyword") type checking

Unification-free ("keyword") type checking

From my perspective, one of the biggest open problems in implementing programming languages is how to add a type system to the language without significantly complicating the implementation.

For example, in my tutorial Fall-from-Grace implementation the type checker logic accounts for over half of the code. In the following lines of code report I’ve highlighted the modules responsible for type-checking with a :

$ cloc --by-file src/Grace/*.hs       

--------------------------------------------------------------------------------
File                                    blank        comment           code
--------------------------------------------------------------------------------
src/Grace/Infer.hs        ‡               499            334           1696
src/Grace/Type.hs         ‡                96             91            633
src/Grace/Syntax.hs                        61            163            543
src/Grace/Parser.hs                       166             15            477
src/Grace/Lexer.hs                         69             25            412
src/Grace/Normalize.hs                     47             48            409
src/Grace/Context.hs      ‡                72            165            249
src/Grace/Import.hs                        38              5            161
src/Grace/REPL.hs                          56              4            148
src/Grace/Interpret.hs                     30             28            114
src/Grace/Pretty.hs                        25             25            108
src/Grace/Monotype.hs     ‡                11             48             61
src/Grace/Location.hs                      16             15             60
src/Grace/TH.hs                            23             32             53
src/Grace/Value.hs                         12             53             53
src/Grace/Input.hs                         10              8             43
src/Grace/Compat.hs                         9              2             32
src/Grace/Existential.hs  ‡                12             23             25
src/Grace/Domain.hs       ‡                 4              7             20
--------------------------------------------------------------------------------
SUM:                                     1256           1091           5297
--------------------------------------------------------------------------------

That’s 2684 lines of code (≈51%) just for type-checking (and believe me: I tried very hard to simplify the type-checking code).

This is the reason why programming language implementers will be pretty keen to just not implement a type-checker for their language, and that’s how we end up with a proliferation of untyped programming languages (e.g. Godot or Nix), or ones that end up with a type system bolted on long after the fact (e.g. TypeScript or Python). You can see why someone would be pretty tempted to skip implementing a type system for their language (especially given that it’s an optional language feature) if it’s going to balloon the size of their codebase.

So I’m extremely keen on implementing a “lean” type checker that has a high power-to-weight ratio. I also believe that a compact type checker is an important foundational step for functional programming to “go viral” and displace imperative programming. This post outlines one approach to this problem that I’ve been experimenting with1.

Unification

The thing that bloats the size of most type-checking implementations is the need to track unification variables. These variables are placeholders for storing as-yet-unknown information about something’s type.

For example, when a functional programming language infers the type of something like this Grace expression:

(λx → x) true

… the way it typically works is that it will infer the type of the function (λx → x) which will be:

λx → x : α → α

… where α is a unification variable (an unsolved type). So you can read the above type annotation as saying “the type of λx → x is a function from some unknown input type (α) to the same output type (α).

Then the type checker will infer the type of the function’s input argument (true) which will be:

true : Bool

… and finally the type checker will combine those two pieces of information and reason about the final type like this:

  • the input to the function (true) is a Bool
  • therefore the function’s input type (α) must also be Bool
  • therefore the function’s output type (α) must also be Bool
  • therefore the entire expression’s type is Bool

… which gives the following conclusion of type inference:

(λx → x) true : Bool

However, managing unification variables like α is a lot trickier than it sounds. There are multiple unification algorithms/frameworks in the wild but the problem with all of them is that you have to essentially implement a bespoke logic programming language (with all of the complexity that entails). Like, geez, I’m already implementing a programming language and I don’t want to have to implement a logic programming language on top of that just to power my type-checker.

So there are a couple of ways I’ve been brainstorming how to address this problem and one idea I had was: what if we could get rid of unification variables altogether?

Deleting unification

Alright, so this is the part of the post that requires some familiarity/experience with implementing a type-checker. If you’re somebody new to programming language theory then you can still keep reading but this is where I have to assume some prior knowledge otherwise this post will get way too long.

The basic idea is that you start from the “Complete and Easy” bidirectional type checking algorithm which is a type checking algorithm that does use unification variables2 but is simpler than most type checking algorithms. The type checking rules look like this (you can just gloss over them):

Now, delete all the rules involving unification variables. Yes, all of them. That means that all of the type-checking judgments from Figures 9 and 10 are gone and also quite a few rules from Figure 11 disappear, too.

Surprisingly, you can still type check a lot of code with what’s left, but you lose two important type inference features if you do this:

  • you can no longer infer the types of lambda arguments

  • you can no longer automatically instantiate polymorphic code

… and I’ll dig into those two issues in more detail.

Inferring lambda argument types

You lose the ability to infer the type of a function like this one when you drop support for unification variables:

λx → x == False

Normally, a type checker that supports unification can infer that the above function has type Bool → Bool, but (in general) a type checker can no longer infer that when you drop unification variables from the implementation.

This loss is not too bad (in fact, it’s a pretty common trade-off proposed in the bidirectional type checking literature) because you can make up for it in a few ways (all of which are easy and efficient to implement in a type checker):

  • You can allow the input type to be inferred if the lambda is given an explicit type annotation, like this:

    λx → x == False : BoolBool

    More generally, you can allow the input type to be inferred if the lambda is checked against an expected type (and a type annotation is one case, but not the only case, where a lambda is checked against an expected type).

    We’re going to lean on this pretty heavily because it’s pretty reasonable to ask users to provide type annotations for function definitions and also because there are many situations where we can infer the expected type of a lambda expression from its immediate context.

  • You can allow the user to explicitly supply the type of the argument

    … like this:

    λ(x : Bool) → x == False

    This is how Dhall works, although it’s not as ergonomic.

  • You can allow the input type to be inferred if the lambda is applied to an argument

    This is not that interesting, but I’m mentioning it for completeness. The reason it’s not interesting is because you won’t often see expressions of the form (λx → e) y in the wild, because they can more idiomatically be rewritten as let x = y in e.

Instantiating polymorphic code

The bigger issue with dropping support for unification variables is: all user-defined polymorphic functions now require explicit type abstraction and explicit type application, which is a major regression in the type system’s user experience.

For example, in a language with unification variables you can write the polymorphic identity function as:

λx → x

… and use it like this3:

let id = λx → x
in  (id true, id 1)

… but when you drop support for unification variables then you have to do something like this:

let id = λ(a : Type) → λ(x : a) → x
in  (id Bool true, id Natural 1)

Most programmers do NOT want to program in a language where they have to explicitly manipulate type variables in this way. In particular, they really hate explicit type application. For example, nobody wants to write:

map { x : Bool, … large record … } Bool (λr → r.x) rs

So we need to figure out some way to work around this limitation.

The trick

However, there is a solution that I believe gives a high power-to-weight ratio, which I will refer to as “keyword” type checking:

  • add a bunch of built-in functions

    Specifically, add enough built-in functions to cover most use cases where users would need a polymorphic function.

  • add special type-checking rules for those built-in functions when they’re fully saturated with all of their arguments

    These special-cased type-checking rules would not require unification variables.

  • still require explicit type abstraction when these built-in functions are not fully saturated

    Alternatively, you can require that built-in polymorphic functions are fully saturated with their arguments and make it a parsing error if they’re not.

  • still require explicit type abstraction and explicit type application for all user-defined (i.e. non-builtin) polymorphic functions

  • optionally, turn these built-in functions into keywords or language constructs

I’ll give a concrete example: the map function for lists. In many functional programming languages this map function is not a built-in function; rather it’s defined within the host language as a function of the following type:

map : ∀(a b : Type) → (a → b) → List a → List b

What I’m proposing is that the map function would now become a built-in function within the language and you would now apply a special type-checking rule when the map function is fully saturated:

Γ ⊢ xs ⇒ List a   Γ ⊢ f ⇐ a → b
───────────────────────────────
Γ ⊢ map f xs ⇐ List b

In other words, we’re essentially treating the map built-in function like a “keyword” in our language (when it’s fully saturated). Just like a keyword, it’s a built-in language feature that has special type-checking rules. Hell, you could even make it an actual keyword or language construct (e.g. a list comprehension) instead of a function call.

I would even argue that you should make each of these special-cased builtin-functions a keyword or a language construct instead of a function call (which is why I call this “keyword type checking” in the first place). When viewed through this lens the restrictions that these polymorphic built-in functions (A) are saturated with their arguments and (B) have a special type checking judgment are no different than the restrictions for ordinary keywords or language constructs (which also must be saturated with their arguments and also require special type checking judgments).

To make an analogy, in many functional programming languages the if/then/else construct has this same “keyword” status. You typically don’t implement it as a user-space function of this type:

ifThenElse : ∀(a : Type) → Bool → a → a → a

Rather, you define if as a language construct and you also add a special type-checking rule for if:

Γ ⊢ b ⇐ Bool   Γ ⊢ x ⇒ a   Γ ⊢ y ⇐ a
────────────────────────────────────
Γ ⊢ if b then x else y ⇒ a

… and what I’m proposing is essentially greatly exploding the number of “keywords” in the implementation of the language by turning a whole bunch of commonly-used polymorphic functions into built-in functions (or keywords, or language constructs) that are given special type-checking treatment.

For example, suppose the user were to create a polymorphic function like this one:

let twice = λ(a : Type) → λ(x : a) → [ x, x ]

in  twice (List Bool) (twice Bool true)

That’s not very ergonomic to define and use, but we also can’t reasonably expect our programming language to provide a twice built-in function. However, our language could provide a generally useful replicate builtin function (like Haskell’s replicate function):

replicate : ∀(a : Type) → Natural → a → List a

… with the following type-checking judgment:

Γ ⊢ n ⇐ Natural   Γ ⊢ x ⇒ a
───────────────────────────
Γ ⊢ replicate n x ⇒ List a

… and then you would tell the user to use replicate directly instead of defining their own twice function:

replicate 2 (replicate 2 true)

… and if the user were to ask you “How do I define a twice synonym for replicate 2” you would just tell them “Don’t do that. Use replicate 2 directly.”

Conclusion

This approach has the major upside that it’s much easier to implement a large number of keywords than it is to implement a unification algorithm, but there are other benefits to doing this, too!

  • It discourages complexity and fragmentation in user-space code

    Built-in polymorphic functions have an ergonomic advantage over user-defined polymorphic functions because under this framework type inference works better for built-in functions. This creates an ergonomic incentive to stick to the “standard library” of built-in polymorphic functions, which in turn promotes an opinionated coding style across all code written in that language.

    You might notice that this approach is somewhat similar in spirit to how Go handles polymorphism which is to say: it doesn’t handle user-defined polymorphic code well. For example, Go provides a few built-in language features that support polymorphism (e.g. the map data structure and for loops) but if users ask for any sort of user-defined polymorphism then the maintainers tell them they’re wrong for wanting that. The main difference here is that (unlike Go) we do actually support user-defined polymorphism; it’s not forbidden, but it is less ergonomic than sticking to the built-in utilities that support polymorphism..

  • It improves error messages

    When you special-case the type-checking logic you can also special-case the error messages, too! With general-purpose unification the error message can often be a bit divorced from the user’s intent, but with “keyword type checking” the error message is not only more local to the problem but it can also suggest highly-specific tips or fixes appropriate for that built-in function (or keyword or language construct).

  • It can in some cases more closely match the expectations of imperative programmers

    What I mean is: most programmers coming from an imperative and typed background are used to languages where (most of the time) polymorphism is “supported” via built-in language constructs and keywords and user-defined polymorphism might be supported but considered “fancy”. Leaning on polymorphism via keywords and language constructs would actually make them more comfortable using polymorphism instead of trying to teach them how to produce and consume user-defined polymorphic functions.

    For example, in a lot of imperative languages the idiomatic solution for how to do anything with a list is “use a for loop” where you can think of a for loop as a built-in keyword that supports polymorphic code. The functional programming equivalent of “just use a for loop” would be something like “just use a list comprehension” (where a list comprehension is a “keyword” that supports polymorphic code that we can give special type checking treatment).

That said, this approach is still more brittle than unification and will require more type annotations in general. The goal here isn’t to completely recover the full power of unification but rather to get something that’s not too bad but significantly easier to implement.

I think this “keyword type checking” can potentially occupy a “low tech” point in the type checking design space for functional programming languages that need to have efficient and compact implementations (e.g. for ease of embedding). Also, this can potentially provide a stop-gap solution for novice language implementers that want some sort of a type system but they’re not willing to commit to implementing a unification-based type system.

There’s also variation on this idea which Verity Scheel has been exploring, which is to provide userland support for defining new functions with special type-checking rules and there’s a post from her outlining how to do that:

User Operators with Implicits & Overloads


  1. The other approach is to create essentially an “ABNF for type checkers” that would let you write type-checking judgments in a standard format that could generate the corresponding type-checking code in multiple languages. That’s still a work-in-progress, though.↩︎

  2. I believe some people might take issue with calling these unification variables because they consider bidirectional type checking as a distinct framework from unification. Moreover, in the original bidirectional type checking paper they’re called “unsolved” variables rather than unification variables. However, I feel that for the purpose of this post it’s still morally correct to refer to these unsolved variables as unification variables since their usage and complexity tradeoffs are essentially identical to unification variables in traditional unification algorithms.↩︎

  3. … assuming let expressions are generalized.↩︎

by Gabriella Gonzalez (noreply@blogger.com) at February 22, 2024 04:04 PM

February 21, 2024

Well-Typed.Com

The Haskell Unfolder Episode 20: Dijkstra's shortest paths

Today, 2024-02-21, at 1930 UTC (11:30 am PST, 2:30 pm EST, 7:30 pm GMT, 20:30 CET, …) we are streaming the 20th episode of the Haskell Unfolder live on YouTube.

The Haskell Unfolder Episode 20: Dijkstra’s shortest paths

In this (beginner-friendly) episode, we will use Dijkstra’s shortest paths algorithm as an example of how one can go about implementing an algorithm given in imperative pseudo-code in idiomatic Haskell. We will focus on readability, not on performance.

About the Haskell Unfolder

The Haskell Unfolder is a YouTube series about all things Haskell hosted by Edsko de Vries and Andres Löh, with episodes appearing approximately every two weeks. All episodes are live-streamed, and we try to respond to audience questions. All episodes are also available as recordings afterwards.

We have a GitHub repository with code samples from the episodes.

And we have a public Google calendar (also available as ICal) listing the planned schedule.

by andres, edsko at February 21, 2024 12:00 AM

February 18, 2024

Haskell Interlude

43: Ivan Perez

In this episode, Wouter and Andres interview Ivan Perez, a senior research scientist at NASA. Ivan tells us about how NASA uses Haskell to develop the Copilot embedded domain specific language for runtime verification, together with some of the obstacles he encounters getting to end users to learn Haskell and adopt such an EDSL.

February 18, 2024 12:00 PM

February 16, 2024

Mark Jason Dominus

Etymology roundup 2024-02

The Recurse Center Zulip chat now has an Etymology channel, courtesy of Jesse Chen, so I have been posting whenever I run into something interesting. This is a summary of some of my recent discoveries. Everything in this article is, to the best of my knowledge, accurate. That is, there are no intentional falsehoods.

Baba ghanouj

I tracked down the meaning of (Arabic) baba ghanouj. It was not what I would have guessed.

Well, sort of. Baba is “father” just like in every language. I had thought of this and dismissed it as unlikely. (What is the connection with eggplants?) But that is what it is.

And ghanouj is …
“coquetry”.

So it's the father of coquetry.

Very mysterious.

Eggnog

Toph asked me if “nog” appeared in any word other than “eggnog”. Is there lemonnog or baconnog? I had looked this up before but couldn't remember what it was except that it was some obsolete word for some sort of drink.

“Nog” is an old Norfolk (England) term for a kind of strong beer which was an ingredient in the original recipe, sometime in the late 17th or early 18th century.

I think modern recipes don't usually include beer.

Wow

“Wow!” appears to be an 18th-century borrowing from an indigenous American language, because most of its early appearances are quotes from indigenous Americans. It is attested in standard English from 1766, spelled “waugh!”, and in Scots English from 1788, spelled “vow!”

Riddles

Katara asked me for examples of words in English like “bear” where there are two completely unrelated meanings. (The word bear like to bear fruit, bear children, or bear a burden is not in any way related to the big brown animal with claws.)

There are a zillion examples of this. They're easy to find in a paper dictionary: you just go down the margin looking for a superscript. When you see “bear¹” and “bear²”, you know you've found an example.

The example I always think of first is “venery” because long, long ago Jed Hartman pointed it out to me: venery can mean stuff pertaining to hunting (it is akin to “venison”) and it can also mean stuff pertaining to sex (akin to “venereal”) and the fact that these two words are spelled the same is a complete coincidence.

Jed said “I bet this is a really rare phenomenon” so I harassed him for the next several years by emailing him examples whenever I happened to think of it.

Anyway, I found an excellent example for Katara that is less obscure than “venery”: “riddle” (like a puzzling question) has nothing to do with when things are riddled with errors. It's a complete coincidence.

The “bear” / “bear” example is a nice simple one, everyone understands it right away. When I was studying Korean I asked my tutor an etymology question, something like whether the “eun” in eunhaeng 은행, “bank”, was the same word as “eun” 은 which means “silver”. He didn't understand the question at first: what did I mean, “is it the same word”?

I gave the bear / bear example, and said that to bear fruit and to bear children are the same word, but the animal with claws is a different word, and just a coincidence that it is spelled the same way. Then he understood what I meant.

(Korean eunhaeng 은행 is a Chinese loanword, from 銀行. 銀 is indeed the word for silver, and 行 is a business-happening-place.)

Right and left

The right arm is the "right" arm because, being the one that is (normally) stronger and more adept, it is the right one to use for most jobs.

But if you ignore the right arm, there is only one left, so that is the "left" arm.

This sounds like a joke, but I looked it up and it isn't.

Leave and left

"Left" is the past tense passive of "leave". As in, I leave the room, I left the room, when I left the room I left my wallet there, my wallet was left, etc.

(As noted above, this is also where we get the left side.)

There are two other words "leave" in English. Leaves like the green things on trees are not related to leaving a room.

(Except I was once at a talk by J.H. Conway in which he was explaining some sort of tree algorithm in which certain nodes were deleted and he called the remaining ones "leaves" because they were the ones that were left. Conway was like that.)

The other "leave" is the one that means "permission" as in "by your leave…". This is the leave we find in "sick leave" or "shore leave". They are not related to the fact that you have left on leave, that is a coincidence.

Normal norms

Latin norma is a carpenter's square, for making sure that things are at right angles to one another.

So something that is normal is something that is aligned the way things are supposed to be aligned, that is to say at right angles. And a norm is a rule or convention or standard that says how things ought to line up.

In mathematics and physics we have terms like “normal vector”, “normal forces” and the like, which means that vectors or forces are at right angles to something. This is puzzling if you think of “normal” as “conventional” or “ordinary” but becomes obvious if you remember the carpenter's square.

In contrast, mathematical “normal forms” have nothing to do with right angles, they are conventional or standard forms. “Normal subgroups” are subgroups that behave properly, the way subgroups ought to.

The names Norman and Norma are not related to this. They are related to the surname Norman which means a person from Normandy. Normandy is so-called because it was inhabited by Vikings (‘northmen’) starting from the 9th century.

Hydrogen and oxygen

Jesse Chen observed that hydrogen means “water-forming”, because when you burn it you get water.

A lot of element names are like this. Oxygen is oxy- (“sharp” or “sour”) because it makes acids, or was thought to make acids. In German the analogous calque is “sauerstoff”.

Nitrogen makes nitre, which is an old name for saltpetre (potassium nitrate). German for nitre seems to be salpeter which doesn't work as well with -stoff.

The halogen gases are ‘salt-making’. (Greek for salt is hals.) Chlorine, for example, is a component of table salt, which is sodium chloride.

In Zulip I added that The capital of Denmark, Copenha-gen, is so-called because in the 11th century is was a major site for the production of koepenha, a Germanic term for a lye compound, used in leather tanning processes, produced from bull dung. I was somewhat ashamed when someone believed this lie despite my mention of bull dung.

Spas, baths, and coaches

Spas (like wellness spa or day spa) are named for the town of Spa, Belgium, which has been famous for its cold mineral springs for thousands of years!

(The town of Bath England is named for its baths, not the other way around.)

The coach is named for the town of Kocs (pronounced “coach”), Hungary, where it was invented. This sounds like something I would make up to prank the kids, but it is not.

Spanish churches

“Iglesia” is Spanish for “church”, and you see it as a surname in Spanish as in English. (I guess, like “Church”, originally the name of someone who lived near a church).

Thinking on this, I realized: “iglesia” is akin to English “ecclesiastic”.

They're both from ἐκκλησία which is an assembly or congregation.

The mysterious Swedish hedgehog

In German, a hedgehog is “Igel”. This is a very ancient word, and several other Germanic languages have similar words. For example, in Frisian it's “ychel”.

In Swedish, “igel” means leech. The hedgehog is “igelkott”.

I tried to find out what -kott was about. “kotte” is a pinecone and may be so-called because “kott” originally meant some rounded object, so igelkott would mean the round igel rather than the blood igel, which is sometimes called blodigel in Swedish.

I was not able to find any other words in Swedish with this sense of -kott. There were some obviously unrelated words like bojkott (“boycott”). And there are a great many Swedish words that end in -skott, which is also unrelated. It means “tail”. For example, the grip of a handgun is revolverskott.

[ Addendum: Gustaf Erikson advises me that I have misunderstood ‑skott; see below. ]

Bonus hedgehog weirdness: In Michael Moorcock's Elric books, Elric's brother is named “Yyrkoon”. The Middle English for a hedgehog is “yrchoun” (variously spelled). Was Moorcock thinking of this? The -ch- in “yrchoun” is t͡ʃ though, which doesn't match the stop consonant in “Yyrkoon”. Also which makes clear that “yrchoun” is just a variant spelling of “urchin”. (Compare “sea urchin”, which is a sea hedgehog. or compare “street urchin”, a small round bristly person who scuttles about in the gutter.)

In Italian a hedgehog is riccio, which I think is also used as a nickname for a curly-haired or bristly-haired person.

Slobs and schlubs

These are not related. Schlub is originally Polish, coming to English via (obviously!) Yiddish. But slob is Irish.

-euse vs. -ice

I tried to guess the French word for a female chiropractor. I guessed “chiropracteuse" by analogy with masseur, masseuse, but I was wrong. It is chiropractrice.

The '‑ice' suffix was clearly descended from the Latin '‑ix' suffix, but I had to look up ‘‑euse’. It's also from a Latin suffix, this time from ‘‑osa’.

Jot

When you jot something down on a notepad, the “jot” is from Greek iota, which is the name of the small, simple letter ι that is easily jotted.

Bonus: This is also the jot that is meant by someone who says “not a jot or a tittle”, for example Matthew 5:18 (KJV):

For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled.

A tittle is the dot above the lowercase ‘i’ or ‘j’. The NIV translates this as “not the smallest letter, not the least stroke of a pen”, which I award an A-plus for translation.

Vilifying villains

I read something that suggested that these were cognate, but they are not.

“Vilify” is from Latin vīlificō which means to vilify. It is a compound of vīlis (of low value or worthless, I suppose the source of “vile”) and faciō (to make, as in “factory” and “manufacture”.)

A villain, on the other hand, was originally just a peasant or serf; that is, a person who lives in a village. “Village” is akin to Latin villa, which originally meant a plantation.

Döner kebab

I had always assumed that “Döner” and its “ö” were German, but they are not, at least not originally. “Döner kebab” is the original Turkish name of the dish, right down to the diaresis on the ‘ö’, which is the normal Turkish spelling; Turkish has an ‘ö’ also. Döner is the Turkish word for a turning-around-thing, because döner kebab meat roasts on a vertical spit from which it is sliced off as needed.

“Döner” was also used in Greek as a loanword but at some point the Greeks decided to use the native Greek word gyro, also a turning-around-thing, instead. Greek is full of Turkish loanwords. (Ottoman Empire, yo.)

“Shawarma”, another variation on the turning-around-vertical-spit dish, is from a different Ottoman Turkish word for a turning-around thing, this time چویرمه (çevirme), which is originally from Arabic.

The Armenian word for shawarma is also shawarma, but despite Armenian being full of Turkish loanwords, this isn't one. They got it from Russian.

Everyone loves that turning-on-a-vertical-spit dish. Lebanese immigrants brought it to Mexico, where it is served in tacos with pineapple and called tacos al pastor (“shepherd style”). I do not know why the Mexicans think that Lebanese turning-around-meat plus pineapples adds up to shepherds. I suppose it must be because the meat is traditionally lamb.

Roll call

To roll is to turn over with a circular motion. This motion might wind a long strip of paper into a roll, or it might roll something into a flat sheet, as with a rolling pin. After rolling out the flat sheet you could then roll it up into a roll.

Dinner rolls are made by rolling up a wad of bread dough.

When you call the roll, it is because you are reading a list of names off a roll of paper.

Theatrical roles are from French rôle which seems to have something to do with rolls but I am not sure what. Maybe because the cast list is a roll (as in roll call).

Wombats and numbats

Both of these are Australian animals. Today it occurred to me to wonder: are the words related? Is -bat a productive morpheme, maybe a generic animal suffix in some Australian language?

The answer is no! The two words are from different (although distantly related) languages. Wombat is from Dharug, a language of the Sydney area. Numbat is from the Nyungar language, spoken on the other end of the continent.

Addendum

Gustaf Erikson advises me that I have misunderstood ‑skott. It is akin to English shoot, and means something that springs forth suddenly, like little green shoots in springtime, or like the shooting of an arrow. In the former sense, it can mean a tail or a sticking-out thing more generally. But in revolverskott is it the latter sense, the firing of a revolver.

by Mark Dominus (mjd@plover.com) at February 16, 2024 09:05 AM

February 06, 2024

Tweag I/O

Evaluating Retrieval in RAGs: A Gentle Introduction

llama on rag
No, not this RAG.

Despite their many capabilities, Large Language Models (LLMs) have a serious limitation: they’re stuck in time and their knowledge is limited to the data they have been trained on.

Updating the knowledge of an LLM can take two forms: fine-tuning, which we will address in a future post, and the ever-present RAG. RAG, short for Retrieval Augmented Generation, has garnered a lot of attention in the GenAI community and for good reasons. You “simply” hook the LLM up to your documents (more on that later), and it can suddenly tackle any question, as long as the answer is somewhere in the documents.

This is almost too good to be true: it offers endless possibilities, a simple concept and, thanks to advances in the tooling ecosystem, a straightforward implementation. It is hard to imagine at first sight how it could go wrong.

Yet wrong it goes, and we have seen it happen consistently with our chatbots, as well as SaaS products that we have tested.

In this article, the first of a series on evaluation in LLMs, we will unpack how retrieval impacts the performance of RAG systems, why we need systematic evaluation and what the different schools and frameworks of evaluation are. If you’ve been wondering about evaluating your own RAG system and needed an introduction, look no further.

The perfect RAG assumptions

Simply put, a RAG system retrieves documents similar to your query and uses them to generate a response (see Figure 1).

RAG data
Figure 1. Data flow in a RAG system.

For this to work perfectly, the following assumptions should hold:

  • In retrieval, you need to retrieve relevant data, all the relevant data and nothing but the relevant data.
  • In generation, the LLM should know enough about the topic to synthesize retrieved documents, yet be capable of changing its knowledge when confronted with conflicting or updated evidence.

Good retrieval is vital for a good RAG system. If you feed garbage into your LLM, you should not be too surprised when it spouts garbage back at you. But good retrieval becomes even more essential when using smaller LLMs. These models are not always the best at identifying and filtering irrelevant context.

Retrieval can indeed be one of the weakest parts of a RAG system. Despite the hype around vector databases and semantic search, the problem of knowledge indexing is still far from being solved.

Retrieval, semantic search and everything in between

Because the context an LLM can take is limited, stuffing your whole knowledge base in a prompt is not an option. Even if it were, LLMs are not as good with extracting information from a long piece of text as they are with shorter contexts. This is why retrieval is needed to find the documents that are most relevant to your query.

While this can be done with good old keyword search, semantic search is becoming increasingly the norm for RAG applications. This makes sense. Suppose you ask, “Why do we need search in RAGs?” Despite the absence of the exact words from the query, semantic search may be able to find the previous paragraph as it is semantically aligned with the query. On the other hand, keyword search will fail as neither “search” nor “RAG” are in the text.

In practice, the process is a bit more involved:

  • Documents in a knowledge base are divided into smaller chunks.
  • An embedding model is used to “vectorize” these chunks.
  • These vectors are indexed into a vector database.

Upon receiving a query:

  • The query is vectorized using the same embedding model.
  • The closest vectors are retrieved from the vector database.

Experiments vs. Eyeballing: or why do we need evaluation anyway?

developer eyeballing
Figure 2. Eyeballing, aka changing the code until it works.

We’ve been to so many demos and presentations where questions about evaluation were answered with a variation of “evaluation is on our future agenda” or “we changed the [prompt|chain|model|temperature] until the answer looked good”1 that we internally coined a term for this: eyeballing™.

When performing “eyeballing”, the most probable scenario is that someone, likely the engineer working on the RAG app, tested the app with some queries. For one or more of those, the generated answer was subpar. The engineer randomly debugs these cases, and finds one or more of the following problems:

  • Retrieved references are not relevant to the query.
  • The answer is not truthful to the retrieved content.
  • The answer does not address the question.

The engineer changes something in the implementation, and now the answer looks better (for some, most probably vague definition of the notion of better).

There are many problems with this approach:

  • No benchmark: There is no guarantee that the introduced change did not degrade performance on other questions.
  • No experiment tracking: Likely none of the intermediate states were committed or properly tracked. So we don’t know what combinations of parameters were tested.
  • No evaluation metrics: In the absence of an evaluation framework that defines the notion of “better”, we cannot numerically compare the current RAG state to any other possible state.

The closest software engineering metaphor to the eyeballing approach is manually testing every change applied to the code without having a proper test suite.

The two schools of evaluation: human vs. machine

schools of evaluation

By now it should be clear why evaluation of RAG systems is a must. The question has been approached from various angles and with different evaluation metrics and strategies. We can distinguish, however, a division along the line of whether the evaluator, or the oracle (as it’s typically referred to in Machine Learning and Expert Systems), is a human or an LLM.

  • In human-based evaluation, a human labeler rates the relevance of retrieved documents, either repeatedly (for every experimental setting), or as a one-off, by creating a benchmark of queries and associated relevant documents.
  • In LLM-based evaluation, it is an LLM, usually one that is powerful enough, that evaluates if and how the retrieved content is relevant to a query.

Building a benchmark

Note that in both cases, you need a benchmark to evaluate the RAG against. With LLM-based evaluation, this is usually a set of queries over the documents database. In human-based evaluation, benchmarks can be more elaborate (more on that further below).

Building a useful benchmark is not an easy task. One should balance the types of queries asked, their statistical incidence over the database and the value in catering to a specific subset of queries as opposed to doing a good job over all queries. Exploring these considerations is beyond the scope of this post.

Human-based evaluation

Human-based evaluation is closer to the evaluation paradigm in classic Machine Learning. One can easily apply evaluation metrics originally devised for Information Retrieval. These should be adapted to the RAG retrieval setting where only the k top documents are passed on to the LLM as context and the order in which these documents are retrieved is not relevant. Instead of raw recall and precision, we should instead think of those as a function of k.

A higher precision at k means less noise is mixed with the signal, while a higher recall means that more relevant information is retrieved. Since k is fixed, these should go hand in hand.

Besides using k as a threshold, we can also consider other parameters such as the threshold of similarity between the query and retrieved documents.

Note that despite this approach being more demanding in time, automation of evaluation is still possible once a one-off benchmark is created and evaluation metrics are defined.

LLM-based evaluation

LLM-based evaluation is easier to set-up and automate since it does not require any human involvement beyond the creation of a benchmark of queries. This is the core of the RAGAS and TruLens evaluation frameworks that we will discuss below.

TruLens

rag triad
Figure 3. The RAG evaluation triad.

TruLens defines a golden triad of RAG evaluation (see Figure 3). Let’s discuss in particular retrieval relevance. The idea is to quantify how much the retrieved content is relevant to the query by computing the ratio of relevant to total sentences in the retrieved documents. It is an LLM that determines whether a sentence is needed to answer a query.

RAGAS

RAGAS defines an evaluation matrix over both retrieval and generation, two of those are retrieval evaluation metrics, namely: context relevancy (which is similar to the one defined by TruLens) and context recall.

Context recall is defined as the ratio of statements in the retrieved documents out of the statements in a “model” answer. This model answer should be provided in a “human”-crafted ground truth and the approach is therefore a hybrid human-LLM one. An LLM is responsible for extracting and comparing statements from the retrieved context and the model answer.

Limitations of LLM-based evaluation

llama not understanding quenya
Llamas do not understand Quenya

A fundamental unspoken assumption that underlies using LLMs to evaluate retrieval is that the LLM knows enough about the question and the context to make a judgement on their relevance. This assumption is hard to justify in the context e.g. of fairly technical documentation that the model has not seen before or in a subject the model is not fluent in.

Take the following passage written in Quenya, a fictional language Tolkien invented in Lord of the Rings:

Alcar i cala elenion ancalima. Varda Elentári, Tintallë, tiris ninqe eleni. Lórien omentieva Yavanna Kementári. Eärendil elenion ancalima, perina i oiolossë.2

And take this query:

Man enyalië Varda Elentári tiris eleni?3

Can you tell if the context is relevant to the query?

This is admittedly a constructed example, but we have seen similar cases in play while evaluating a chatbot over Bazel documentation.

This approach has the additional pitfall of not taking into account that, even if the retrieved context is relevant to the query, this does not measure recall: how many of the existing relevant documents are in the database, or how much information required to answer the question is retrieved. While the RAGAS recall metric attempts to mitigate this, crafting answers to fairly technical topics or those that require intimate knowledge of a domain or a knowledge base is both hard and time-consuming. It also does not take into account that the crafted answer might be correct without necessarily including all relevant bits of relevant information in a knowledge base.

Conclusion

The evaluation of RAG systems presents a unique sets of challenges but its value in building usable apps cannot be overstated.

The evaluation frameworks we discussed, both human-based and LLM-based, each have their own advantages and limitations. Human-based evaluations, while thorough and more trustworthy are labor-intensive and hard to repeat. LLM-based evaluations, on the other hand, are much more scalable and can easily be repeated but they rely heavily on LLMs, which have their own biases and limitations.

Stay tuned for the next post in this series, where we present our in-house evaluation framework and share insights and results from real-world cases.


  1. I (Nour) have been to a conference lately where someone said they were using RAGAS, and I had a hard time containing my excitement.
  2. “The glory of the light of the stars is brightest. Varda, Queen of the Stars, Kindler, watches over the sparkling stars. Lórien met Yavanna, Queen of the Earth. Eärendil, brightest of stars, sailed on the everlasting night.”
  3. “Who called Varda, the star queen, watcher of the stars?”

February 06, 2024 12:00 AM

February 03, 2024

Magnus Therning

Bending Warp

In the past I've noticed that Warp both writes to stdout at times and produces some default HTTP responses, but I've never bothered taking the time to look up what possibilities it offers to changes this behaviour. I've also always thought that I ought to find out how Warp handles signals.

If you wonder why this would be interesting to know there are three main points:

  1. The environments where the services run are set up to handle structured logging. In our case it should be JSONL written to stdout, i.e. one JSON object per line.
  2. We've decided that the error responses we produce in our code should be JSON, so it's irritating to have to document some special cases where this isn't true just because Warp has a few default error responses.
  3. Signal handling is, IMHO, a very important part of writing a service that runs well in k8s as it uses signals to handle the lifetime of pods.

Looking through the Warp API

Browsing through the API documentation for Warp it wasn't too difficult to find the interesting pieces, and that Warp follows a fairly common pattern in Haskell libraries

  • There's a function called runSettings that takes an argument of type Settings.
  • The default settings are available in a variable called defaultSettings (not very surprising).
  • There are several functions for modifying the settings and they all have the same shape

    setX :: X -> Settings -> Settings.
    

    which makes it easy to chain them together.

  • The functions I'm interested in now are
    setOnException
    the default handler, defaultOnException, prints the exception to stdout using its Show instance
    setOnExceptionResponse
    the default responses are produced by defaultOnExceptionResponse and contain plain text response bodies
    setInstallShutdownHandler
    the default behaviour is to wait for all ongoing requests and then shut done
    setGracefulShutdownTimeout
    sets the number of seconds to wait for ongoing requests to finnish, the default is to wait indefinitely

Some experimenting

In order to experiment with these I put together a small API using servant, app, with a main function using runSettings and stringing together a bunch of modifications to defaultSettings.

main :: IO ()
main = Log.withLogger $ \logger -> do
    Log.infoIO logger "starting the server"
    runSettings (mySettings logger defaultSettings) (app logger)
    Log.infoIO logger "stopped the server"
  where
    mySettings logger = myShutdownHandler logger . myOnException logger . myOnExceptionResponse

myOnException logs JSON objects (using the logging I've written about before, here and here). It decides wether to log or not using defaultShouldDisplayException, something I copied from defaultOnException.

myOnException :: Log.Logger -> Settings -> Settings
myOnException logger = setOnException handler
  where
    handler mr e = when (defaultShouldDisplayException e) $ case mr of
        Nothing -> Log.warnIO logger $ lm $ "exception: " <> T.pack (show e)
        Just _ -> do
            Log.warnIO logger $ lm $ "exception with request: " <> T.pack (show e)

myExceptionResponse responds with JSON objects. It's simpler than defaultOnExceptionResponse, but it suffices for my learning.

myOnExceptionResponse :: Settings -> Settings
myOnExceptionResponse = setOnExceptionResponse handler
  where
    handler _ =
        responseLBS
            H.internalServerError500
            [(H.hContentType, "application/json; charset=utf-8")]
            (encode $ object ["error" .= ("Something went wrong" :: String)])

Finally, myShutdownHandler installs a handler for SIGTERM that logs and then shuts down.

myShutdownHandler :: Log.Logger -> Settings -> Settings
myShutdownHandler logger = setInstallShutdownHandler shutdownHandler
  where
    shutdownAction = Log.infoIO logger "closing down"
    shutdownHandler closeSocket = void $ installHandler sigTERM (Catch $ shutdownAction >> closeSocket) Nothing

Conclusion

I really ought to have looked into this sooner, especially as it turns out that Warp offers all the knobs and dials I could wish for to control these aspects of its behaviour. The next step is to take this and put it to use in one of the services at $DAYJOB

February 03, 2024 09:16 PM

January 31, 2024

Haskell Interlude

42 : Jezen Thomas

Jezen Thomas is co-founder and CTO of Supercede, a company applying Haskell in the reinsurance industry. In this episode, Jezen, Wouter and Joachim talk about his experience using Haskell in industry, growing a diverse and remote team of developers, and starting a company to create your own Haskell job.

by Haskell Podcast at January 31, 2024 10:00 AM

Well-Typed.Com

The Haskell Unfolder Episode 19: a new perspective on foldl'

Today, 2024-01-31, at 1930 UTC (11:30 am PST, 2:30 pm EST, 7:30 pm GMT, 20:30 CET, …) we are streaming the 19th episode of the Haskell Unfolder live on YouTube.

The Haskell Unfolder Episode 19: a new perspective on foldl’

In this beginner-oriented episode we introduce a useful combinator called repeatedly, which captures the concept “repeatedly execute an action to a bunch of arguments.” We will discuss both how to implement this combinator as well as some use cases.

About the Haskell Unfolder

The Haskell Unfolder is a YouTube series about all things Haskell hosted by Edsko de Vries and Andres Löh, with episodes appearing approximately every two weeks. All episodes are live-streamed, and we try to respond to audience questions. All episodes are also available as recordings afterwards.

We have a GitHub repository with code samples from the episodes.

And we have a public Google calendar (also available as ICal) listing the planned schedule.

by andres, edsko at January 31, 2024 12:00 AM

January 25, 2024

Joachim Breitner

GHC Steering Committee Retrospective

After seven years of service as member and secretary on the GHC Steering Committee, I have resigned from that role. So this is a good time to look back and retrace the formation of the GHC proposal process and committee.

In my memory, I helped define and shape the proposal process, optimizing it for effectiveness and throughput, but memory can be misleading, and judging from the paper trail in my email archives, this was indeed mostly Ben Gamari’s and Richard Eisenberg’s achievement: Already in Summer of 2016, Ben Gamari set up the ghc-proposals Github repository with a sketch of a process and sent out a call for nominations on the GHC user’s mailing list, which I replied to. The Simons picked the first set of members, and in the fall of 2016 we discussed the committee’s by-laws and procedures. As so often, Richard was an influential shaping force here.

Three ingredients

For example, it was him that suggested that for each proposal we have one committee member be the “Shepherd�, overseeing the discussion. I believe this was one ingredient for the process effectiveness: There is always one person in charge, and thus we avoid the delays incurred when any one of a non-singleton set of volunteers have to do the next step (and everyone hopes someone else does it).

The next ingredient was that we do not usually require a vote among all members (again, not easy with volunteers with limited bandwidth and occasional phases of absence). Instead, the shepherd makes a recommendation (accept/reject), and if the other committee members do not complain, this silence is taken as consent, and we come to a decision. It seems this idea can also be traced back on Richard, who suggested that “once a decision is requested, the shepherd [generates] consensus. If consensus is elusive, then we vote.�

At the end of the year we agreed and wrote down these rules, created the mailing list for our internal, but publicly archived committee discussions, and began accepting proposals, starting with Adam Gundry’s OverloadedRecordFields.

At that point, there was no “secretary� role yet, so how I did become one? It seems that in February 2017 I started to clean-up and refine the process documentation, fixing “bugs in the process� (like requiring authors to set Github labels when they don’t even have permissions to do that). This in particular meant that someone from the committee had to manually handle submissions and so on, and by the aforementioned principle that at every step there ought to be exactly one person in change, the role of a secretary followed naturally. In the email in which I described that role I wrote:

Simon already shoved me towards picking up the “secretary� hat, to reduce load on Ben.

So when I merged the updated process documentation, I already listed myself “secretary�.

It wasn’t just Simon’s shoving that put my into the role, though. I dug out my original self-nomination email to Ben, and among other things I wrote:

I also hope that there is going to be clear responsibilities and a clear workflow among the committee. E.g. someone (possibly rotating), maybe called the secretary, who is in charge of having an initial look at proposals and then assigning it to a member who shepherds the proposal.

So it is hardly a surprise that I became secretary, when it was dear to my heart to have a smooth continuous process here.

I am rather content with the result: These three ingredients – single secretary, per-proposal shepherds, silence-is-consent – helped the committee to be effective throughout its existence, even as every once in a while individual members dropped out.

Ulterior motivation

I must admit, however, there was an ulterior motivation behind me grabbing the secretary role: Yes, I did want the committee to succeed, and I did want that authors receive timely, good and decisive feedback on their proposals – but I did not really want to have to do that part.

I am, in fact, a lousy proposal reviewer. I am too generous when reading proposals, and more likely mentally fill gaps in a specification rather than spotting them. Always optimistically assuming that the authors surely know what they are doing, rather than critically assessing the impact, the implementation cost and the interaction with other language features.

And, maybe more importantly: why should I know which changes are good and which are not so good in the long run? Clearly, the authors cared enough about a proposal to put it forward, so there is some need… and I do believe that Haskell should stay an evolving and innovating language… but how does this help me decide about this or that particular feature.

I even, during the formation of the committee, explicitly asked that we write down some guidance on “Vision and Guideline�; do we want to foster change or innovation, or be selective gatekeepers? Should we accept features that are proven to be useful, or should we accept features so that they can prove to be useful? This discussion, however, did not lead to a concrete result, and the assessment of proposals relied on the sum of each member’s personal preference, expertise and gut feeling. I am not saying that this was a mistake: It is hard to come up with a general guideline here, and even harder to find one that does justice to each individual proposal.

So the secret motivation for me to grab the secretary post was that I could contribute without having to judge proposals. Being secretary allowed me to assign most proposals to others to shepherd, and only once in a while myself took care of a proposal, when it seemed to be very straight-forward. Sneaky, ain’t it?

7 Years later

For years to come I happily played secretary: When an author finished their proposal and public discussion ebbed down they would ping me on GitHub, I would pick a suitable shepherd among the committee and ask them to judge the proposal. Eventually, the committee would come to a conclusion, usually by implicit consent, sometimes by voting, and I’d merge the pull request and update the metadata thereon. Every few months I’d summarize the current state of affairs to the committee (what happened since the last update, which proposals are currently on our plate), and once per year gathered the data for Simon Peyton Jones’ annually GHC Status Report. Sometimes some members needed a nudge or two to act. Some would eventually step down, and I’d sent around a call for nominations and when the nominations came in, distributed them off-list among the committee and tallied the votes.

Initially, that was exciting. For a long while it was a pleasant and rewarding routine. Eventually, it became a mere chore. I noticed that I didn’t quite care so much anymore about some of the discussion, and there was a decent amount of naval-gazing, meta-discussions and some wrangling about claims of authority that was probably useful and necessary, but wasn’t particularly fun.

I also began to notice weaknesses in the processes that I helped shape: We could really use some more automation for showing proposal statuses, notifying people when they have to act, and nudging them when they don’t. The whole silence-is-assent approach is good for throughput, but not necessary great for quality, and maybe the committee members need to be pushed more firmly to engage with each proposal. Like GHC itself, the committee processes deserve continuous refinement and refactoring, and since I could not muster the motivation to change my now well-trod secretarial ways, it was time for me to step down.

Luckily, Adam Gundry volunteered to take over, and that makes me feel much less bad for quitting. Thanks for that!

And although I am for my day job now enjoying a language that has many of the things out of the box that for Haskell are still only language extensions or even just future proposals (dependent types, BlockArguments, do notation with (� foo) expressions and 💜 Unicode), I’m still around, hosting the Haskell Interlude Podcast, writing on this blog and hanging out at ZuriHac etc.

by Joachim Breitner (mail@joachim-breitner.de) at January 25, 2024 12:21 AM

Well-Typed.Com

Eras profiling for GHC

Memory detectives now have many avenues of investigation when looking into memory usage problems in Haskell programs. You might start by looking at what has been allocated: which types of closures and which constructors are contributing significantly to the problem. Then perhaps it’s prudent to look at why a closure has been allocated by the info table provenance information. This will tell you from which point in the source code your allocations are coming from. But if if you then turned to investigate when a closure was allocated during the lifecycle of your program, you end up being stuck.

Existing Haskell heap profiling tools work by taking regular samples of the heap to generate a graph of heap usage over time. This can give an aggregate view, but makes it difficult to determine when an individual closure was allocated.

Eras profiling is a new GHC profiling mode that will be available in GHC 9.10 (!11903). For each closure it records the “era” during which it was allocated, thereby making it possible to analyse the points at which closures are allocated much more precisely.

In this post, we are going to explore this profiling mode that makes it easier to find space leaks and identify suspicious long lived objects on the heap. We have discussed ghc-debug before, and we are going to make use of it to explore the new profiling mode using some new features added to the ghc-debug-brick TUI.

Introduction to eras profiling

The idea of eras profiling is to mark each closure with the era it was allocated. An era is simply a Word. The era can then be used to create heap profiles by era and also inspected by ghc-debug.

To enable eras profiling, you compile programs with profiling enabled and run with the +RTS -he option.

The era starts at 1, then there are two means by which it can be changed:

  • User: The user program has control to set the era explicitly (by using functions in GHC.Profiling.Eras).
  • Automatic: The era is incremented at each major garbage collection (enabled by --automatic-era-increment).

The user mode is most useful as this allows you to provide domain specific eras. There are three new primitive functions exported from GHC.Profiling.Eras for manipulating the era:

setUserEra :: Word -> IO ()
getUserEra :: IO Word
incrementUserEra :: Word -> IO Word

Note that the current era is a program global variable, so if your program is multi-threaded then setting the era will apply to all threads.

Below is an example of an eras profile rendered using eventlog2html. The eras have been increased by the user functions, and the programmer has defined 4 distinct eras.

eventlog2html eras graph, sample
eventlog2html eras graph, sample

Diagnosing a GHCi memory leak

Recently, we came across a regression in GHCi’s memory behaviour (#24116), where reloading a project would use double the amount of expected memory. During each reload of a project, the memory usage would uniformly increase, only to return to the expected level after the reload had concluded.

Reproducing the problem

In order to investigate the issue we loaded Agda into a GHCi session and reloaded the project a few times. Agda is the kind of project we regularly use to analyse compiler performance as it’s a typical example of a medium size Haskell application.

eventlog2html detail chart, agda
eventlog2html detail chart, agda

The profile starts with the initial load of Agda into GHCi, then each subsequent vertical line represents a :reload call.

We can see that while loading the project a second time, GHCi seems to hold on to all of the in-memory compilation artifacts from the first round of compilation, before releasing them right as the load finishes. This is not expected behaviour for GHCi, and ideally it should stay at a roughly constant level of heap usage during a reload.

During a reload, GHCi should either

  1. Keep compilation artifacts from the previous build in memory if it determines that they are still valid after recompilation avoidance checking
  2. Release them as soon as it realizes they are out of date, replacing them with fresh artifacts

In either case, the total heap usage shouldn’t change, since we are either keeping old artifacts or replacing them with new ones of roughly similar size. This task is a perfect fit for eras profiling, if we can get assign a different era for each reload then we should be able to easily confirm our hypothesis about the memory usage pattern.

Instrumenting GHCi

First we instrument the program so that the era is incremented on each reload. We do this by modifying the load function in GHC to increment the era on each call:

--- a/compiler/GHC/Driver/Make.hs
+++ b/compiler/GHC/Driver/Make.hs
@@ -153,6 +153,7 @@ import GHC.Utils.Constants
 import GHC.Types.Unique.DFM (udfmRestrictKeysSet)
 import GHC.Types.Unique
 import GHC.Iface.Errors.Types
+import GHC.Profiling.Eras

 import qualified GHC.Data.Word64Set as W

@@ -702,6 +703,8 @@ load' mhmi_cache how_much diag_wrapper mHscMessage mod_graph = do
     -- In normal usage plugins are initialised already by ghc/Main.hs this is protective
     -- for any client who might interact with GHC via load'.
     -- See Note [Timing of plugin initialization]
+    era <- liftIO getUserEra
+    liftIO $ setUserEra (era + 1)
     initializeSessionPlugins
     modifySession $ \hsc_env -> hsc_env { hsc_mod_graph = mod_graph }
     guessOutputFile

Then when running the benchmark with eras profiling enabled, the profile looks as follows:

eventlog2html eras graph, agda
eventlog2html eras graph, agda

Now we can clearly see that after the reload (the vertical line), all the memory which has been allocated during era 2 remains alive as newly allocated memory belongs to era 3.

Identifying a culprit closure

With the general memory usage pattern established, it’s time to look more closely at the culprits. By performing an info table profile and looking at the detailed tab, we can identify a specific closure which contributes to the bad memory usage pattern.

The GRE closure is one of the top entries in the info table profile, and we can see that its pattern of heap usage matches the overall shape of the graph, which means that we are probably incorrectly holding on to GREs from the first round of compilation.

eventlog2html screenshot, detailed
eventlog2html screenshot, detailed

Now we can turn to more precise debugging tools in order to actually determine where the memory leak is.

Looking closer with ghc-debug

We decided to investigate the leak using ghc-debug. After instrumenting the GHC executable we can connect to a running ghc process with ghc-debug-brick and explore its heap objects using a TUI interface.

Tracing retainers with ghc-debug

To capture the leak, we pause the GHC process right before it finished the reload, while it is compiling the final few modules in Agda. Remember that all the memory is released after the end of the reload to return to the expected baseline.

In order to check for the cause of the leak, we do a search for the retainers of GRE closure in ghc-debug-brick. We are searching for GRE because the info table profile indicated that this was one type of closure which was leaking.

Before eras profiling, if we tried to use this knowledge and ghc-debug-brick to find out why the GREs are being retained then we got a bit stuck. Looking at the interface we can’t distinguish between the two distinct classes of live GREs:

  1. Fresh GREs from the current load (era 3), which we really do need in memory.
  2. Stale GREs from the first load (era 2), which shouldn’t be live anymore and should have be released.
ghc-debug-brick screenshot, no eras
ghc-debug-brick screenshot, no eras

This is the retainer view of ghc-debug, where all closures matching our search (constructor is GRE) are listed, and expanding a closure shows a tree with a retainer stack of all the heap objects which retain the closure. Reading this stack downwards you can determine a chain of references through which any particular closure is retained, going all the way back to a GC root. Inspecting the retainer stack can shed light on why your program is holding on to a particular object.

We can try to scroll through the list of GREs in the TUI, carefully inspecting the retainer stack of each in turn and using our domain knowledge to classify each GRE closure as belonging to one of the two categories above.

However, this process is tedious and error prone, especially given that we have such a large number of potentially leaking objects to inspect. Depending on the order that ghc-debug happened to traverse the heap, we might find leaking entries after inspecting the first few items, or we may be very unlucky and all the leaking items might be hundreds or thousands of entries deep into the list.

ghc-debug supercharged with eras profiling

Now with eras profiling there are two extensions to ghc-debug which make it easy to distinguish these two cases since we already distinguished the era of the objects.

  • Filtering By Era: You can filter the results of any search to only include objects allocated during particular eras, given by ranges.
  • Colouring by Era: You can also enable colouring by era, so that the background colour of entries in ghc-debug-brick is selected based on the era the object was allocated in, making it easy to visually partition the heap and quickly identify leaking objects.

So now, if we enable filtering by era, it’s easy to distinguish the new and old closures.

With the new filtering mode, we search for retainers of GREs which were allocated in era 2. Now we can inspect the retainer stacks of any one of these closures with the confidence that it has leaked. Because ghc-debug-brick is also colouring by era, we can also easily identify roughly where in the retainer stack the leak occurs, because we can see new objects (from era 3) holding on to objects from the previous era (era 2).

We can see that a GRE from era 2 (green) is being retained, through a thunk from GHC.Driver.Make allocated in era 3 (yellow):

ghc-debug-brick screenshot, with closures coloured by era
ghc-debug-brick screenshot, with closures coloured by era

The location of the thunk tells us the exact location of the leak, and it is now just of matter of understanding why this code is retaining on to the unwanted objects and plugging the leak. For more details on the actual fix, see !11608.

Conclusion

We hope the new ghc-debug features and the eras profiling mode will be useful to others investigating the memory behaviour of their programs and easily identifying leaking objects which should not be retained in memory.

This work has been performed in collaboration with Mercury. Mercury have a long-term commitment to the scalability and robustness of the Haskell ecosystem and are supporting the development of memory profiling tools to aid with these goals.

Well-Typed are always interested in projects and looking for funding to improve GHC and other Haskell tools. Please contact info@well-typed.com if we might be able to work with you!

by matthew, zubin at January 25, 2024 12:00 AM

January 18, 2024

Tweag I/O

A look under GHC's hood: desugaring linear types

I recently merged linear let- and where-bindings in GHC. Which means that we’ll have these in GHC 9.10, which is cause for celebration for me. Though they are much overdue, so maybe I should instead apologise to you.

Anyway, I thought I’d take the opportunity to discuss some of GHC’s inner workings and how they explain some of the features of linear types in Haskell. We’ll be discussing a frequently asked question: why can’t Ur be a newtype? And some new questions such as: why must linear let-bindings have a !? But first…

GHC’s frontend part 1: typechecking

Like a nature documentary, let’s follow the adventure of a single declaration as it goes through GHC’s frontend. Say

f True x y = [ toUpper c | c <- show x ++ show y ]
f False x y = show x ++ y

There’s a lot left implicit here: What are the types of x, y, and c? Where do the instances of Show required by show come from? (Magic from outer space?)

Figuring all this out is the job of the typechecker. The typechecker is carefully designed; it’s actually very easy to come up with a sound type system which is undecidable. But even then, decidability is not enough: we don’t want inversion of the True and the False equations of f to change the result of the typechecker. Nor should pulling part of a term in a let let s = show x ++ show y in [ toUpper c | c <- s ]. We want the algorithm to be stable and predictable.

GHC’s typechecker largely follows the algorithm described in the paper OutsideIn(x). It would walk the declaration as:

  • x is used as an argument of show, so it has some type tx and we need Show tx.
  • y is used as an argument of show, so it has some type ty and we need Show ty.
  • c is bound to the return of show, so it has some type tc such that [tc] is [Char] (from which we know that tc is Char).
  • c is used as an argument of toUpper so tc is Char (which we already knew from the previous point, so there’s no type error).
  • From all this we deduce that f :: Bool -> tx -> ty -> [Char].
  • Then the typechecker proceeds through the second equation and discovers that ty is [Char].
  • So the type of f is Bool -> tx -> [Char] -> [Char].
  • Only now, we look at tx and notice that we don’t know anything about it, except that it must have Show tx. So tx is generalised and we get the final type for f: f :: forall a. Show a => Bool -> a -> [Char] -> [Char].

Where we’re allowed to generalise is very important: if the typechecking algorithm could generalise everywhere it’d be undecidable. So the typechecker only generalises at bindings (it even only generalises at top-level bindings with -XMonoLocalBinds), and where it knows that it needs a forall (usually: because of a type signature).

Of course, there’s no way I could give a faithful account of GHC’s typechecker in a handful of paragraphs. I’m sweeping a ton under the carpet (such as bidirectional typechecking or the separation of constraint generation and constraint solving).

GHC’s frontend part 2: desugaring

Now that we know all of this type information, our declaration is going to get desugared. In GHC, this means that the expression is translated to a language called Core. Our expression looks like this after desugaring:

f = \(@a :: Type) -> \(d :: Show a) -> \(b :: Bool) -> \(x :: a) -> \(y :: [Char]) ->
  case b of
  { True -> map @Char (\(c::Char) -> toUpper c) ((++) @Char (show d x) (show (showList showChar) y))
  , False -> (++) @Char (show d x) y
  }

There are a few things to notice:

  • This language is typed.
  • Nothing is left implicit: all the types are tediously specified on each variable binding, on each forall, and on each application of a polymorphic function.
  • Core is a language with far fewer features than Haskell. Gone are the type classes (replaced with actual function arguments), gone are list comprehensions, gone are equations, in fact gone are deep pattern matching, “do” notation, and so many other things.

For the optimiser (Core is the language where most of the optimisations take place), this means dealing with fewer constructs. So there’s less interaction to consider. Types are used to drive some optimisations such as specialisation. But by the time we reach the optimiser, we’ve already left the frontend, and it’s a story for another blog post maybe.

For language authors, this means something very interesting: when defining a new language construction, we can describe its semantics completely by specifying its desugaring. This is a powerful technique, which is used, for instance, in many GHC proposals.

Ur can’t be a newtype

Let me use this idea that a feature’s semantics is given by its desugaring to answer a common question: why must Ur be boxed, couldn’t it be a newtype instead?

To this end, consider the following function, which illustrates the meaning of Ur

f :: forall a. Ur a %1 -> Bool
f (Ur _) = True

That is, Ur a is a wrapper over a that lets me use the inner a as many times as I wish (here, none at all).

The desugaring is very straightforward:

f = \(@a :: Type) -> \(x :: Ur a) ->
  case x of
    Ur y -> True

Because Ur a is just a wrapper, it looks as though it could be defined as a newtype. But newtypes are actually desugared quite differently and this will turn out to be a problem. That is, if Ur a was a newtype then it would have the same underlying representation as a, the desugaring reflects that we can always use one in place of the other. Specifically, desugaring a newtype is done via a coercion of the form ur_co :: Ur a ~R a, which lets Core convert back and forth between types Ur a and a (the R stands for representational). With a newtype the desugaring of f would be:

f_newtype = \(@a :: Type) -> \(x :: Ur a) ->
  let
    y :: a
    y = x `cast` ur_co
  in
  True

Because y isn’t used at all, this is optimised to:

f_newtype = \(@a :: Type) -> \(x :: Ur a) -> True

See the problem yet?

Consider f_newtype destroyTheOneRing. This evaluates to True; the one ring is never destroyed! Contrast with f destroyTheOneRing which reduces to:

case destroyTheOneRing of
  Ur y -> True

This forces me to destroy the one ring before returning True (in our story, the ring is destroyed when the Ur constructor is forced).

This is why Ur can’t be a newtype and why GHC’s typechecker rejects attempts to define an Ur newtype: such a newtype would semantically violate the linearity discipline.

Exercise: this objection wouldn’t apply to a strict language. Do you see why?

No linear lazy patterns

In Haskell, a pattern in a let-binding is lazy. That is:

let (x, y) = bakeTurkey in body

Desugars to:

let
  xy = bakeTurkey
  x = fst xy
  y = snd xy
in
body

Is this linear in bakeTurkey? You can make this argument: although it’s forced several times (once when forcing x and once when forcing y), laziness will ensure that you don’t actually bake your turkey twice. There are some systems, such as Linearity and laziness [pdf], which treat lazy patterns as linear.

But it’s awfully tricky to treat the desugared code as linear. A typechecker that does so would have to understand that x and y are disjoint (so as to make sure that we don’t bake parts of the turkey twice) and that they actually cover all of xy (so that no part of the turkey is left unbaked). And because Core is typed, we do have to make sure that the desugaring is linear.

At this point, this is a complication that we don’t want to tackle. If we wanted to, it would require some additional research; it’s not just a matter of engineering time. As a consequence, patterns in linear let-bindings must be annotated with a !, making them strict:

let !(x, y) = bakeTurkey in body

which desugars to:1

case bakeTurkey of
  (x, y) -> body

Which is understood as linear in Core.

This is really only for actual patterns. Variables let x = rhs in body, which desugar to themselves, are fine.

No linear polymorphic patterns

Finally, let’s look at a trickier interaction. Consider the following:

let (x, y) = ((\z -> z), (\z -> z)) in
case b of
  True -> (x 'a', y False)
  False -> (x True, y 'b')

If MonoLocalBind is turned off, then x and y are inferred to have the type forall a. a -> a. What’s really interesting is the desugaring:

let
  xy = \@a -> ((\(z::a) -> z), (\(z::a) -> z))
  x = \@a -> fst (xy @a)
  y = \@a -> snd (xy @a)
in
case b of
  True -> (x @Char 'a', y @Bool False)
  False -> (x @Bool True, y @Char 'b')

An alternative desugaring might have been:

let
  xy = ((\@a (z::a) -> z), (\@a (z::a) -> z))
  x = fst xy
  y = snd xy
in
case b of
  True -> (x @Char 'a', y @Bool False)
  False -> (x @Bool True, y @Char 'b')

Which does indeed give the same type to x and y. But this alternative desugaring can’t be produced because of how the typechecker works: remember that the typechecker only considers generalisation (which corresponds to adding \@a -> in the desugaring) at the root of a let-binding, which isn’t the case of the alternative desugaring.

So, what does this all have to do with us anyway? Well, here’s the thing. If the alternative desugaring can be transformed into a case (in the strict case):

case ((\@a (z::a) -> z), (\@a (z::a) -> z)) of
  (x, y)  ->
    case b of
      True -> (x @Char 'a', y @Bool False)
      False -> (x @Bool True, y @Char 'b')

Then the actual desugaring cannot:

-- There's no type to use for `??`
case (\@a -> ((\(z::a) -> z), (\(z::a) -> z))) @?? of
  (x, y) ->
    -- Neither `x` nor `y` is polymorphic so the result doesn't
    -- type either.
    case b of
      True -> (x @Char 'a', y @Bool False)
      False -> (x @Bool True, y @Char 'b')

So we’re back to the situation of the lazy pattern matching: we might be able to make the desugaring linear without endangering soundness, but it’s too much effort.

So pattern let-bindings can’t be both linear and polymorphic (here again, variables are fine). This means in practice that:

  • In let %1 (x, y) = the type of x and y is never generalised.
  • Starting with GHC 9.10, LinearTypes will imply MonoLocalBinds.
  • If you turn MonoLocalBinds off, however, and the typechecker infers a polymorphic type for let (x, y) = rhs then rhs will count as being used Many times.

What’s really interesting, here, is that this limitation follows mainly from the typechecking algorithm, which can’t infer every permissible type otherwise it wouldn’t always terminate.

Wrapping up

This is the end of our little journey in the interactions of typechecking, desugaring, and linear types. Maybe you’ve learned a thing or two about what makes GHC tick. Hopefully you’ve learned something about why the design of linear types is how it is.

What I’d like to stress before I leave you is that the design of GHC is profound. We’re not speaking about accidents of history that makes some features harder to implement than they need to. It’s quite possible that a future exists where we can lift some of the limitations that I discussed (not Ur being a newtype, though, this one is simply unsound). But the reason why they are hard to lift is that they require actual research to understand what the relaxed condition implies. They are hard to lift because they are fundamentally hard problems! GHC forces you to think about it. It forces you to deeply understand the features you implement, and tell you to go do your research if you don’t.


  1. This is not actually how strict lets used to desugar. Instead they desugared to:

    let
      xy = bakeTurkey
      x = fst xy
      y = snd xy
    in
    xy `seq` body

    Which will eventually optimise to the appropriate case expression. This desugaring generalises to (mutually) recursive strict bindings, but these aren’t relevant for us as recursive definition lets aren’t linear.

    As part of the linear let patch, I did change the desugaring to target a case directly in the non-recursive case, in order to support linear lets.

January 18, 2024 12:00 AM

Michael Snoyman

My Best and Worst Deadlock in Rust

We're going to build up a deadlock together. If you're unfamiliar with Rust and/or its multithreaded concepts, you'll probably learn a lot from this. If you are familiar with Rust's multithreading capabilities, my guess is you'll be as surprised by this deadlock as I was. And if you spot the deadlock immediately, you get a figurative hat-tip from me.

As to the title, this deadlock was the worst I ever experienced because of how subtle it was. It was the best because of the tooling told me exactly where the problem was. You'll see both points come out below.

Access control

If you've read much of my writing, you'll know I almost always introduce a data structure that looks like this:

struct Person {
    name: String,
    age: u32,
}

So we'll do something very similar here! I'm going to simulate some kind of an access control program that allows multiple threads to use some shared, mutable state representing a person. And we'll make two sets of accesses to this state:

  • A read-only thread that checks if the user has access
  • A writer thread that will simulate a birthday and make the person 1 year older

Our access control is really simple: we grant access to people 18 years or older. One way to write this program looks like this:

use std::sync::Arc;

use parking_lot::RwLock;

#[derive(Clone)]
struct Person {
    inner: Arc<RwLock<PersonInner>>,
}

struct PersonInner {
    name: String,
    age: u32,
}

impl Person {
    fn can_access(&self) -> bool {
        const MIN_AGE: u32 = 18;

        self.inner.read().age >= MIN_AGE
    }

    /// Returns the new age
    fn birthday(&self) -> u32 {
        let mut guard = self.inner.write();
        guard.age += 1;
        guard.age
    }
}

fn main() {
    let alice = Person {
        inner: Arc::new(RwLock::new(PersonInner {
            name: "Alice".to_owned(),
            age: 15,
        })),
    };

    let alice_clone = alice.clone();
    std::thread::spawn(move || loop {
        println!("Does the person have access? {}", alice_clone.can_access());
        std::thread::sleep(std::time::Duration::from_secs(1));
    });

    for _ in 0..10 {
        std::thread::sleep(std::time::Duration::from_secs(1));
        let new_age = alice.birthday();

        println!("Happy birthday! Person is now {new_age} years old.");
    }
}

We're using the wonderful parking-lot crate for this example. Since we have one thread which will exclusively read, an RwLock seems like the right data structure to use. It will allow us to take multiple concurrent read locks or one exclusive write lock at a time. For those familiar with it, this is very similar to the general Rust borrow rules, which allow for multiple read-only (or shared) references or a single mutable (or exclusive) reference.

Anyway, we follow a common pattern with our Person data type. It has a single inner field, which contains an Arc and RwLock wrapping around our inner data structure, which contains the actual name and age. Now we can cheaply clone the Person, keep a single shared piece of data in memory for multiple threads, and either read or mutate the values inside.

Next up, to provide nicely encapsulated access, we provide a series of methods on Person that handle the logic of getting read or write locks. In particular, the can_access method takes a read lock, gets the current age, and compares it to the constant value 18. The birthday method takes a write lock and increments the age, returning the new value.

If you run this on your computer, you'll see something like the following output:

Does the person have access? false
Happy birthday! Person is now 16 years old.
Does the person have access? false
Happy birthday! Person is now 17 years old.
Does the person have access? false
Does the person have access? false
Happy birthday! Person is now 18 years old.
Does the person have access? true
Happy birthday! Person is now 19 years old.
Does the person have access? true
Happy birthday! Person is now 20 years old.
Does the person have access? true
Happy birthday! Person is now 21 years old.
Does the person have access? true
Happy birthday! Person is now 22 years old.
Does the person have access? true
Happy birthday! Person is now 23 years old.
Does the person have access? true
Happy birthday! Person is now 24 years old.
Does the person have access? true
Happy birthday! Person is now 25 years old.

The output may look slightly different due to timing differences, but you get the idea. The person, whoever that happens to be, suddenly has access starting at age 18.

NOTE TO READER I'm not going to keep asking this, but I encourage you to look at each code sample and ask: is this the one that introduces the deadlock? I'll give you the answers towards the end of the post.

What's in a name?

It's pretty annoying having now idea who has access. Alice has a name! We should use it. Let's implement a helper method for getting the person's name:

fn get_name(&self) -> &String {
    &self.inner.read().name
}

While this looks nice, it doesn't actually compile:

error[E0515]: cannot return value referencing temporary value
  --> src/main.rs:30:9
   |
30 |         &self.inner.read().name
   |         ^-----------------^^^^^
   |         ||
   |         |temporary value created here
   |         returns a value referencing data owned by the current function

You see, the way an RwLock's read method works is that it returns a RwLockReadGuard. This implements all the borrow rules we want to see at runtime via value creation and dropping. Said more directly: when you call read, it does something like the following:

  1. Waits until it's allowed to take a read guard. For example, if there's an existing write guard active, it will block until that write guard finishes.
  2. Increments a counter somewhere indicating that there's a new active read guard.
  3. Constructs the RwLockReadGuard value.
  4. When that value gets dropped, its Drop impl will decrement that counter.

And this is basically how many interior mutability primitives in Rust work, whether it's an RwLock, Mutex, or RefCell.

The problem with our implementation of get_name is that it tries to take a lock and then borrow a value through the lock. However, when we exit the get_name method it's still holding a reference to the RwLockReadGuard which we're trying to drop. So how do we implement this method? There are a few possibilities:

  • Return the RwLockReadGuard<PersonInner>. This is no longer a get_name method, but now a general purpose "get a read lock" method. It's also unsatisfying because it requires exposing the innards of our inner data structure.
  • Clone the inner String, which is unnecessary allocation.
  • Wrap the name field with an Arc and clone the Arc, which is probably cheaper than cloning the String.

There are really interesting API design points implied by all this, and it would be fun to explore them another time. However, right now, I've got a tight deadline from my boss on the really important feature of print out the person's name, so I better throw together something really quick and direct. And the easiest thing to do is to just lock the RwLock directly wherever we want a name.

We'll make a small tweak to our spawned thread's closure:

std::thread::spawn(move || loop {
    let guard = alice_clone.inner.read();
    println!(
        "Does the {} have access? {}",
        guard.name,
        alice_clone.can_access()
    );
    std::thread::sleep(std::time::Duration::from_secs(1));
});

Delays

The definition of insanity is doing the same thing over and over and expecting different results

- Somebody, but almost certainly not Albert Einstein

By the above definition of insanity, many have pointed out that multithreaded programming is asking the programmer to become insane. You need to expect different results for different runs of a program. That's because the interleaving of actions between two different threads is non-deterministic. Random delays, scheduling differences, and much more can cause a program to behave correctly on one run and completely incorrectly on another. Which is what makes deadlocks so infuriatingly difficult to diagnose and fix.

So let's simulate some of those random delays in our program by pretending that we need to download some super cute loading image while checking access. I've done so with a println call and an extra sleep to simulate the network request time:

    std::thread::spawn(move || loop {
        let guard = alice_clone.inner.read();
        println!("Downloading a cute loading image, please wait...");
        std::thread::sleep(std::time::Duration::from_secs(1));
        println!(
            "Does the {} have access? {}",
            guard.name,
            alice_clone.can_access()
        );
        std::thread::sleep(std::time::Duration::from_secs(1));
    });

And when I run my program, lo and behold, output stops after printing Downloading a cute loading image, please wait.... Maybe the output will be a bit different on your computer, maybe not. That's the nature of the non-deterministic beast. But this appears to be a deadlock.

The best deadlock experience ever

It turns out that the parking-lot crate provides an experimental feature: deadlock detection. When we were facing the real-life deadlock in our production systems, Sibi found this feature and added it to our executable. And boom! The next time our program deadlocked, we immediately got a backtrace pointing us to the exact function where the deadlock occurred. Since it was a release build, we didn't get line numbers, since those had been stripped out. But since I'm doing a debug build for this blog post, we're going to get something even better here.

Let's add in the following code to the top of our main function:

    std::thread::spawn(move || loop {
        std::thread::sleep(std::time::Duration::from_secs(2));
        for deadlock in parking_lot::deadlock::check_deadlock() {
            for deadlock in deadlock {
                println!(
                    "Found a deadlock! {}:\n{:?}",
                    deadlock.thread_id(),
                    deadlock.backtrace()
                );
            }
        }
    });

Every 2 seconds, this background thread will check if parking-lot has detected any deadlocks and print out the thread they occurred in and the full backtrace. (Why 2 seconds? Totally arbitrary. You could use any sleep amount you want.) When I add this to my program, I get some very helpful output. I'll slightly trim the output to not bother with a bunch of uninteresting backtraces outside of the main function:

Found a deadlock! 140559740036800:
   0: parking_lot_core::parking_lot::deadlock_impl::on_unpark
             at /home/michael/.cargo/registry/src/index.crates.io-6f17d22bba15001f/parking_lot_core-0.9.9/src/parking_lot.rs:1211:32
   1: parking_lot_core::parking_lot::deadlock::on_unpark
             at /home/michael/.cargo/registry/src/index.crates.io-6f17d22bba15001f/parking_lot_core-0.9.9/src/parking_lot.rs:1144:9
   2: parking_lot_core::parking_lot::park::{{closure}}
             at /home/michael/.cargo/registry/src/index.crates.io-6f17d22bba15001f/parking_lot_core-0.9.9/src/parking_lot.rs:637:17
   3: parking_lot_core::parking_lot::with_thread_data
             at /home/michael/.cargo/registry/src/index.crates.io-6f17d22bba15001f/parking_lot_core-0.9.9/src/parking_lot.rs:207:5
      parking_lot_core::parking_lot::park
             at /home/michael/.cargo/registry/src/index.crates.io-6f17d22bba15001f/parking_lot_core-0.9.9/src/parking_lot.rs:600:5
   4: parking_lot::raw_rwlock::RawRwLock::lock_common
             at /home/michael/.cargo/registry/src/index.crates.io-6f17d22bba15001f/parking_lot-0.12.1/src/raw_rwlock.rs:1115:17
   5: parking_lot::raw_rwlock::RawRwLock::lock_shared_slow
             at /home/michael/.cargo/registry/src/index.crates.io-6f17d22bba15001f/parking_lot-0.12.1/src/raw_rwlock.rs:719:9
   6: <parking_lot::raw_rwlock::RawRwLock as lock_api::rwlock::RawRwLock>::lock_shared
             at /home/michael/.cargo/registry/src/index.crates.io-6f17d22bba15001f/parking_lot-0.12.1/src/raw_rwlock.rs:109:26
   7: lock_api::rwlock::RwLock<R,T>::read
             at /home/michael/.cargo/registry/src/index.crates.io-6f17d22bba15001f/lock_api-0.4.11/src/rwlock.rs:459:9
   8: access_control::Person::can_access
             at src/main.rs:19:9
   9: access_control::main::{{closure}}
             at src/main.rs:59:13
  10: std::sys_common::backtrace::__rust_begin_short_backtrace
             at /rustc/79e9716c980570bfd1f666e3b16ac583f0168962/library/std/src/sys_common/backtrace.rs:154:18

Wow, this gave us a direct pointer to where in our codebase the problem occurs. The deadlock happens in the can_access method, which is called from our println! macro call in main.

In a program of this size, getting a direct link to the relevant code isn't terribly helpful. There were only a few lines that could have possibly caused the deadlock. However, in our production codebase, we have thousands of lines of code in the program itself that could have possibly been related. And it turns out the program itself wasn't even the culprit, it was one of the support libraries we wrote!

Being able to get such direct information on a deadlock is a complete gamechanger for debugging problems of this variety. Absolutely huge props and thanks to the parking-lot team for providing this.

But what's the problem?

OK, now it's time for the worst. We still need to identify what's causing the deadlock. Let's start off with the actual deadlock location: the can_access method:

fn can_access(&self) -> bool {
    const MIN_AGE: u32 = 18;

    self.inner.read().age >= MIN_AGE
}

Is this code, on its own, buggy? Try as I might, I can't possibly find a bug in this code. And there isn't one. This is completely legitimate usage of a read lock. In fact, it's a great demonstration of best practices: we take the lock for as little time as needed, ensuring we free the lock and avoiding contention.

So let's go up the call stack and look at the body of our subthread infinite loop:

let guard = alice_clone.inner.read();
println!("Downloading a cute loading image, please wait...");
std::thread::sleep(std::time::Duration::from_secs(1));
println!(
    "Does the {} have access? {}",
    guard.name,
    alice_clone.can_access()
);
std::thread::sleep(std::time::Duration::from_secs(1));

This code is already pretty suspicious. The first thing that pops out to me when reading this code is the sleeps. We're doing something very inappropriate: holding onto a read lock while sleeping. This is a sure-fire way to cause contention for locks. It would be far superior to only take the locks for a limited period of time. Because lexical scoping leads to drops, and drops lead to freeing locks, one possible implementation would look like this:

println!("Downloading a cute loading image, please wait...");
std::thread::sleep(std::time::Duration::from_secs(1));
{
    let guard = alice_clone.inner.read();
    println!(
        "Does the {} have access? {}",
        guard.name,
        alice_clone.can_access()
    );
}
std::thread::sleep(std::time::Duration::from_secs(1));

This version of the code is an improvement. We've eliminated a legitimate performance issue of over-locking a value. And if you run it, you might see output like the following:

Downloading a cute loading image, please wait...
Happy birthday! Person is now 16 years old.
Does the Alice have access? false
Happy birthday! Person is now 17 years old.
Downloading a cute loading image, please wait...
Happy birthday! Person is now 18 years old.
Does the Alice have access? true
Happy birthday! Person is now 19 years old.
Downloading a cute loading image, please wait...
Happy birthday! Person is now 20 years old.
Does the Alice have access? true
Downloading a cute loading image, please wait...
Happy birthday! Person is now 21 years old.
Happy birthday! Person is now 22 years old.
Does the Alice have access? true
Downloading a cute loading image, please wait...
Happy birthday! Person is now 23 years old.
Does the Alice have access? true
Happy birthday! Person is now 24 years old.
Happy birthday! Person is now 25 years old.
Downloading a cute loading image, please wait...

However, you may also see another deadlock message! So our change is a performance improvement, and makes it more likely for our program to complete without hitting the deadlock. But the deadlock is still present. But where???

Why I thought this isn't a deadlock

It's worth pausing one quick moment before explaining where the deadlock is. (And figurative hat-tip if you already know.) Our program has three threads of execution:

  1. The deadlock detection thread. We know this isn't the cause of the deadlock, because we added that thread after we saw the deadlock. (Though "deadlock detection thread leads to deadlock" would be an appropriately mind-breaking statement to make.)
  2. The access check thread, which only does read locks.
  3. The main thread, where we do the birthday updates. We'll call it the birthday thread instead. This thread takes write locks.

And my assumption going into our debugging adventure is that this is perfectly fine. The birthday thread will keep blocking waiting for a write lock. It will block as long as the access check thread is holding a read lock. OK, that's part of a deadlock: thread B is waiting on thread A. And the check access thread will wait for the birthday thread to release its write lock before it can grab a read lock. That's another component of a deadlock. But it seems like each thread can always complete its locking without waiting on the other thread.

If you don't know what the deadlock is yet, and want to try to figure it out for yourself, go check out the RwLock docs from the standard library. But we'll continue the analysis here.

How many read locks?

At this point in our real-life debugging, Sibi observed something: our code was less efficient than it should be. Focus on this bit of code:

let guard = alice_clone.inner.read();
println!(
    "Does the {} have access? {}",
    guard.name,
    alice_clone.can_access()
);

If we inline the definition of can_access, the problem becomes more obvious:

let guard = alice_clone.inner.read();
println!("Does the {} have access? {}", guard.name, {
    const MIN_AGE: u32 = 18;

    alice_clone.inner.read().age >= MIN_AGE
});

The inefficiency is that we're taking two read locks instead of one! We already read-lock inner to get the name, and then we call alice_clone.can_access() which makes its own lock. This is good from a code reuse standpoint. But it's not good from a resource standpoint. During our debugging session, I agreed that this warranted further investigation, but we continued looking for the deadlock.

Turns out, I was completely wrong. This wasn't just an inefficiency. This is the deadlock. But how? It turns out, I'd missed a very important piece of the documentation for RwLock.

This lock uses a task-fair locking policy which avoids both reader and writer starvation. This means that readers trying to acquire the lock will block even if the lock is unlocked when there are writers waiting to acquire the lock. Because of this, attempts to recursively acquire a read lock within a single thread may result in a deadlock.

Or, to copy from std's docs, we have a demonstration of how to generate a potential deadlock with seemingly innocuous code:

// Thread 1             |  // Thread 2
let _rg = lock.read();  |
                        |  // will block
                        |  let _wg = lock.write();
// may deadlock         |
let _rg = lock.read();  |

This is exactly what our code above was doing: the access check thread took a first read lock to get the name, then took a second read lock inside the can_access method to check the age. By introducing a sleep in between these two actions, we increased the likelihood of the deadlock occurring by giving a wider timespan when the write lock from the birthday thread could come in between those two locks. But the sleep was not the bug. The bug was taking two read locks in the first place!

Let's first try to understand why RwLock behaves like this, and then put together some fixes.

Fairness and starvation

Imagine that, instead of a single access check thread, we had a million of them. Each of them is written so that it grabs a read lock, holds onto it for about 200 milliseconds, and then releases it. With a million such threads, there's a fairly high chance that the birthday thread will never be able to get a write lock. There will always be at least one read lock active.

This problem is starvation: one of the workers in a system is never able to get a lock, and therefore it's starved from doing any work. This can be more than just a performance issue, it can completely undermine the expected behavior of a system. In our case, Alice would remain 15 for the entire lifetime of the program and never be able to access the system.

The solution to starvation is fairness, where you make sure all workers get a chance to do some work. With a simpler data structure like a Mutex, this is relatively easy to think about: everyone who wants a lock stands in line and takes the lock one at a time.

However, RwLocks are more complex. They have both read and write locks, so there's not really just one line to stand in. A naive implementation--meaning what I would have implemented before reading the docs from std and parking-lot--would look like this:

  • read blocks until all write locks are released
  • write blocks until all read and write locks are released

However, the actual implementation with fairness accounted for looks something like this:

  • read blocks if there's an active write lock, or if another thread is waiting for a write lock
  • write blocks until all read and write locks are released

And now we can see the deadlock directly:

  1. Access check thread takes a read lock (for reading the name)
  2. Birthday thread tries to take a write lock, but it can't because there's already a read lock. It stands in line waiting its turn.
  3. Access check thread tries to take a read lock (for checking the age). It sees that there's a write lock waiting in line, and to avoid starving it, stands in line behind the birthday thread
  4. The access check thread is blocked until the birthday thread releases its lock. The birthday thread is blocked until the access check thread releases its first lock. Neither thread can make progress. Deadlock!

This, to me, is the worst deadlock I've encountered. Every single step of this process is logical. The standard library and parking-lot both made the correct decisions about implementation. And it still led to confusing behavior at runtime. Yes, the answer is "you should have read the docs," which I've now done. Consider this blog post an attempt to make sure that everyone else reads the docs too.

OK, so how do we resolve this problem? Let's check out two approaches.

Easiest: read_recursive

The parking-lot crate provides a read_recursive method. Unlike the normal read method, this method will not check if there's a waiting write lock. It will simply grab a read lock. By using read_recursive in our can_access method, we don't have a deadlock anymore. And in this program, we also don't have a risk of starvation, because the read_recursive call is always gated after our thread already got a read lock.

However, this isn't a good general purpose solution. It's essentially undermining all the fairness work that's gone into RwLock. Instead, even though it requires a bit more code change, there's a more idiomatic solution.

Just take one lock

This is the best approach we can take. We only need to take one read lock inside our access check thread. One way to make this work is to move the can_access method from Person to PersonInner, and then call can_access on the guard, like so:

impl PersonInner {
    fn can_access(&self) -> bool {
        const MIN_AGE: u32 = 18;

        self.age >= MIN_AGE
    }
}

// ...


let guard = alice_clone.inner.read();
println!("Downloading a cute loading image, please wait...");
std::thread::sleep(std::time::Duration::from_secs(1));
println!(
    "Does the {} have access? {}",
    guard.name,
    guard.can_access()
);
std::thread::sleep(std::time::Duration::from_secs(1));

This fully resolves the deadlock issue. There are still questions about exposing the innards of our data structure. We could come up with a more complex API that keeps some level of encapsulation, e.g.:

use std::sync::Arc;

use parking_lot::{RwLock, RwLockReadGuard};

#[derive(Clone)]
struct Person {
    inner: Arc<RwLock<PersonInner>>,
}

struct PersonInner {
    name: String,
    age: u32,
}

struct PersonReadGuard<'a> {
    guard: RwLockReadGuard<'a, PersonInner>,
}

impl Person {
    fn read(&self) -> PersonReadGuard {
        PersonReadGuard {
            guard: self.inner.read(),
        }
    }

    /// Returns the new age
    fn birthday(&self) -> u32 {
        let mut guard = self.inner.write();
        guard.age += 1;
        guard.age
    }
}

impl PersonReadGuard<'_> {
    fn can_access(&self) -> bool {
        const MIN_AGE: u32 = 18;

        self.guard.age >= MIN_AGE
    }

    fn get_name(&self) -> &String {
        &self.guard.name
    }
}

fn main() {
    std::thread::spawn(move || loop {
        std::thread::sleep(std::time::Duration::from_secs(2));
        for deadlock in parking_lot::deadlock::check_deadlock() {
            for deadlock in deadlock {
                println!(
                    "Found a deadlock! {}:\n{:?}",
                    deadlock.thread_id(),
                    deadlock.backtrace()
                );
            }
        }
    });

    let alice = Person {
        inner: Arc::new(RwLock::new(PersonInner {
            name: "Alice".to_owned(),
            age: 15,
        })),
    };

    let alice_clone = alice.clone();
    std::thread::spawn(move || loop {
        let guard = alice_clone.read();
        println!("Downloading a cute loading image, please wait...");
        std::thread::sleep(std::time::Duration::from_secs(1));
        println!(
            "Does the {} have access? {}",
            guard.get_name(),
            guard.can_access()
        );
        std::thread::sleep(std::time::Duration::from_secs(1));
    });

    for _ in 0..10 {
        std::thread::sleep(std::time::Duration::from_secs(1));
        let new_age = alice.birthday();

        println!("Happy birthday! Person is now {new_age} years old.");
    }
}

Is this kind of overhead warranted? Definitely not for this case. But such an approach might make sense for larger programs.

So when did we introduce the bug?

Just to fully answer the question I led with: we introduced the deadlock in the section title "What's in a name". In the real life production code, the bug came into existance in almost exactly the same way I described above. We had an existing helper method that took a read lock, then ended up introducing another method that took a read lock on its own and, while that lock was held, called into the existing helper method.

It's very easy to introduce a bug like that. (Or at least that's what I'm telling myself to feel like less of an idiot.) Besides the deadlock problem, it also introduces other race conditions. For example, if I had taken-and-released the read lock in the parent function before calling the helper function, I'd have a different kind of race condition: I'd be pulling data from the same RwLock in a non-atomic manner. Consider if, for example, Alice's name changes to "Alice the Adult" when she turns 18. In the program above, it's entirely possible to imagine a scenario where we say that "Alice the Adult" doesn't have access.

All of this to say: any time you're dealing with locking, you need to be careful to avoid potential data races. Rust makes it so much nicer than many other languages to avoid race conditions through things like RwLockReadGuard, the Send and Sync traits, mutable borrow checking, and other techniques. But it's still not a panacea.

January 18, 2024 12:00 AM

January 15, 2024

Monday Morning Haskell

Functional Programming vs. Object Oriented Programming

Functional Programming (FP) and Object Oriented Programming (OOP) are the two most important programming paradigms in use today. In this article, we'll discuss these two different programming paradigms and compare their key differences, strengths and weaknesses. We'll also highlight a few specific ways Haskell fits into this discussion. Here's a quick outline if you want to skip around a bit!

What is a Programming Paradigm?

A paradigm is a way of thinking about a subject. It's a model against which we can compare examples of something.

In programming, there are many ways to write code to solve a particular task. Our tasks normally involve taking some kind of input, whether data from a database or commands from a user. A program's job is then to produce outputs of some kind, like updates in that database or images on the user's screen.

Programming paradigms help us to organize our thinking so that we can rapidly select an implementation path that makes sense to us and other developers looking at the code. Paradigms also provide mechanisms for reusing code, so that we don't have to start from scratch every time we write a new program.

The two dominant paradigms in programming today are Object Oriented Programming (OOP) and Functional Programming (FP).

The Object Oriented Paradigm

In object oriented programming, our program's main job is to maintain objects. Objects almost always store data, and they have particular ways of acting on other objects and being acted on by other objects (these are the object's methods). Objects often have mutable data - many actions you take on your objects are capable of changing some of the object's underlying data.

Object oriented programming allows code reuse through a system called inheritance. Objects belong to classes which share the same kinds of data and actions. Classes can inherit from a parent class (or multiple classes, depending on the language), so that they also have access to the data from the base class and some of the same code that manipulates it.

The Functional Paradigm

In functional programming, we think about programming in terms of functions. This idea is rooted in the mathematical idea of a function. A function in math is a process which takes some input (or a series of different inputs) and produces some kind of output. A simple example would be a function that takes an input number and produces the square of that number. Many functional languages emphasize pure functions, which produce the exact same output every time when given the same input.

In programming, we may view our entire program as a function. It is a means by which some kind of input (file data or user commands), is transformed into some kind of output (new files, messages on our terminal). Individual functions within our program might take smaller portions of this input and produce some piece of our output, or some intermediate result that is needed to eventually produce this output.

In functional programming, we still need to organize our data in some way. So some of the ideas of objects/classes are still used to combine separate pieces of data in meaningful ways. However, we generally do not attach "actions" to data in the same way that classes do in OOP languages.

Since we don't perform actions directly on our data, functional languages are more likely to use immutable data as a default, rather than mutable data. (We should note though that both paradigms use both kinds of data in their own ways).

Functional Programming vs. OOP

The main point of separation between these paradigms is the question of "what is the fundamental building block of my program?" In object oriented programming, our programs are structured around objects. Functions are things we can do to an object or with an object.

In functional programming, functions are always first class citizens - the main building block of our code. In object oriented programming, functions can be first class citizens, but they do not need to be. Even in languages where they can be, they often are not used in this way, since this isn't as natural within the object oriented paradigm.

Object Oriented Programming Languages

Many of the most popular programming languages are OOP languages. Java, for a long time the most widely used language, is perhaps the most archetypal OO language. All code must exist within an object, even in a simple "Hello World" program:

class MyProgram {
  public static void main(String[] args) {
    System.out.println("Hello World!");
  }
}

In this example, we could not write our 'main' function on its own, without the use of 'class MyProgram'.

Java has a single basic 'Object' class, and all other classes (including any new classes you write) must inherit from it for basic behaviors like memory allocation. Java classes only allow single inheritance. This means that a class cannot inherit from multiple different types. Thus, all Java classes you would use can be mapped out on a tree structure with 'Object' as the root of the tree.

Other object oriented languages use the general ideas of classes, objects, and inheritance, but with some differences. C++ and Python both allow multiple inheritance, so that a class can inherit behavior from multiple existing classes. While these are both OOP languages, they are also more flexible in allowing functions to exist outside of classes. A basic script in either of these languages need not use any classes. In Python, we'd just write:

if __name__ == "__main__":
  print("Hello World!")

In C++, this looks like:

int main() {
  std::cout << "Hello World!" << std::endl;
}

These languages also don't have such a strictly defined inheritance structure. You can create classes that do not inherit from anything else, and they'll still work.

FP Languages

Haskell is perhaps the language that is most identifiable with the functional paradigm. Its type system and compiler really force you to adopt functional ideas, especially around immutable data, pure functions, and tail call optimization. It also embraces lazy evaluation, which is aligned with FP principles, but not a requirement for a functional language.

There are several other programming languages that generally get associated with the functional paradigm include Clojure, OCaml, Lisp, Scala and Rust. These languages aren't all functional in the same way as Haskell; there are many notable differences. Lisp bills itself specifically as a multi-paradigm language, and Scala is built to cross-compile with Java! Meanwhile Rust's syntax looks more object oriented, but its inheritance system (traits) feel much more like Haskell. However, on balance, these languages express functional programming ideas much more than their counterparts.

Amongst the languages mentioned in the object oriented section, Python has the most FP features. It is more natural to write functions outside of your class objects, and concepts like higher order functions and lambda expressions are more idiomatic than in C++ or Java. This is part of the reason Python is often recommended for beginners, with another reason being that its syntax makes it a relatively simple language to learn.

Advantages of Functional Programming

Fewer Bugs

FP code has a deserved reputation for having fewer bugs. Anecdotally, I certainly find I have a much easier time writing bug free code in Haskell than Python. Many bugs in object oriented code are caused by the proliferation of mutable state. You might pass an object to a method and expect your object to come back unchanged...only to find that the method does in fact change your object's state. With objects, it's also very easy for unstated pre-conditions to pop up in class methods. If your object is not in the state you expect when the method is called, you'll end up with behavior you didn't intend.

A lot of function-based code makes these errors impossible by imposing immutable objects as the default, if not making it a near requirement, as Haskell does. When the function is the building block of your code, you must specify precisely what the inputs of the function are. This gives you more opportunities to determine pre-conditions for this data. It also ensures that the return results of the function are the primary way you affect the rest of your program.

Functions also tend to be easier to test than objects. It is often tricky to create objects with the precise state you want to assess in a unit test, whereas to test a function you only need to reproduce the inputs.

More Expressive, Reasonable Design

The more you work with functions as your building blocks, and the more you try to fill your code with pure functions, the easier it will be to reason about your code. Imagine you have a couple dozen fields on an object in OO code. If someone calls a function on that object, any of those fields could impact the result of the method call.

Functions give you the opportunity to narrow things down to the precise values that you actually need to perform the computation. They let you separate the essential information from superfluous information, making it more obvious what the responsibilities are for each part of your code.

Multithreading

You can do parallel programming no matter what programming language you're using, but the functional programming paradigm aligns very well with parallel processing. To kick off a new thread in any language, you pretty much always have to pass a function as an argument, and this is more natural in FP. And with pure functions that don't modify shared mutable objects, FP is generally much easier to break into parallelizable pieces that don't require complex locking schemes.

Disadvantages of Functional Programming

Intuition of Complete Objects

Functional programming can feel less intuitive than object oriented programming. Perhaps one reason for this is that object oriented programming allows us to reason about "complete" objects, whose state at any given time is properly defined.

Functions are, in a sense, incomplete. A function is not a what that you can hold as a picture in your head. A function is a how. Given some inputs, how do you produce the outputs? In other words, it's a procedure. And a procedure can only really be imagined as a concrete object once you've filled in its inputs. This is best exemplified by the fact that functions have no native 'Show' instance in Haskell.

>> show (+)
No instance for Show (Integer -> Integer -> Integer) arising from a use of 'show'

If you apply the '+' function to arguments (and so create what could be called an "object"), then we can print it. But until then, it doesn't make much sense. If objects are the building block of your code though, you could, hypothetically, print the state of the objects in your code every step of the way.

Mutable State can be Useful!

As much as mutable state can cause a lot of bugs, it is nonetheless a useful tool for many problems, and decidedly more intuitive for certain data structures. If we just imagine something like the "Snake" game, it has a 2D grid that remains mostly the same from tick to tick, with just a couple things updating. This is easier to capture with mutable data.

Web development is another area where mutable objects are extremely useful. Anytime the user enters information on the page, some object has to change! Web development in FP almost requires its own paradigm (see "Functional Reactive Programming"). Haskell can represent mutable data, but the syntax is more cumbersome; you essentially need a separate data structure. Likewise, other functional languages might make mutability easier than Haskell, but mutability is still, again, more intuitive when objects are your fundamental building block, rather than functions on those objects.

We can see this even with something as simple as loops. Haskell doesn't perform "for-loops" in the same way as other languages, because most for loops essentially rely on the notion that there is some kind of state updating on each iteration of the loop, even if that state is only the integer counter. To write loops in Haskell, you have to learn concepts like maps and folds, which require you to get very used to writing new functions on the fly.

A Full Introduction to Haskell (and its Functional Aspects)

So functional programming languages are perhaps a bit more difficult to learn, but can offer a significant payoff if you put in the time to master the skills. Ultimately, you can use either paradigm for most kinds of projects and keep your development productive. It's down to your personal preference which you try while building software.

If you really want to dive into functional programming though, Haskell is a great language, since it will force you to learn FP principles more than other functional languages. For a complete introduction to Haskell, you should take a look at Haskell From Scratch, our beginner-level course for those new to the language. It will teach you everything you need to know about syntax and fundamental concepts, while providing you with a ton of hands-on practice through exercises and projects.

Haskell From Scratch also includes Making Sense of Monads, our course that shows the more functional side of Haskell by teaching you about the critical concept of monads. With these two courses under your belt, you'll be well on your way to mastery of functional programming! Head over here to learn more about these courses!

by James Bowen at January 15, 2024 04:00 PM

Haskell Interlude

41: Moritz Angermann

Today, Matthías and Joachim are interviewing Moritz Angermann. Moritz knew he wanted to use Haskell before he knew Haskell, fixed cross-compilation as his first GHC contribution. We’ll talk more about cross-compilation to Windows and mobile platforms, why Template Haskell is the cause of most headaches, why you should be careful if your sister calls and tells you to cabal install a package, and finally how we can reduce the fear of new GHC releases, by improving stability.

by Haskell Podcast at January 15, 2024 08:00 AM

Derek Elkins

The Pullback Lemma in Gory Detail (Redux)

Introduction

Andrej Bauer has a paper titled The pullback lemma in gory detail that goes over the proof of the pullback lemma in full detail. This is a basic result of category theory and most introductions leave it as an exercise. It is a good exercise, and you should prove it yourself before reading this article or Andrej Bauer’s.

Andrej Bauer’s proof is what most introductions are expecting you to produce. I very much like the representability perspective on category theory and like to see what proofs look like using this perspective.

So this is a proof of the pullback lemma from the perspective of representability.

Preliminaries

The key thing we need here is a characterization of pullbacks in terms of representability. To just jump to the end, we have for |f : A \to C| and |g : B \to C|, |A \times_{f,g} B| is the pullback of |f| and |g| if and only if it represents the functor \[\{(h, k) \in \mathrm{Hom}({-}, A) \times \mathrm{Hom}({-}, B) \mid f \circ h = g \circ k \}\]

That is to say we have the natural isomorphism \[ \mathrm{Hom}({-}, A \times_{f,g} B) \cong \{(h, k) \in \mathrm{Hom}({-}, A) \times \mathrm{Hom}({-}, B) \mid f \circ h = g \circ k \} \]

We’ll write the left to right direction of the isomorphism as |\langle u,v\rangle : U \to A \times_{f,g} B| where |u : U \to A| and |v : U \to B| and they satisfy |f \circ u = g \circ v|. Applying the isomorphism right to left on the identity arrow gives us two arrows |p_1 : A \times_{f,g} B \to A| and |p_2 : A \times_{f,g} B \to B| satisfying |p_1 \circ \langle u, v\rangle = u| and |p_2 \circ \langle u,v \rangle = v|. (Exercise: Show that this follows from being a natural isomorphism.)

One nice thing about representability is that it reduces categorical reasoning to set-theoretic reasoning that you are probably already used to, as we’ll see. You can connect this definition to a typical universal property based definition used in Andrej Bauer’s article. Here we’re taking it as the definition of the pullback.

Proof

The claim to be proven is if the right square in the below diagram is a pullback square, then the left square is a pullback square if and only if the whole rectangle is a pullback square. \[ \xymatrix { A \ar[d]_{q_1} \ar[r]^{q_2} & B \ar[d]_{p_1} \ar[r]^{p_2} & C \ar[d]^{h} \\ X \ar[r]_{f} & Y \ar[r]_{g} & Z }\]

Rewriting the diagram as equations, we have:

Theorem: If |f \circ q_1 = p_1 \circ q_2|, |g \circ p_1 = h \circ p_2|, and |(B, p_1, p_2)| is a pullback of |g| and |h|, then |(A, q_1, q_2)| is a pullback of |f| and |p_1| if and only if |(A, q_1, p_2 \circ q_2)| is a pullback of |g \circ f| and |h|.

Proof: If |(A, q_1, q_2)| was a pullback of |f| and |p_1| then we’d have the following.

\[\begin{align} \mathrm{Hom}({-}, A) & \cong \{(u_1, u_2) \in \mathrm{Hom}({-}, X)\times\mathrm{Hom}({-}, B) \mid f \circ u_1 = p_1 \circ u_2 \} \\ & \cong \{(u_1, (v_1, v_2)) \in \mathrm{Hom}({-}, X)\times\mathrm{Hom}({-}, Y)\times\mathrm{Hom}({-}, C) \mid f \circ u_1 = p_1 \circ \langle v_1, v_2\rangle \land g \circ v_1 = h \circ v_2 \} \\ & = \{(u_1, (v_1, v_2)) \in \mathrm{Hom}({-}, X)\times\mathrm{Hom}({-}, Y)\times\mathrm{Hom}({-}, C) \mid f \circ u_1 = v_1 \land g \circ v_1 = h \circ v_2 \} \\ & = \{(u_1, v_2) \in \mathrm{Hom}({-}, X)\times\mathrm{Hom}({-}, C) \mid g \circ f \circ u_1 = h \circ v_2 \} \end{align}\]

The second isomorphism is |B| being a pullback and |u_2| is an arrow into |B| so it’s necessarily of the form |\langle v_1, v_2\rangle|. The first equality is just |p_1 \circ \langle v_1, v_2\rangle = v_1| mentioned earlier. The second equality merely eliminates the use of |v_1| using the equation |f \circ u_1 = v_1|.

This overall natural isomorphism, however, is exactly what it means for |A| to be a pullback of |g \circ f| and |h|. We verify the projections are what we expect by pushing |id_A| through the isomorphism. By assumption, |u_1| and |u_2| will be |q_1| and |q_2| respectively in the first isomorphism. We see that |v_2 = p_2 \circ \langle v_1, v_2\rangle = p_2 \circ q_2|.

We simply run the isomorphism backwards to get the other direction of the if and only if. |\square|

The simplicity and compactness of this proof demonstrates why I like representability.

January 15, 2024 01:33 AM

January 11, 2024

Chris Reade

Graphs, Kites and Darts

Graphs, Kites and Darts

Figure 1: Three Coloured Patches
Figure 1: Three Coloured Patches

Non-periodic tilings with Penrose’s kites and darts

(An updated version, since original posting on Jan 6, 2022)

We continue our investigation of the tilings using Haskell with Haskell Diagrams. What is new is the introduction of a planar graph representation. This allows us to define more operations on finite tilings, in particular forcing and composing.

Previously in Diagrams for Penrose Tiles we implemented tools to create and draw finite patches of Penrose kites and darts (such as the samples depicted in figure 1). The code for this and for the new graph representation and tools described here can be found on GitHub https://github.com/chrisreade/PenroseKiteDart.

To describe the tiling operations it is convenient to work with the half-tiles: LD (left dart), RD (right dart), LK (left kite), RK (right kite) using a polymorphic type HalfTile (defined in a module HalfTile)

data HalfTile rep 
 = LD rep | RD rep | LK rep | RK rep   deriving (Show,Eq)

Here rep is a type variable for a representation to be chosen. For drawing purposes, we chose two-dimensional vectors (V2 Double) and called these Pieces.

type Piece = HalfTile (V2 Double)

The vector represents the join edge of the half tile (see figure 2) and thus the scale and orientation are determined (the other tile edges are derived from this when producing a diagram).

Figure 2: The (half-tile) pieces showing join edges (dashed) and origin vertices (red dots)
Figure 2: The (half-tile) pieces showing join edges (dashed) and origin vertices (red dots)

Finite tilings or patches are then lists of located pieces.

type Patch = [Located Piece]

Both Piece and Patch are made transformable so rotate, and scale can be applied to both and translate can be applied to a Patch. (Translate has no effect on a Piece unless it is located.)

In Diagrams for Penrose Tiles we also discussed the rules for legal tilings and specifically the problem of incorrect tilings which are legal but get stuck so cannot continue to infinity. In order to create correct tilings we implemented the decompose operation on patches.

The vector representation that we use for drawing is not well suited to exploring properties of a patch such as neighbours of pieces. Knowing about neighbouring tiles is important for being able to reason about composition of patches (inverting a decomposition) and to find which pieces are determined (forced) on the boundary of a patch.

However, the polymorphic type HalfTile allows us to introduce our alternative graph representation alongside Pieces.

Tile Graphs

In the module Tgraph.Prelude, we have the new representation which treats half tiles as triangular faces of a planar graph – a TileFace – by specialising HalfTile with a triple of vertices (clockwise starting with the tile origin). For example

LD (1,3,4)       RK (6,4,3)
type Vertex = Int
type TileFace = HalfTile (Vertex,Vertex,Vertex)

When we need to refer to particular vertices from a TileFace we use originV (the first vertex – red dot in figure 2), oppV (the vertex at the opposite end of the join edge – dashed edge in figure 2), wingV (the remaining vertex not on the join edge).

originV, oppV, wingV :: TileFace -> Vertex

Tgraphs

The Tile Graphs implementation uses a newtype Tgraph which is a list of tile faces.

newtype Tgraph = Tgraph [TileFace]
                 deriving (Show)

faces :: Tgraph -> [TileFace]
faces (Tgraph fcs) = fcs

For example, fool (short for a fool’s kite) is a Tgraph with 6 faces (and 7 vertices), shown in figure 3.

fool = Tgraph [RD (1,2,3),LD (1,3,4),RK (6,2,5)
              ,LK (6,3,2),RK (6,4,3),LK (6,7,4)
              ]

(The fool is also called an ace in the literature)

Figure 3: fool
Figure 3: fool

With this representation we can investigate how composition works with whole patches. Figure 4 shows a twice decomposed sun on the left and a once decomposed sun on the right (both with vertex labels). In addition to decomposing the right Tgraph to form the left Tgraph, we can also compose the left Tgraph to get the right Tgraph.

Figure 4: sunD2 and sunD
Figure 4: sunD2 and sunD

After implementing composition, we also explore a force operation and an emplace operation to extend tilings.

There are some constraints we impose on Tgraphs.

  • No spurious vertices. The vertices of a Tgraph are the vertices that occur in the faces of the Tgraph (and maxV is the largest number occurring).
  • Connected. The collection of faces must be a single connected component.
  • No crossing boundaries. By this we mean that vertices on the boundary are incident with exactly two boundary edges. The boundary consists of the edges between the Tgraph faces and exterior region(s). This is important for adding faces.
  • Tile connected. Roughly, this means that if we collect the faces of a Tgraph by starting from any single face and then add faces which share an edge with those already collected, we get all the Tgraph faces. This is important for drawing purposes.

In fact, if a Tgraph is connected with no crossing boundaries, then it must be tile connected. (We could define tile connected to mean that the dual graph excluding exterior regions is connected.)

Figure 5 shows two excluded graphs which have crossing boundaries at 4 (left graph) and 13 (right graph). The left graph is still tile connected but the right is not tile connected (the two faces at the top right do not have an edge in common with the rest of the faces.)

Although we have allowed for Tgraphs with holes (multiple exterior regions), we note that such holes cannot be created by adding faces one at a time without creating a crossing boundary. They can be created by removing faces from a Tgraph without necessarily creating a crossing boundary.

Important We are using face as an abbreviation for half-tile face of a Tgraph here, and we do not count the exterior of a patch of faces to be a face. The exterior can also be disconnected when we have holes in a patch of faces and the holes are not counted as faces either. In graph theory, the term face would generally include these other regions, but we will call them exterior regions rather than faces.

Figure 5: A tile-connected graph with crossing boundaries at 4, and a non tile-connected graph
Figure 5: A tile-connected graph with crossing boundaries at 4, and a non tile-connected graph

In addition to the constructor Tgraph we also use

checkedTgraph:: [TileFace] -> Tgraph

which creates a Tgraph from a list of faces, but also performs checks on the required properties of Tgraphs. We can then remove or select faces from a Tgraph and then use checkedTgraph to ensure the resulting Tgraph still satisfies the required properties.

selectFaces, removeFaces  :: [TileFace] -> Tgraph -> Tgraph
selectFaces fcs g = checkedTgraph (faces g `intersect` fcs)
removeFaces fcs g = checkedTgraph (faces g \\ fcs)

Edges and Directed Edges

We do not explicitly record edges as part of a Tgraph, but calculate them as needed. Implicitly we are requiring

  • No spurious edges. The edges of a Tgraph are the edges of the faces of the Tgraph.

To represent edges, a pair of vertices (a,b) is regarded as a directed edge from a to b. A list of such pairs will usually be regarded as a directed edge list. In the special case that the list is symmetrically closed [(b,a) is in the list whenever (a,b) is in the list] we will refer to this as an edge list rather than a directed edge list.

The following functions on TileFaces all produce directed edges (going clockwise round a face).

type Dedge = (Vertex,Vertex)

joinE  :: TileFace -> Dedge  -- join edge - dashed in figure 2
shortE :: TileFace -> Dedge  -- the short edge which is not a join edge
longE  :: TileFace -> Dedge  -- the long edge which is not a join edge
faceDedges :: TileFace -> [Dedge]
  -- all three directed edges clockwise from origin

For the whole Tgraph, we often want a list of all the directed edges of all the faces.

graphDedges :: Tgraph -> [Dedge]
graphDedges = concatMap faceDedges . faces

Because our graphs represent tilings they are planar (can be embedded in a plane) so we know that at most two faces can share an edge and they will have opposite directions of the edge. No two faces can have the same directed edge. So from graphDedges g we can easily calculate internal edges (edges shared by 2 faces) and boundary directed edges (directed edges round the external regions).

internalEdges, boundaryDedges :: Tgraph -> [Dedge]

The internal edges of g are those edges which occur in both directions in graphDedges g. The boundary directed edges of g are the missing reverse directions in graphDedges g.

We also refer to all the long edges of a Tgraph (including kite join edges) as phiEdges (both directions of these edges).

phiEdges :: Tgraph -> [Dedge]

This is so named because, when drawn, these long edges are phi times the length of the short edges (phi being the golden ratio which is approximately 1.618).

Drawing Tgraphs (Patches and VPatches)

The module Tgraph.Convert contains functions to convert a Tgraph to our previous vector representation (Patch) defined in TileLib so we can use the existing tools to produce diagrams.

However, it is convenient to have an intermediate stage (a VPatch = Vertex Patch) which contains both faces and calculated vertex locations (a finite map from vertices to locations). This allows vertex labels to be drawn and for faces to be identified and retained/excluded after the location information is calculated.

data VPatch = VPatch { vLocs :: VertexLocMap
                     , vpFaces::[TileFace]
                     } deriving Show

The conversion functions include

makeVP   :: Tgraph -> VPatch

For drawing purposes we introduced a class Drawable which has a means to create a diagram when given a function to draw Pieces.

class Drawable a where
  drawWith :: (Piece -> Diagram B) -> a -> Diagram B

This allows us to make Patch, VPatch and Tgraph instances of Drawable, and we can define special cases for the most frequently used drawing tools.

draw :: Drawable a => a -> Diagram B
draw = drawWith drawPiece

drawj :: Drawable a => a -> Diagram B
drawj = drawWith dashjPiece

We also need to be able to create diagrams with vertex labels, so we use a draw function modifier

class DrawableLabelled a where
  labelSize :: Measure Double -> (VPatch -> Diagram B) -> a -> Diagram B

Both VPatch and Tgraph are made instances (but not Patch as this no longer has vertex information). The type Measure is defined in Diagrams, but we generally use a default measure for labels to define

labelled :: DrawableLabelled a => (VPatch -> Diagram B) -> a -> Diagram B
labelled = labelSize (normalized 0.018)

This allows us to use, for example (where g is a Tgraph or VPatch)

labelled draw g
labelled drawj g

One consequence of using abstract graphs is that there is no unique predefined way to orient or scale or position the VPatch (and Patch) arising from a Tgraph representation. Our implementation selects a particular join edge and aligns it along the x-axis (unit length for a dart, philength for a kite) and tile-connectedness ensures the rest of the VPatch (and Patch) can be calculated from this.

We also have functions to re-orient a VPatch and lists of VPatchs using chosen pairs of vertices. [Simply doing rotations on the final diagrams can cause problems if these include vertex labels. We do not, in general, want to rotate the labels – so we need to orient the VPatch before converting to a diagram]

Decomposing Graphs

We previously implemented decomposition for patches which splits each half-tile into two or three smaller scale half-tiles.

decompPatch :: Patch -> Patch

We now have a Tgraph version of decomposition in the module Tgraph.Decompose:

decompose :: Tgraph -> Tgraph

Graph decomposition is particularly simple. We start by introducing one new vertex for each long edge (the phiEdges) of the Tgraph. We then build the new faces from each old face using the new vertices.

As a running example we take fool (mentioned above) and its decomposition foolD

*Main> foolD = decompose fool

*Main> foolD
Tgraph [LK (1,8,3),RD (2,3,8),RK (1,3,9)
       ,LD (4,9,3),RK (5,13,2),LK (5,10,13)
       ,RD (6,13,10),LK (3,2,13),RK (3,13,11)
       ,LD (6,11,13),RK (3,14,4),LK (3,11,14)
       ,RD (6,14,11),LK (7,4,14),RK (7,14,12)
       ,LD (6,12,14)
       ]

which are best seen together (fool followed by foolD) in figure 6.

Figure 6: fool and foolD (= decomposeG fool)
Figure 6: fool and foolD (= decompose fool)

Composing Tgraphs, and Unknowns

Composing is meant to be an inverse to decomposing, and one of the main reasons for introducing our graph representation. In the literature, decomposition and composition are defined for infinite tilings and in that context they are unique inverses to each other. For finite patches, however, we will see that composition is not always uniquely determined.

In figure 7 (Two Levels) we have emphasised the larger scale faces on top of the smaller scale faces.

Figure 7: Two Levels
Figure 7: Two Levels

How do we identify the composed tiles? We start by classifying vertices which are at the wing tips of the (smaller) darts as these determine how things compose. In the interior of a graph/patch (e.g in figure 7), a dart wing tip always coincides with a second dart wing tip, and either

  1. the 2 dart halves share a long edge. The shared wing tip is then classified as a largeKiteCentre and is at the centre of a larger kite. (See left vertex type in figure 8), or
  2. the 2 dart halves touch at their wing tips without sharing an edge. This shared wing tip is classified as a largeDartBase and is the base of a larger dart. (See right vertex type in figure 8)
Figure 8: largeKiteCentre (left) and largeDartBase (right)
Figure 8: largeKiteCentre (left) and largeDartBase (right)

[We also call these (respectively) a deuce vertex type and a jack vertex type later in figure 10]

Around the boundary of a Tgraph, the dart wing tips may not share with a second dart. Sometimes the wing tip has to be classified as unknown but often it can be decided by looking at neighbouring tiles. In this example of a four times decomposed sun (sunD4), it is possible to classify all the dart wing tips as a largeKiteCentre or a largeDartBase so there are no unknowns.

If there are no unknowns, then we have a function to produce the unique composed Tgraph.

compose:: Tgraph -> Tgraph

Any correct decomposed Tgraph without unknowns will necessarily compose back to its original. This makes compose a left inverse to decompose provided there are no unknowns.

For example, with an (n times) decomposed sun we will have no unknowns, so these will all compose back up to a sun after n applications of compose. For n=4 (sunD4 – the smaller scale shown in figure 7) the dart wing classification returns 70 largeKiteCentres, 45 largeDartBases, and no unknowns.

Similarly with the simpler foolD example, if we classsify the dart wings we get

largeKiteCentres = [14,13]
largeDartBases = [3]
unknowns = []

In foolD (the right hand Tgraph in figure 6), nodes 14 and 13 are new kite centres and node 3 is a new dart base. There are no unknowns so we can use compose safely

*Main> compose foolD
Tgraph [RD (1,2,3),LD (1,3,4),RK (6,2,5)
       ,RK (6,4,3),LK (6,3,2),LK (6,7,4)
       ]

which reproduces the original fool (left hand Tgraph in figure 6).

However, if we now check out unknowns for fool we get

largeKiteCentres = []
largeDartBases = []
unknowns = [4,2]    

So both nodes 2 and 4 are unknowns. It had looked as though fool would simply compose into two half kites back-to-back (sharing their long edge not their join), but the unknowns show there are other possible choices. Each unknown could become a largeKiteCentre or a largeDartBase.

The question is then what to do with unknowns.

Partial Compositions

In fact our compose resolves two problems when dealing with finite patches. One is the unknowns and the other is critical missing faces needed to make up a new face (e.g the absence of any half dart).

It is implemented using an intermediary function for partial composition

partCompose:: Tgraph -> ([TileFace],Tgraph) 

partCompose will compose everything that is uniquely determined, but will leave out faces round the boundary which cannot be determined or cannot be included in a new face. It returns the faces of the argument Tgraph that were not used, along with the composed Tgraph.

Figure 9 shows the result of partCompose applied to two graphs. [These are force kiteD3 and force dartD3 on the left. Force is described later]. In each case, the excluded faces of the starting Tgraph are shown in pale green, overlaid by the composed Tgraph on the right.

Figure 9: partCompose for two graphs (force kiteD3 top row and force dartD3 bottom row)
Figure 9: partCompose for two graphs (force kiteD3 top row and force dartD3 bottom row)

Then compose is simply defined to keep the composed faces and ignore the unused faces produced by partCompose.

compose:: Tgraph -> Tgraph
compose = snd . partCompose 

This approach avoids making a decision about unknowns when composing, but it may lose some information by throwing away the uncomposed faces.

For correct Tgraphs g, if decompose g has no unknowns, then compose is a left inverse to decompose. However, if we take g to be two kite halves sharing their long edge (not their join edge), then these decompose to fool which produces an empty Tgraph when recomposed. Thus we do not have g = compose (decompose g) in general. On the other hand we do have g = compose (decompose g) for correct whole-tile Tgraphs g (whole-tile means all half-tiles of g have their matching half-tile on their join edge in g)

Later (figure 21) we show another exception to g = compose (decompose g) with an incorrect tiling.

We make use of

selectFacesVP    :: [TileFace] -> VPatch -> VPatch
removeFacesVP    :: [TileFace] -> VPatch -> VPatch

for creating VPatches from selected tile faces of a Tgraph or VPatch. This allows us to represent and draw a list of faces which need not be connected nor satisfy the no crossing boundaries property provided the Tgraph it was derived from had these properties.

Forcing

When building up a tiling, following the rules, there is often no choice about what tile can be added alongside certain tile edges at the boundary. Such additions are forced by the existing patch of tiles and the rules. For example, if a half tile has its join edge on the boundary, the unique mirror half tile is the only possibility for adding a face to that edge. Similarly, the short edge of a left (respectively, right) dart can only be matched with the short edge of a right (respectively, left) kite. We also make use of the fact that only 7 types of vertex can appear in (the interior of) a patch, so on a boundary vertex we sometimes have enough of the faces to determine the vertex type. These are given the following names in the literature (shown in figure 10): sun, star, jack (=largeDartBase), queen, king, ace, deuce (=largeKiteCentre).

Figure 10: Vertex types
Figure 10: Vertex types

The function

force :: Tgraph -> Tgraph

will add some faces on the boundary that are forced (i.e new faces where there is exactly one possible choice). For example:

  • When a join edge is on the boundary – add the missing half tile to make a whole tile.
  • When a half dart has its short edge on the boundary – add the half kite that must be on the short edge.
  • When a vertex is both a dart origin and a kite wing (it must be a queen or king vertex) – if there is a boundary short edge of a kite half at the vertex, add another kite half sharing the short edge, (this converts 1 kite to 2 and 3 kites to 4 in combination with the first rule).
  • When two half kites share a short edge their common oppV vertex must be a deuce vertex – add any missing half darts needed to complete the vertex.

Figure 11 shows foolDminus (which is foolD with 3 faces removed) on the left and the result of forcing, ie force foolDminus on the right which is the same Tgraph we get from force foolD (modulo vertex renumbering).

foolDminus = 
    removeFaces [RD(6,14,11), LD(6,12,14), RK(5,13,2)] foolD
Figure 11: foolDminus and force foolDminus = force foolD
Figure 11: foolDminus and force foolDminus = force foolD

Figures 12, 13 and 14 illustrate the result of forcing a 5-times decomposed kite, a 5-times decomposed dart, and a 5-times decomposed sun (respectively). The first two figures reproduce diagrams from an article by Roger Penrose illustrating the extent of influence of tiles round a decomposed kite and dart. [Penrose R Tilings and quasi-crystals; a non-local growth problem? in Aperiodicity and Order 2, edited by Jarich M, Academic Press, 1989. (fig 14)].

Figure 12: force kiteD5 with kiteD5 shown in red
Figure 12: force kiteD5 with kiteD5 shown in red
Figure 13: force dartD5 with dartD5 shown in red
Figure 13: force dartD5 with dartD5 shown in red
Figure 14: force sunD5 with sunD5 shown in red
Figure 14: force sunD5 with sunD5 shown in red

In figure 15, the bottom row shows successive decompositions of a dart (dashed blue arrows from right to left), so applying compose to each dart will go back (green arrows from left to right). The black vertical arrows are force. The solid blue arrows from right to left are (force . decompose) being applied to the successive forced Tgraphs. The green arrows in the reverse direction are compose again and the intermediate (partCompose) figures are shown in the top row with the remainder faces in pale green.

Figure 15: Arrows: black = force, green = composeG, solid blue = (force . decomposeG)
Figure 15: Arrows: black = force, green = compose, solid blue = (force . decompose)

Figure 16 shows the forced graphs of the seven vertex types (with the starting Tgraphs in red) along with a kite (top right).

Figure 16: Relating the forced seven vertex types and the kite
Figure 16: Relating the forced seven vertex types and the kite

These are related to each other as shown in the columns. Each Tgraph composes to the one above (an empty Tgraph for the ones in the top row) and the Tgraph below is its forced decomposition. [The rows have been scaled differently to make the vertex types easier to see.]

Adding Faces to a Tgraph

This is technically tricky because we need to discover what vertices (and implicitly edges) need to be newly created and which ones already exist in the Tgraph. This goes beyond a simple graph operation and requires use of the geometry of the faces. We have chosen not to do a full conversion to vectors to work out all the geometry, but instead we introduce a local representation of relative directions of edges at a vertex allowing a simple equality test.

Edge directions

All directions are integer multiples of 1/10th turn (mod 10) so we use these integers for face internal angles and boundary external angles. The face adding process always adds to the right of a given directed edge (a,b) which must be a boundary directed edge. [Adding to the left of an edge (a,b) would mean that (b,a) will be the boundary direction and so we are really adding to the right of (b,a)]. Face adding looks to see if either of the two other edges already exist in the Tgraph by considering the end points a and b to which the new face is to be added, and checking angles.

This allows an edge in a particular sought direction to be discovered. If it is not found it is assumed not to exist. However, the search will be undermined if there are crossing boundaries. In such a case there will be more than two boundary directed edges at the vertex and there is no unique external angle.

Establishing the no crossing boundaries property ensures these failures cannot occur. We can easily check this property for newly created Tgraphs (with checkedTgraph) and the face adding operations cannot create crossing boundaries.

Touching Vertices and Crossing Boundaries

When a new face to be added on (a,b) has neither of the other two edges already in the Tgraph, the third vertex needs to be created. However it could already exist in the Tgraph – it is not on an edge coming from a or b but from another non-local part of the Tgraph. We call this a touching vertex. If we simply added a new vertex without checking for a clash this would create a non-sensible Tgraph. However, if we do check and find an existing vertex, we still cannot add the face using this because it would create a crossing boundary.

Our version of forcing prevents face additions that would create a touching vertex/crossing boundary by calculating the positions of boundary vertices.

No conflicting edges

There is a final (simple) check when adding a new face, to prevent a long edge (phiEdge) sharing with a short edge. This can arise if we force an incorrect Tgraph (as we will see later).

Implementing Forcing

Our order of forcing prioritises updates (face additions) which do not introduce a new vertex. Such safe updates are easy to recognise and they do not require a touching vertex check. Surprisingly, this pretty much removes the problem of touching vertices altogether.

As an illustration, consider foolDMinus again on the left of figure 11. Adding the left dart onto edge (12,14) is not a safe addition (and would create a crossing boundary at 6). However, adding the right dart RD(6,14,11) is safe and creates the new edge (6,14) which then makes the left dart addition safe. In fact it takes some contrivance to come up with a Tgraph with an update that could fail the check during forcing when safe cases are always done first. Figure 17 shows such a contrived Tgraph formed by removing the faces shown in green from a twice decomposed sun on the left. The forced result is shown on the right. When there are no safe cases, we need to try an unsafe one. The four green faces at the bottom are blocked by the touching vertex check. This leaves any one of 9 half-kites at the centre which would pass the check. But after just one of these is added, the check is not needed again. There is always a safe addition to be done at each step until all the green faces are added.

Figure 17: A contrived example requiring a touching vertex check
Figure 17: A contrived example requiring a touching vertex check

Boundary information

The implementation of forcing has been made more efficient by calculating some boundary information in advance. This boundary information uses a type BoundaryState

data BoundaryState
  = BoundaryState
    { boundary    :: [Dedge]
    , bvFacesMap  :: Mapping Vertex [TileFace]
    , bvLocMap    :: Mapping Vertex (Point V2 Double)
    , allFaces    :: [TileFace]
    , nextVertex  :: Vertex
    } deriving (Show)

This records the boundary directed edges (boundary) plus a mapping of the boundary vertices to their incident faces (bvFacesMap) plus a mapping of the boundary vertices to their positions (bvLocMap). It also keeps track of all the faces and the vertex number to use when adding a vertex. The boundary information is easily incremented for each face addition without being recalculated from scratch, and a final Tgraph with all the new faces is easily recovered from the boundary information when there are no more updates.

makeBoundaryState  :: Tgraph -> BoundaryState
recoverGraph  :: BoundaryState -> Tgraph

The saving that comes from using boundary information lies in efficient incremental changes to the boundary information and, of course, in avoiding the need to consider internal faces. As a further optimisation we keep track of updates in a mapping from boundary directed edges to updates, and supply a list of affected edges after an update so the update calculator (update generator) need only revise these. The boundary and mapping are combined in a ForceState.

type UpdateMap = Mapping Dedge Update
type UpdateGenerator = BoundaryState -> [Dedge] -> UpdateMap
data ForceState = ForceState 
       { boundaryState:: BoundaryState
       , updateMap:: UpdateMap 
       }

Forcing then involves using a specific update generator (allUGenerator) and initialising the state, then using the recursive forceAll which keeps doing updates until there are no more, before recovering the final Tgraph.

force:: Tgraph -> Tgraph
force = forceWith allUGenerator

forceWith:: UpdateGenerator -> Tgraph -> Tgraph
forceWith uGen = recoverGraph . boundaryState . 
                 forceAll uGen . initForceState uGen

forceAll :: UpdateGenerator -> ForceState -> ForceState
initForceState :: UpdateGenerator -> Tgraph -> ForceState

In addition to force we can easily define

wholeTiles:: Tgraph -> Tgraph
wholeTiles = forceWith wholeTileUpdates 

which just uses the first forcing rule to make sure every half-tile has a matching other half.

We also have a version of force which counts to a specific number of face additions.

stepForce :: Int -> ForceState -> ForceState

This proved essential in uncovering problems of accumulated inaccuracy in calculating boundary positions (now fixed).

Some Other Experiments

Below we describe results of some experiments using the tools introduced above. Specifically: emplacements, sub-Tgraphs, incorrect tilings, and composition choices.

Emplacements

The finite number of rules used in forcing are based on local boundary vertex and edge information only. We thought we may be able to improve on this by considering a composition and forcing at the next level up before decomposing and forcing again. This thus considers slightly broader local information. In fact we can iterate this process to all the higher levels of composition. Some Tgraphs produce an empty Tgraph when composed so we can regard those as maximal compositions. For example compose fool produces an empty Tgraph.

The idea was to take an arbitrary Tgraph and apply (compose . force) repeatedly to find its maximally composed (non-empty) Tgraph, before applying (force . decompose) repeatedly back down to the starting level (so the same number of decompositions as compositions).

We called the function emplace, and called the result the emplacement of the starting Tgraph as it shows a region of influence around the starting Tgraph.

With earlier versions of forcing when we had fewer rules, emplace g often extended force g for a Tgraph g. This allowed the identification of some new rules. However, since adding the new rules we have not found Tgraphs where the result of force had fewer faces than the result of emplace.

[As an important update, we have now found examples where the result of force strictly includes the result of emplace (modulo vertex renumbering).

Sub-Tgraphs

In figure 18 on the left we have a four times decomposed dart dartD4 followed by two sub-Tgraphs brokenDart and badlyBrokenDart which are constructed by removing faces from dartD4 (but retaining the connectedness condition and the no crossing boundaries condition). These all produce the same forced result (depicted middle row left in figure 15).

Figure 18: dartD4, brokenDart, badlyBrokenDart
Figure 18: dartD4, brokenDart, badlyBrokenDart

However, if we do compositions without forcing first we find badlyBrokenDart fails because it produces a graph with crossing boundaries after 3 compositions. So compose on its own is not always safe, where safe means guaranteed to produce a valid Tgraph from a valid correct Tgraph.

In other experiments we tried force on Tgraphs with holes and on incomplete boundaries around a potential hole. For example, we have taken the boundary faces of a forced, 5 times decomposed dart, then removed a few more faces to make a gap (which is still a valid Tgraph). This is shown at the top in figure 19. The result of forcing reconstructs the complete original forced graph. The bottom figure shows an intermediate stage after 2200 face additions. The gap cannot be closed off to make a hole as this would create a crossing boundary, but the channel does get filled and eventually closes the gap without creating a hole.

Figure 19: Forcing boundary faces with a gap (after 2200 steps)
Figure 19: Forcing boundary faces with a gap (after 2200 steps)

Incorrect Tilings

When we say a Tgraph g is correct (respectively: incorrect), we mean g represents a correct tiling (respectively: incorrect tiling). A simple example of an incorrect Tgraph is a kite with a dart on each side (referred to as a mistake by Penrose) shown on the left of figure 20.

*Main> mistake
Tgraph [RK (1,2,4),LK (1,3,2),RD (3,1,5)
       ,LD (4,6,1),LD (3,5,7),RD (4,8,6)
       ]

If we try to force (or emplace) this Tgraph it produces an error in construction which is detected by the test for conflicting edge types (a phiEdge sharing with a non-phiEdge).

*Main> force mistake
... *** Exception: doUpdate:(incorrect tiling)
Conflicting new face RK (11,1,6)
with neighbouring faces
[RK (9,1,11),LK (9,5,1),RK (1,2,4),LK (1,3,2),RD (3,1,5),LD (4,6,1),RD (4,8,6)]
in boundary
BoundaryState ...

In figure 20 on the right, we see that after successfully constructing the two whole kites on the top dart short edges, there is an attempt to add an RK on edge (1,6). The process finds an existing edge (1,11) in the correct direction for one of the new edges so tries to add the erroneous RK (11,1,6) which fails a noConflicts test.

Figure 20: An incorrect Tgraph (mistake), and the point at which force mistake fails
Figure 20: An incorrect Tgraph (mistake), and the point at which force mistake fails

So it is certainly true that incorrect Tgraphs may fail on forcing, but forcing cannot create an incorrect Tgraph from a correct Tgraph.

If we apply decompose to mistake it produces another incorrect Tgraph (which is similarly detected if we apply force), but will nevertheless still compose back to mistake if we do not try to force.

Interestingly, though, the incorrectness of a Tgraph is not always preserved by decompose. If we start with mistake1 which is mistake with just two of the half darts (and also incorrect) we still get a similar failure on forcing, but decompose mistake1 is no longer incorrect. If we apply compose to the result or force then compose the mistake is thrown away to leave just a kite (see figure 21). This is an example where compose is not a left inverse to either decompose or (force . decompose).

Figure 21: mistake1 with its decomposition, forced decomposition, and recomposed.
Figure 21: mistake1 with its decomposition, forced decomposition, and recomposed.

Composing with Choices

We know that unknowns indicate possible choices (although some choices may lead to incorrect Tgraphs). As an experiment we introduce

makeChoices :: Tgraph -> [Tgraph]

which produces 2^n alternatives for the 2 choices of each of n unknowns (prior to composing). This uses forceLDB which forces an unknown to be a largeDartBase by adding an appropriate joined half dart at the node, and forceLKC which forces an unknown to be a largeKiteCentre by adding a half dart and a whole kite at the node (making up the 3 pieces for a larger half kite).

Figure 22 illustrates the four choices for composing fool this way. The top row has the four choices of makeChoices fool (with the fool shown embeded in red in each case). The bottom row shows the result of applying compose to each choice.

Figure 22: makeChoices fool (top row) and compose of each choice (bottom row)
Figure 22: makeChoices fool (top row) and compose of each choice (bottom row)

In this case, all four compositions are correct tilings. The problem is that, in general, some of the choices may lead to incorrect tilings. More specifically, a choice of one unknown can determine what other unknowns have to become with constraints such as

  • a and b have to be opposite choices
  • a and b have to be the same choice
  • a and b cannot both be largeKiteCentres
  • a and b cannot both be largeDartBases

This analysis of constraints on unknowns is not trivial. The potential exponential results from choices suggests we should compose and force as much as possible and only consider unknowns of a maximal Tgraph.

For calculating the emplacement of a Tgraph, we first find the forced maximal Tgraph before decomposing. We could also consider using makeChoices at this top step when there are unknowns, i.e a version of emplace which produces these alternative results (emplaceChoices)

The result of emplaceChoices is illustrated for foolD in figure 23. The first force and composition is unique producing the fool level at which point we get 4 alternatives each of which compose further as previously illustrated in figure 22. Each of these are forced, then decomposed and forced, decomposed and forced again back down to the starting level. In figure 23 foolD is overlaid on the 4 alternative results. What they have in common is (as you might expect) emplace foolD which equals force foolD and is the graph shown on the right of figure 11.

Figure 23: emplaceChoices foolD
Figure 23: emplaceChoices foolD

Future Work

I am collaborating with Stephen Huggett who suggested the use of graphs for exploring properties of the tilings. We now have some tools to experiment with but we would also like to complete some formalisation and proofs.

It would also be good to establish whether it is true that g is incorrect iff force g fails.

We have other conjectures relating to subgraph ordering of Tgraphs and Galois connections to explore.

by readerunner at January 11, 2024 12:51 PM

January 10, 2024

Chris Reade

Diagrams for Penrose Tiles

Penrose Kite and Dart Tilings with Haskell Diagrams

Revised version (no longer the full program in this literate Haskell)

Infinite non-periodic tessellations of Roger Penrose’s kite and dart tiles.

leftFilledSun6
leftFilledSun6

As part of a collaboration with Stephen Huggett, working on some mathematical properties of Penrose tilings, I recognised the need for quick renderings of tilings. I thought Haskell diagrams would be helpful here, and that turned out to be an excellent choice. Two dimensional vectors were well-suited to describing tiling operations and these are included as part of the diagrams package.

This literate Haskell uses the Haskell diagrams package to draw tilings with kites and darts. It also implements the main operations of compChoices and decompPatch which are used for constructing tilings (explained below).

Firstly, these 5 lines are needed in Haskell to use the diagrams package:

{-# LANGUAGE NoMonomorphismRestriction #-}
{-# LANGUAGE FlexibleContexts          #-}
{-# LANGUAGE TypeFamilies              #-}
import Diagrams.Prelude
import Diagrams.Backend.SVG.CmdLine

and we will also import a module for half tiles (explained later)

import HalfTile

These are the kite and dart tiles.

Kite and Dart
Kite and Dart

The red line marking here on the right hand copies, is purely to illustrate rules about how tiles can be put together for legal (non-periodic) tilings. Obviously edges can only be put together when they have the same length. If all the tiles are marked with red lines as illustrated on the right, the vertices where tiles meet must all have a red line or none must have a red line at that vertex. This prevents us from forming a simple rombus by placing a kite top at the base of a dart and thus enabling periodic tilings.

All edges are powers of the golden section \phi which we write as phi.

phi::Double
phi = (1.0 + sqrt 5.0) / 2.0

So if the shorter edges are unit length, then the longer edges have length phi. We also have the interesting property of the golden section that phi^2 = phi + 1 and so 1/phi = phi-1, phi^3 = 2phi +1 and 1/phi^2 = 2-phi.

All angles in the figures are multiples of tt which is 36 deg or 1/10 turn. We use ttangle to express such angles (e.g 180 degrees is ttangle 5).

ttangle:: Int -> Angle Double
ttangle n = (fromIntegral (n `mod` 10))*^tt
             where tt = 1/10 @@ turn

Pieces

In order to implement compChoices and decompPatch, we need to work with half tiles. We now define these in the separately imported module HalfTile with constructors for Left Dart, Right Dart, Left Kite, Right Kite

data HalfTile rep = LD rep -- defined in HalfTile module
                  | RD rep
                  | LK rep
                  | RK rep

where rep is a type variable allowing for different representations. However, here, we want to use a more specific type which we will call Piece:

type Piece = HalfTile (V2 Double)

where the half tiles have a simple 2D vector representation to provide orientation and scale. The vector represents the join edge of each half tile where halves come together. The origin for a dart is the tip, and the origin for a kite is the acute angle tip (marked in the figure with a red dot).

These are the only 4 pieces we use (oriented along the x axis)

ldart,rdart,lkite,rkite:: Piece
ldart = LD unitX
rdart = RD unitX
lkite = LK (phi*^unitX)
rkite = RK (phi*^unitX)
pieces
pieces

Perhaps confusingly, we regard left and right of a dart differently from left and right of a kite when viewed from the origin. The diagram shows the left dart before the right dart and the left kite before the right kite. Thus in a complete tile, going clockwise round the origin the right dart comes before the left dart, but the left kite comes before the right kite.

When it comes to drawing pieces, for the simplest case, we just want to show the two tile edges of each piece (and not the join edge). These edges are calculated as a list of 2 new vectors, using the join edge vector v. They are ordered clockwise from the origin of each piece

pieceEdges:: Piece -> [V2 Double]
pieceEdges (LD v) = [v',v ^-^ v'] where v' = phi*^rotate (ttangle 9) v
pieceEdges (RD v) = [v',v ^-^ v'] where v' = phi*^rotate (ttangle 1) v
pieceEdges (RK v) = [v',v ^-^ v'] where v' = rotate (ttangle 9) v
pieceEdges (LK v) = [v',v ^-^ v'] where v' = rotate (ttangle 1) v

Now drawing lines for the 2 outer edges of a piece is simply

drawPiece:: Piece -> Diagram B
drawPiece = strokeLine . fromOffsets . pieceEdges

and drawing all 3 edges round a piece is

drawRoundPiece:: Piece -> Diagram B
drawRoundPiece = strokeLoop . closeLine . fromOffsets . pieceEdges

To fill half tile pieces, we can use fillOnlyPiece which fills without showing edges of a half tile (by using line width none).

fillOnlyPiece:: Colour Double -> Piece -> Diagram B
fillOnlyPiece col piece = drawRoundPiece piece # fc col # lw none

We also use fillPieceDK which fills darts and kites with given colours and also draws edges using drawPiece.

fillPieceDK:: Colour Double -> Colour Double -> Piece -> Diagram B
fillPieceDK dcol kcol piece = drawPiece piece <> fillOnlyPiece col piece where
    col = case piece of (LD _) -> dcol
                        (RD _) -> dcol
                        (LK _) -> kcol
                        (RK _) -> kcol

For an alternative fill operation on whole tiles, it is useful to calculate a list of the 4 tile edges of a completed half-tile piece clockwise from the origin of the tile. (This will allow colour filling a whole tile)

wholeTileEdges:: Piece -> [V2 Double]
wholeTileEdges (LD v) = pieceEdges (RD v) ++ map negated (reverse (pieceEdges (LD v)))
wholeTileEdges (RD v) = wholeTileEdges (LD v)
wholeTileEdges (LK v) = pieceEdges (LK v) ++ map negated (reverse (pieceEdges (RK v)))
wholeTileEdges (RK v) = wholeTileEdges (LK v)

To fill whole tiles with colours, darts with dcol and kites with kcol we can now use leftFillPieceDK. This uses only the left pieces to identify the whole tile and ignores right pieces so that a tile is not filled twice.

leftFillPieceDK:: Colour Double -> Colour Double -> Piece -> Diagram B
leftFillPieceDK dcol kcol c = case c of 
  (LD _) -> (strokeLoop $ glueLine $ fromOffsets $ wholeTileEdges c)  # fc dcol
  (LK _) -> (strokeLoop $ glueLine $ fromOffsets $ wholeTileEdges c)  # fc kcol
  _      -> mempty

By making Pieces transformable we can reuse generic transform operations. These 4 lines of code are required to do this

type instance N (HalfTile a) = N a
type instance V (HalfTile a) = V a
instance Transformable a => Transformable (HalfTile a) where
    transform t ht = fmap (transform t) ht

So we can also scale and rotate a piece by an angle. (Positive rotations are in the anticlockwise direction.)

scale :: Double -> Piece -> Piece
rotate :: Angle Double -> Piece -> Piece

Patches

A patch is a list of located pieces (each with a 2D point)

type Patch = [Located Piece]

To turn a whole patch into a diagram using some function pd for drawing the pieces, we use

drawPatchWith:: (Piece -> Diagram B) -> Patch -> Diagram B 
drawPatchWith pd patch = position $ fmap (viewLoc . mapLoc pd) patch

Here mapLoc applies a function to the piece in a located piece – producing a located diagram in this case, and viewLoc returns the pair of point and diagram from a located diagram. Finally position forms a single diagram from the list of pairs of points and diagrams.

Update: We now use a class for drawable tilings, making Patch an instance

class Drawable a where
 drawWith :: (Piece -> Diagram B) -> a -> Diagram B
instance Drawable Patch where
 drawWith = drawPatchWith

We then introduce special cases:

draw :: Drawable a => a -> Diagram B
draw = drawWith drawPiece
fillDK:: Drawable a => Colour Double -> Colour Double -> a -> Diagram B
fillDK c1 c2 = drawWith (fillPieceDK c1 c2)

Patches are automatically inferred to be transformable now Pieces are transformable, so we can also scale a patch, translate a patch by a vector, and rotate a patch by an angle.

scale :: Double -> Patch -> Patch
rotate :: Angle Double -> Patch -> Patch
translate:: V2 Double -> Patch -> Patch

As an aid to creating patches with 5-fold rotational symmetry, we combine 5 copies of a basic patch (rotated by multiples of ttangle 2 successively).

penta:: Patch -> Patch
penta p = concatMap copy [0..4] 
            where copy n = rotate (ttangle (2*n)) p

This must be used with care to avoid nonsense patches. But two special cases are

sun,star::Patch         
sun =  penta [rkite `at` origin, lkite `at` origin]
star = penta [rdart `at` origin, ldart `at` origin]

This figure shows some example patches, drawn with draw The first is a star and the second is a sun.

tile patches
tile patches

The tools so far for creating patches may seem limited (and do not help with ensuring legal tilings), but there is an even bigger problem.

Correct Tilings

Unfortunately, correct tilings – that is, tilings which can be extended to infinity – are not as simple as just legal tilings. It is not enough to have a legal tiling, because an apparent (legal) choice of placing one tile can have non-local consequences, causing a conflict with a choice made far away in a patch of tiles, resulting in a patch which cannot be extended. This suggests that constructing correct patches is far from trivial.

The infinite number of possible infinite tilings do have some remarkable properties. Any finite patch from one of them, will occur in all the others (infinitely many times) and within a relatively small radius of any point in an infinite tiling. (For details of this see links at the end)

This is why we need a different approach to constructing larger patches. There are two significant processes used for creating patches, namely inflate (also called compose) and decompose.

To understand these processes, take a look at the following figure.

experiment
experiment

Here the small pieces have been drawn in an unusual way. The edges have been drawn with dashed lines, but long edges of kites have been emphasised with a solid line and the join edges of darts marked with a red line. From this you may be able to make out a patch of larger scale kites and darts. This is an inflated patch arising from the smaller scale patch. Conversely, the larger kites and darts decompose to the smaller scale ones.

Decomposition

Since the rule for decomposition is uniquely determined, we can express it as a simple function on patches.

decompPatch :: Patch -> Patch
decompPatch = concatMap decompPiece

where the function decompPiece acts on located pieces and produces a list of the smaller located pieces contained in the piece. For example, a larger right dart will produce both a smaller right dart and a smaller left kite. Decomposing a located piece also takes care of the location, scale and rotation of the new pieces.

decompPiece lp = case viewLoc lp of
  (p, RD vd)-> [ LK vd  `at` p
               , RD vd' `at` (p .+^ v')
               ] where v'  = phi*^rotate (ttangle 1) vd
                       vd' = (2-phi) *^ (negated v') -- (2-phi) = 1/phi^2
  (p, LD vd)-> [ RK vd `at` p
               , LD vd' `at` (p .+^ v')
               ]  where v'  = phi*^rotate (ttangle 9) vd
                        vd' = (2-phi) *^ (negated v')  -- (2-phi) = 1/phi^2
  (p, RK vk)-> [ RD vd' `at` p
               , LK vk' `at` (p .+^ v')
               , RK vk' `at` (p .+^ v')
               ] where v'  = rotate (ttangle 9) vk
                       vd' = (2-phi) *^ v' -- v'/phi^2
                       vk' = ((phi-1) *^ vk) ^-^ v' -- (phi-1) = 1/phi
  (p, LK vk)-> [ LD vd' `at` p
               , RK vk' `at` (p .+^ v')
               , LK vk' `at` (p .+^ v')
               ] where v'  = rotate (ttangle 1) vk
                       vd' = (2-phi) *^ v' -- v'/phi^2
                       vk' = ((phi-1) *^ vk) ^-^ v' -- (phi-1) = 1/phi

This is illustrated in the following figure for the cases of a right dart and a right kite.

explanation
explanation

The symmetric diagrams for left pieces are easy to work out from these, so they are not illustrated.

With the decompPatch operation we can start with a simple correct patch, and decompose repeatedly to get more and more detailed patches. (Each decomposition scales the tiles down by a factor of 1/phi but we can rescale at any time.)

This figure illustrates how each piece decomposes with 4 decomposition steps below each one.

four decompositions of pieces
four decompositions of pieces
thePieces =  [ldart, rdart, lkite, rkite]  
fourDecomps = hsep 1 $ fmap decomps thePieces # lw thin where
        decomps pc = vsep 1 $ fmap draw $ take 5 $ decompositionsP [pc `at` origin] 

We have made use of the fact that we can create an infinite list of finer and finer decompositions of any patch, using:

decompositionsP:: Patch -> [Patch]
decompositionsP = iterate decompPatch

We could get the n-fold decomposition of a patch as just the nth item in a list of decompositions.

For example, here is an infinite list of decomposed versions of sun.

suns = decompositionsP sun

The coloured tiling shown at the beginning is simply 6 decompositions of sun displayed using leftFillPieceDK

leftFilledSun6 :: Diagram B
leftFilledSun6 = drawWith (leftFillPieceDK red blue) (suns !!6) # lw thin

The earlier figure illustrating larger kites and darts emphasised from the smaller ones is also suns!!6 but this time pieces are drawn with experiment.

experimentFig = drawWith experiment (suns!!6) # lw thin
experiment:: Piece -> Diagram B
experiment pc = emph pc <> (drawRoundPiece pc # dashingN [0.002,0.002] 0 # lw ultraThin)
  where emph pc = case pc of
          (LD v) -> (strokeLine . fromOffsets) [v] # lc red   -- emphasise join edge of darts in red
          (RD v) -> (strokeLine . fromOffsets) [v] # lc red 
          (LK v) -> (strokeLine . fromOffsets) [rotate (ttangle 1) v] -- emphasise long edge for kites
          (RK v) -> (strokeLine . fromOffsets) [rotate (ttangle 9) v]

Compose Choices

You might expect composition (also called inflation) to be a kind of inverse to decomposition, but it is a bit more complicated than that. With our current representation of pieces, we can only compose single pieces. This amounts to embedding the piece into a larger piece that matches how the larger piece decomposes. There is thus a choice at each inflation step as to which of several possibilities we select as the larger half-tile. We represent this choice as a list of alternatives. This list should not be confused with a patch. It only makes sense to select one of the alternatives giving a new single piece.

The earlier diagram illustrating how decompositions are calculated also shows the two choices for embedding a right dart into either a right kite or a larger right dart. There will be two symmetric choices for a left dart, and three choices for left and right kites.

Once again we work with located pieces to ensure the resulting larger piece contains the original in its original position in a decomposition.

compChoices :: Located Piece -> [Located Piece]
compChoices lp = case viewLoc lp of
  (p, RD vd)-> [ RD vd' `at` (p .+^ v')
               , RK vk  `at` p
               ] where v'  = (phi+1) *^ vd                  -- vd*phi^2
                       vd' = rotate (ttangle 9) (vd ^-^ v')
                       vk  = rotate (ttangle 1) v'
  (p, LD vd)-> [ LD vd' `at` (p .+^ v')
               , LK vk `at` p
               ] where v'  = (phi+1) *^ vd                  -- vd*phi^2
                       vd' = rotate (ttangle 1) (vd ^-^ v')
                       vk  = rotate (ttangle 9) v'
  (p, RK vk)-> [ LD vk  `at` p
               , LK lvk' `at` (p .+^ lv') 
               , RK rvk' `at` (p .+^ rv')
               ] where lv'  = phi*^rotate (ttangle 9) vk
                       rv'  = phi*^rotate (ttangle 1) vk
                       rvk' = phi*^rotate (ttangle 7) vk
                       lvk' = phi*^rotate (ttangle 3) vk
  (p, LK vk)-> [ RD vk  `at` p
               , RK rvk' `at` (p .+^ rv')
               , LK lvk' `at` (p .+^ lv')
               ] where v0 = rotate (ttangle 1) vk
                       lv'  = phi*^rotate (ttangle 9) vk
                       rv'  = phi*^rotate (ttangle 1) vk
                       rvk' = phi*^rotate (ttangle 7) vk
                       lvk' = phi*^rotate (ttangle 3) vk

As the result is a list of alternatives, we need to select one to do further inflations. We can express all the alternatives after n steps as compNChoices n where

compNChoices :: Int -> Located Piece -> [Located Piece]
compNChoices 0 lp = [lp]
compNChoices n lp = do
    lp' <- inflate lp
    inflations (n-1) lp'

This figure illustrates 5 consecutive choices for inflating a left dart to produce a left kite. On the left, the finishing piece is shown with the starting piece embedded, and on the right the 5-fold decomposition of the result is shown.

five inflations
five inflations
fiveCompChoices = hsep 1 $ [ draw [ld] <> draw [lk']
                           , draw (decompositionsP [lk'] !!5)
                           ] where
  ld  = (ldart `at` origin)
  lk  = compChoices ld  !!1
  rk  = compChoices lk  !!1
  rk' = compChoices rk  !!2
  ld' = compChoices rk' !!0
  lk' = compChoices ld' !!1

Finally, at the end of this literate haskell program we choose which figure to draw as output.

fig :: Diagram B
fig = leftFilledSun6
main = mainWith fig

That’s it. But, What about composing whole patches?, I hear you ask. Unfortunately we need to answer questions like what pieces are adjacent to a piece in a patch and whether there is a corresponding other half for a piece. These cannot be done with our simple vector representations. We would need some form of planar graph representation, which is much more involved. That is another story.

Many thanks to Stephen Huggett for his inspirations concerning the tilings. A library version of the above code is available on GitHub

Further reading on Penrose Tilings

As well as the Wikipedia entry Penrose Tilings I recommend two articles in Scientific American from 2005 by David Austin Penrose Tiles Talk Across Miles and Penrose Tilings Tied up in Ribbons.

There is also a very interesting article by Roger Penrose himself: Penrose R Tilings and quasi-crystals; a non-local growth problem? in Aperiodicity and Order 2, edited by Jarich M, Academic Press, 1989.

More information about the diagrams package can be found from the home page Haskell diagrams

by readerunner at January 10, 2024 04:48 PM

January 09, 2024

GHC Developer Blog

GHC 9.6.4 is now available

GHC 9.6.4 is now available

Zubin Duggal - 2024-01-09

The GHC developers are happy to announce the availability of GHC 9.6.4. Binary distributions, source distributions, and documentation are available on the release page.

This release is primarily a bugfix release addressing a few issues found in the 9.6 series. These include:

  • A fix for a bug where certain warnings flags were not recognised (#24071)
  • Fixes for a number of simplifier bugs (#23952, #23862).
  • Fixes for compiler panics with certain package databases involving unusable units and module reexports (#21097, #16996, #11050).
  • A fix for a typechecker crash (#24083).
  • A fix for a code generator bug on AArch64 platforms resulting in invalid conditional jumps (#23746).
  • Fixes for some memory leaks in GHCi (#24107, #24118)
  • And a few other fixes

A full accounting of changes can be found in the release notes. As some of the fixed issues do affect correctness users are encouraged to upgrade promptly.

We would like to thank Microsoft Azure, GitHub, IOG, the Zw3rk stake pool, Well-Typed, Tweag I/O, Serokell, Equinix, SimSpace, Haskell Foundation, and other anonymous contributors whose on-going financial and in-kind support has facilitated GHC maintenance and release management over the years. Finally, this release would not have been possible without the hundreds of open-source contributors whose work comprise this release.

As always, do give this release a try and open a ticket if you see anything amiss.

Enjoy!

-Zubin

by ghc-devs at January 09, 2024 12:00 AM

January 08, 2024

Monday Morning Haskell

How to Write Comments in Haskell

Comments are often a simple item to learn, but there's a few ways we can get more sophisticated with them! This article is all about writing comments in Haskell. Here's a quick outline to get you started!

  • What is a Comment?
  • Single Line Comments
  • Multi-Line Comments
  • Inline Comments
  • Writing Formal Documentation Comments
  • Intro to Haddock
  • Basic Haddock Comments
  • Creating Our Haskell Report
  • Documenting the Module Header
  • Module Header Fields
  • Haddock Comments Below
  • Commenting Type Signatures
  • Commenting Constructors
  • Commenting Record Fields
  • Commenting Class Definitions
  • A Complete Introduction to the Haskell Programming Language

    What is a Comment?

    A comment is non-code note you write in a code file. You write it to explain what the code does or how it works, in order to help someone else reading it. Comments are ignored by a language's compiler or interpreter. There is usually some kind of syntax to comments to distinguish them from code. Writing comments in Haskell isn't much different from other programming languages. But in this article, we'll look extensively at Haddock, a more advanced program for writing nice-looking documentation.

    Single Line Comments

    The basic syntax for comments in Haskell is easy, even if it is unusual compared to more common programming languages. In languages like Java, Javascript and C++, you use two forward slashes to start a single line comment:

    int main() {
    // This line will print the string value "Hello, World!" to the console
    std::cerr << "Hello, World!" << std::endl;
    }

    But in Haskell, single line comments start with two hyphens, '--':

    -- This is our 'main' function, which will print a string value to the console
    main :: IO ()
    main = putStrLn "Hello World!"

    You can have these take up an entire line by themselves, or you can add a comment after a line of code. In this simple "Hello World" program, we place a comment at the end of the first line of code, giving instructions on what would need to happen if you extended the program.

    main :: IO ()
    main = -- Add 'do' to this line if you add another 'putStrLn' statement!
    putStrLn "Hello World!"

    Multi-Line Comments

    While you can always start multiple consecutive lines with whatever a comment line starts with in your language, many languages also have a specific way to make multiline comments. And generally speaking, this method has a "start" and an "end" sequence. For example, in C++ or Java, you start a multi line comment block with the characters '/' and end it with '/'

    /*
    This function returns a new list
    that is a reversed copy of the input. 
    
    It iterates through each value in the input 
    and uses 'push_front' on the new copy.
    */
    std::list<int> reverseList(const std::list<int>& ints) {
    std::list<int> result;
    for (const auto& i : ints) {
      result.push_front(i);
    }
    return result;
    }

    In Haskell, it is very similar. You use the brace and a hyphen character to open ('{-') and then the reverse to close the block ('-}').

    {- This function returns a new list
     that is a reversed copy of the input.
    
     It uses a tail recursive helper function.
    -}
    reverse :: [a] -> [a]
    reverse = reverseTail []
    where
      reverseTail acc [] = acc
      reverseTail acc (x : xs) = reverseTail (x : acc) xs

    Notice we don't have to start every line in the comment with double hyphens. Everything in there is part of the comment, until we reach the closing character sequence. Comments like these with multiple lines are also known as "block comments". They are useful because it is easy to add more information to the comment without adding any more formatting.

    Inline Comments

    While you generally use the brace/hyphen sequence to write a multiline comment, this format is surprisingly also useful for a particular form of single line comments. You can write an "inline" comment, where the content is in between operational code on that line.

    reverse :: [a] -> [a]
    reverse = reverseTail []
    where
      reverseTail {- Base Case -}      acc [] = acc
      reverseTail {- Recursive Case -} acc (x : xs) = reverseTail (x : acc) xs

    The fact that our code has a start and end sequence means that the compiler knows where the real code starts up again. This is impossible when you use double hyphens to signify a comment.

    Writing Formal Documentation Comments

    If the only people using this code will be you or a small team, the two above techniques are all you really need. They tell people looking at your source code (including your future self) why you have written things in a certain way, and how they should work. However, if other people will be using your code as a library without necessarily looking at the source code, there's a much deeper area you can explore. In these cases, you will want to write formal documentation comments. A documentation comment tells someone what a function does, generally without going into the details of how it works. More importantly, documentation comments are usually compiled into a format for someone to look at outside of the source code. These sorts of comments are aimed at people using your code as a library. They'll import your module into their own programs, rather than modifying it themselves. You need to answer questions they'll have like "How do I use this feature?", or "What argument do I need to provide for this function to work"? You should also consider having examples in this kind of documentation, since these can explain your library much better than plain statements. A simple code snippet often provides way more clarification than a long document of function descriptions.

    Intro to Haddock

    As I mentioned above, formal documentation needs to be compiled into a format that is more readable than source code. In most cases, this requires an additional tool. Doxygen, for example, is one tool that supports many programming languages, like C++ and Python. Haskell has a special tool called Haddock. Luckily, you probably don't need to go through any additional effort to install Haddock. If you used GHCup to install Haskell, then Haddock comes along with it automatically. (For a full walkthrough on getting Haskell installed, you can read our Startup Guide). It also integrates well with Haskell's package tools, Stack and Cabal. In this article we'll use it through Stack. So if you want to follow along, you should create a new Haskell project on your machine with Stack, calling it 'HaddockTest'. Then build the code before we add comments so you don't have to wait for it later:

    >> stack new HaddockTest
    >> cd HaddockTest
    >> stack build

    You can write all the code from the rest of the article in the file 'src/Lib.hs', which Stack creates by default.

    Basic Haddock Comments

    Now let's see how easy it is to write Haddock comments! To write basic comments, you just have to add a vertical bar character after the two hyphens:

    -- | Get the "block" distance of two 2D coordinate pairs
    manhattanDistance :: (Int, Int) -> (Int, Int) -> Int
    manhattanDistance (x1, y1) (x2, y2) = abs (x2 - x1) + abs (y2 - y1)

    It still works even if you add a second line without the vertical bar. All comment lines until the type signature or function definition will be considered part of the Haddock comment.

    -- | Get the "block" distance of two 2D coordinate pairs
    -- This is the sum of the absolute difference in x and y values.
    manhattanDistance :: (Int, Int) -> (Int, Int) -> Int
    manhattanDistance (x1, y1) (x2, y2) = abs (x2 - x1) + abs (y2 - y1)

    You can also make a block comment in the Haddock style. It involves the same character sequences as multi line comments, but once again, you just add a vertical bar after the start sequence. The end sequence does not need the bar:

    {-| Get the "block" distance of two 2D coordinate pairs
     This is the sum of the absolute difference in x and y values.
    -}
    manhattanDistance :: (Int, Int) -> (Int, Int) -> Int
    manhattanDistance (x1, y1) (x2, y2) = abs (x2 - x1) + abs (y2 - y1)

    No matter which of these options you use, your comment will look the same in the final document. Next, we'll see how to generate our Haddock document. To contrast Haddock comments with normal comments, we'll add a second function in our code with a "normal" single line comment. We also need to add both functions to the export list of our module at the top: `haskell module Lib ( someFunc, , manhattanDistance , euclidenDistance ) where

...

-- Get the Euclidean distance of two 2D coordinate pairs (not Haddock) euclideanDistance :: (Double, Double) -> (Double, Double) -> Double euclideanDistance (x1, y1) (x2, y2) = sqrt ((x2 - x1) ^ 2 + (y2 - y1) ^ 2)

Now let's create our document!
## Creating Our Haskell Report
To generate our document, we just use the following command:
```bash
>> stack haddock

This will compile our code. At the end of the process, it will also inform us about what percentage of the elements in our code used Haddock comments. For example:

25% (  1 /  4) in 'Lib'
  Missing documentation for:
    Module header
    someFunc (src/Lib.hs:7)
    euclideanDistance (src/Lib.hs:17)

As expected, 'euclideanDistance' is not considered to have a Haddock comment. We also haven't defined a Haddock comment for our module header. We'll do that in the next section. We'll get rid of the 'someFunc' expression, which is just a stub. This command will generate HTML files for us, most importantly an index file! They get generated in the '.stack-work' directory, usually in a folder that looks like '{project}/.stack-work/install/{os}/{hash}/{ghc_version}/doc/'. For example, the full path of my index file in this example is:

/home/HaddockTest/.stack-work/install/x86_64-linux-tinfo6/6af01190efdb20c14a771b6e2823b492cb22572e9ec30114989156919ec4ab3a/9.6.3/doc/index.html

You can open the file with your web browser, and you'll find a mostly blank page listing the modules in your project, which at this point should only be 'Lib'. If you click on 'Lib', it will take you to a page that looks like this:

We can see that all three expressions from our file are there, but only 'manhattanDistance' has its comment visible on the page. What's neat is that the type links all connect to documentation for the base libraries. If we click on 'Int', it will take us to the page for the 'base' package module 'Data.Int', giving documentation on 'Int' and other integer types.

Documenting the Module Header

In the picture above, you'll see a blank space between our module name and the 'Documentation' section. This is where the module header documentation should go. Let's see how to add this into our code. Just as Haddock comments for functions should go above their type signatures, the module comment should go above the module declaration. You can start it with the same format as you would have with other Haddock block comments:

{-| This module exposes a couple functions
    related to 2D distance calculation.
-}
module Lib
  ( manhattanDistance
  , euclideanDistance
  ) where

...

If you rerun 'stack haddock' and refresh your Haddock page, this comment will now appear under 'Lib' and above 'Documentation'. This is the simplest thing you can do to provide general information about the module.

Module Header Fields

However, there are also additional fields you can add to the header that Haddock will specifically highlight on the page. Suppose we update our block comment to have these lines:

{-|
Module: Lib
Description: A module for distance functions.
Copyright: (c) Monday Morning Haskell, 2023
License: MIT
Maintainer: person@mmhaskell.com

The module has two functions. One calculates the "Manhattan" distance, or "block" distance on integer 2D coordinates. The other calculates the Euclidean distance for a floating-point coordinate system.
-}
module Lib
  ( manhattanDistance
  , euclideanDistance
  ) where

...

At the bottom of the multi line comment, after all the lines for the fields, we can put a longer description, as you see. After adding this, removing 'someFunc', and making our prior comment on Euclidean distance a Haddock comment, we now get 100% marks on the documentation for this module when we recompile it:

100% (  3 /  3) in 'Lib'

And here's what our HTML page looks like now. Note how the fields we entered are populated in the small box in the upper right.

Note that the short description we gave is now visible next to the module name on the index page. This page still only contains the description below the fields.

Haddock Comments Below

So far, we've been using the vertical bar character to place Haddock comments above our type signatures. However, it is also possible to place comments below the type signatures, and this will introduce us to a new syntax technique that we'll use for other areas. The general idea is that we can use a caret character '^' instead of the vertical bar, indicating that the item we are commenting is "above" or "before" the comment. We can do this either with single line comments or block comments. Here's how we would use this technique with our existing functions:

manhattanDistance :: (Int, Int) -> (Int, Int) -> Int
-- ^ Get the "blocK" distance of two 2D coordinate pairs
manhattanDistance (x1, y1) (x2, y2) = abs (x2 - x1) + abs (y2 - y1)

euclideanDistance :: (Double, Double) -> (Double, Double) -> Double
{- ^ Get the Euclidean distance of two 2D coordinate pairs
     This uses the Pythagorean formula.
-}
euclideanDistance (x1, y1) (x2, y2) = sqrt ((x2 - x1) ^ 2 + (y2 - y1) ^ 2)

The comments will appear the same in the final documentation.

Commenting Type Signatures

The comments we've written so far have described each function as a unit. However, sometimes you want to make notes on specific function arguments. The most common way to write these comments in Haskell with Haddock is with the "above" style. Each argument goes on its own line with a "caret" Haddock comment after it. Here's an example:

-- | Given a base point and a list of other points, returns
-- the shortest distance from the base point to a point in the list.
shortestDistance ::
  (Double, Double) -> -- ^ The base point we are measuring from
  [(Double, Double)] -> -- ^ The list of alternative points
  Double
shortestDistance base [] = -1.0
shorestDistance base rest = minimum $ (map (euclideanDistance base) rest)

It is also possible to write these with the vertical bar above each argument, but then you will need a second line for the comment.

-- | Given a base point and a list of other points, returns
-- the shortest distance from the base point to a point in the list.
shortestDistance ::
  -- | The base point we are measuring from
  (Double, Double) ->
  -- | The list of alternative points
  [(Double, Double)] -> 
  Double
shortestDistance base [] = -1.0
shorestDistance base rest = minimum $ (map (euclideanDistance base) rest)

It is even possible to write the comments before AND on the same line as inline comments. However, this is less common since developers usually prefer seeing the type as the first thing on the line.

Commenting Constructors

You can also use Haddock comments for type definitions. Here is an example of a data type with different constructors. Each gets a comment.

data Direction =
  DUp    | -- ^ Positive y direction
  DRight | -- ^ Positive x direction
  DDown  | -- ^ Negative y direction
  DLeft    -- ^ Negative x direction

Commenting Record Fields

You can also comment record fields within a single constructor.

data Movement = Movement
  { direction :: Direction -- ^ Which way we are moving
  , distance  :: Int       -- ^ How far we are moving
  }

An important note is that if you have a constructor on the same line as its fields, a single caret comment will refer to the constructor, not to its last field.

data Point =
  Point2I Int Int       |      -- ^ 2d integral coordinate
  Point2D Double Double |      -- ^ 2d floating point coordinate
  Point3I Int Int Int   |      -- ^ 3d integral coordinate
  Point3D Double Double Double -- ^ 3d floating point coordinate

Commenting Class Definitions

As one final feature, we can add these sorts of comments to class definitions as well. With class functions, it is usually better to use "before" comments with the vertical bar. Unlike constructors and fields, an "after" comment will get associated with the argument, not the method.

{-| The Polar class describes objects which can be described
    in "polar" coordinates, with a magnitude and angle
-}
class Polar a where
  -- | The total length of the item
  magnitude :: a -> Double 
  -- | The angle (in radians) of the point around the z-axis
  angle :: a -> Double

Here's what all these new pieces look like in our documentation:

You can see the way that each comment is associated with a particular field or argument.

A Complete Introduction to the Haskell Programming Language

Of course, comments are useless if you have no code or projects to write them in! If you're a beginner to Haskell, the fastest way to get up to writing project-level code is our course, Haskell From Scratch! This course features hours of video lectures, over 100 programming exercises, and a final project to test your skills! Learn more about it on this page!

by James Bowen at January 08, 2024 04:00 PM

January 04, 2024

Stackage Blog

LTS 22 release for ghc-9.6 and Nightly now on ghc-9.8

Stackage LTS 22 has been released

The Stackage team is happy to announce that Stackage LTS version 22 was released last month, based on GHC stable version 9.6.3.

LTS 22 includes many package changes, and has over 3300 packages! Thank you for all the nightly contributions that made this release possible: the release was made by Mihai Maruseac. (The closest nightly snapshot to lts-22.0 is nightly-2023-12-17.)

If your package is missing from LTS 22 and builds there, you can easily request to have it added by (new) opening a PR in the lts-haskell project to the build-constraints/lts-22-build-constraints.yaml file. The new LTS workflow was implemented by Adam Bergmark and first appeared in lts-22.1: we are in the process of updating our documentation to cover the new nightly-style workflow for LTS snapshots.

Stackage Nightly updated to ghc-9.8.1

At the same time we are excited to have moved Stackage Nightly to GHC 9.8.1: the initial snapshot being nightly-2023-12-27. Current nightly has over 2400 packages, but we expect that number to continue to grow over the coming days, weeks, and months: we very much welcome your contributions and help with this. You can see all the changes made relative to the preceding last 9.6 nightly snapshot. The initial snapshot was done by Alexey Zabelin and Jens.

Thank you to all those who have already done work updating their packages to ghc-9.8.

Adding or enabling your package for Nightly is just a simple pull request to the large build-constraints.yaml file.

If you have questions you can also ask in the Slack #stackage channel.

New HF server

We would also like to take this opportunity to thank the Haskell Foundation for providing the new upgraded Stackage build-server (setup by Bryan Richter, along with other stackage.org migration), which has greatly helped our daily work with much increased performance and storage.

January 04, 2024 04:00 AM

January 03, 2024

Derek Elkins

Universal Quantification and Infinite Conjunction

Introduction

It is not uncommon for universal quantification to be described as (potentially) infinite conjunction1. Quoting Wikipedia’s Quantifier_(logic) page (my emphasis):

For a finite domain of discourse |D = \{a_1,\dots,a_n\}|, the universal quantifier is equivalent to a logical conjunction of propositions with singular terms |a_i| (having the form |Pa_i| for monadic predicates).

The existential quantifier is equivalent to a logical disjunction of propositions having the same structure as before. For infinite domains of discourse, the equivalences are similar.

While there’s a small grain of truth to this, I think it is wrong and/or misleading far more often than it’s useful or correct. Indeed, it takes a bit of effort to even get a statement that makes sense at all. There’s a bit of conflation between syntax and semantics that’s required to have it naively make sense, unless you’re working (quite unusually) in an infinitary logic where it is typically outright false.

What harm does this confusion do? The most obvious harm is that this view does not generalize to non-classical logics. I’ll focus on constructive logics, in particular. Besides causing problems in these contexts, which maybe you think you don’t care about, it betrays a significant gap in understanding of what universal quantification actually is. Even in purely classical contexts, this confusion often manifests, e.g., in confusion about |\omega|-inconsistency.

So what is the difference between universal quantification and infinite conjunction? Well, the most obvious difference is that infinite conjunction is indexed by some (meta-theoretic) set that doesn’t have anything to do with the domain the universal quantifier quantifies over. However, even if these sets happened to coincide2 there are still differences between universal quantification and infinite conjunction. The key is that universal quantification requires the predicate being quantified over to hold uniformly, while infinite conjunction does not. It just so happens that for the standard set-theoretic semantics of classical first-order logic this “uniformity” constraint is degenerate. However, even for classical first-order logic, this notion of uniformity will be relevant.

Classical Semantic View

I want to start in the context where this identification is closest to being true, so I can show where the idea comes from. The summary of this section is that the standard, classical, set-theoretic semantics of universal quantification is equivalent to an infinitary generalization of the semantics of conjunction. The issue is “infinitary generalization of the semantics of conjunction” isn’t the same as “semantics of infinitary conjunction”.

The standard set-theoretic semantics of classical first-order logic interprets each formula, |\varphi|, as a subset of |D^{\mathsf{fv}(\varphi)}| where |D| is a given domain set and |\mathsf{fv}| computes the (necessarily finite) set of free variables of |\varphi|. Traditionally, |D^{\mathsf{fv}(\varphi)}| would be identified with |D^n| where |n| is the cardinality of |\mathsf{fv}(\varphi)|. This involves an arbitrary mapping of the free variables of |\varphi| to the numbers |1| to |n|. The semantics of a formula then becomes an |n|-ary set-theoretic relation.

The interpretation of binary conjunction is straightforward:

\[\den{\varphi \land \psi} = \den{\varphi} \cap \den{\psi}\]

where |\den{\varphi}| stands for the interpretation of the formula |\varphi|. To be even more explicit, I should index this notation by a structure which specifies the domain, |D|, as well as the interpretations of any predicate or function symbols, but we’ll just consider this fixed but unspecified.

The interpretation of universal quantification is more complicated but still fairly straightforward:

\[\den{\forall x.\varphi} = \bigcap_{d \in D}\left\{\bar y|_{\mathsf{fv}(\varphi) \setminus \{x\}} \mid \bar y \in \den{\varphi} \land \bar y(x) = d\right\}\]

Set-theoretically, we have:

\[\begin{align} \bar z \in \bigcap_{d \in D}\left\{\bar y|_{\mathsf{fv}(\varphi) \setminus \{x\}} \mid \bar y \in \den{\varphi} \land \bar y(x) = d\right\} \iff & \forall d \in D. \bar z \in \left\{\bar y|_{\mathsf{fv}(\varphi) \setminus \{x\}} \mid \bar y \in \den{\varphi} \land \bar y(x) = d\right\} \\ \iff & \forall d \in D. \exists \bar y \in \den{\varphi}. \bar z = \bar y|_{\mathsf{fv}(\varphi) \setminus \{x\}} \land \bar y(x) = d \\ \iff & \forall d \in D. \bar z[x \mapsto d] \in \den{\varphi} \end{align}\]

where |f[x \mapsto c]| extends a function |f \in D^{S}| to a function in |D^{S \cup \{x\}}| via |f[x \mapsto c](v) = \begin{cases}c, &\textrm{ if }v = x \\ f(v), &\textrm{ if }v \neq x\end{cases}|. The final |\iff| arises because |\bar z[x \mapsto d]| is the unique function which extends |\bar z| to the desired domain such that |x| is mapped to |d|. Altogether, this illustrates our desired semantics of the interpretation of |\forall x.\varphi| being the interpretations of |\varphi| which hold when |x| is interpreted as any element of the domain.

This demonstrates the summary that the semantics of quantification is an infinitary version of the semantics of conjunction, as |\bigcap| is an infinitary version of |\cap|. But even here there are substantial cracks in this perspective.

Infinitary Logic

The first problem is that we don’t have an infinitary conjunction so saying universal quantification is essentially infinitary conjunction doesn’t make sense. However, it’s easy enough to formulate the syntax and semantics of infinitary conjunction (assuming we have a meta-theoretic notion of sets).

Syntactically, for a (meta-theoretic) set |I| and an |I|-indexed family of formulas |\{\varphi_i\}_{i \in I}|, we have the infinitary conjunction |\bigwedge_{i \in I} \varphi_i|.

The set-theoretic semantics of this connective is a direct generalization of the binary conjunction case:

\[\bigden{\bigwedge_{i \in I}\varphi_i} = \bigcap_{i \in I}\den{\varphi_i}\]

If |I = \{1,2\}|, we recover exactly the binary conjunction case.

Equipped with a semantics of actual infinite conjunction, we can compare to the semantics of universal quantification case and see where things go wrong.

The first problem is that it makes no sense to choose |I| to be |D|. The formula |\bigwedge_{i \in I} \varphi_i| can be interpreted with respect to many different domains. So any particular choice of |D| would be wrong for most semantics. This is assuming that our syntax’s meta-theoretic sets were the same as our semantics’ meta-theoretic sets, which need not be the case at all3.

An even bigger problem is that infinitary conjunction expects a family of formulas while with universal quantification has just one. This is one facet of the uniformity I mentioned. Universal quantification has one formula that is interpreted a single way (with respect to the given structure). The infinitary intersection expression is computing a set out of this singular interpretation. Infinitary conjunction, on the other hand, has a family of formulas which need have no relation to each other. Each of these formulas is independently interpreted and then all those separate interpretations are combined with an infinitary intersection. The problem we have is that there’s generally no way to take a formula |\varphi| with free variable |x| and an element |d \in D| and make a formula |\varphi_d| with |x| not free such that |\bar y[x \mapsto d] \in \den{\varphi} \iff \bar y \in \den{\varphi_d}|. A simple cardinality argument shows that: there are only countably many (finitary) formulas, but there are plenty of uncountable domains. This is why |\omega|-inconsistency is possible. We can easily have elements in the domain which cannot be captured by any formula.

Syntactic View

Instead of taking a semantic view, let’s take a syntactic view of universal quantification and infinitary conjunction, i.e. let’s compare the rules that characterize them. As before, the first problem we have is that traditional first-order logic does not have infinitary conjunction, but we can easily formulate what the rules would be.

The elimination rules are superficially similar but have subtle but important distinctions:

\[\frac{\Gamma \vdash \forall x.\varphi}{\Gamma \vdash \varphi[x \mapsto t]}\forall E,t \qquad \frac{\Gamma \vdash \bigwedge_{i \in I} \varphi_i}{\Gamma \vdash \varphi_j}{\wedge}E,j\] where |t| is a term, |j| is an element of |I|, and |\varphi[x \mapsto t]| corresponds to syntactically substituting |t| for |x| in |\varphi| in a capture-avoiding way. A first, not-so-subtle distinction is if |I| is an infinite set, then |\bigwedge_{i \in I}\varphi_i| is an infinitely large formula. Another pretty obvious issue is universal quantification is restricted to instantiating terms while |I| stands for either an arbitrary (meta-theoretic) set or it may stand for some particular (meta-theoretic) set, e.g. |\mathbb N|. Either way, it is typically not the set of terms of the logic.

Arguably, this isn’t an issue since the claim isn’t that every infinite conjunction corresponds to a universal quantification, but only that universal quantification corresponds to some infinite conjunction. The set of terms is a possible choice for |I|, so that shouldn’t be a problem. Well, whether it’s a problem or not depends on how you set up the syntax of the language. In my preferred way of handling the syntax of logical formulas, I index each formula by the set of free variables that may occur in that formula. This means the set of terms varies with the set of possible free variables. Writing |\vdash_V \varphi| to mean |\varphi| is well-formed and provable in a context with free variables |V|, then we would want the following rule:

\[\frac{\vdash_V \varphi}{\vdash_U \varphi}\] where |V \subseteq U|. This simply states that if a formula is provable, it should remain provable even if we add more (unused) free variables. This causes a problem with having an infinitary conjunction indexed by terms. Writing |\mathsf{Term}(V)| for the set of terms with (potential) free variables in |V|, then while |\vdash_V \bigwedge_{t \in \mathsf{Term}(V)}\varphi_t| might be okay, this would also lead to |\vdash_U \bigwedge_{t \in \mathsf{Term}(V)}\varphi_t| which would also hold but would no longer correspond to universal quantification in a context with free variables in |U|. This really makes a difference. For example, for many theories, such as the usual presentation of ZFC, |\mathsf{Term}(\varnothing) = \varnothing|, i.e. there are no closed terms. As such, |\vdash_\varnothing \forall x.\bot| is neither provable (which we wouldn’t expect it to be) nor refutable without additional axioms. On the other hand, |\bigwedge_{i \in \varnothing}\bot| is |\top| and thus trivially provable. If we consider |\vdash_{\{y\}} \forall x.\bot| next, it becomes refutable. This doesn’t contradict our earlier rule about adding free variables because |\vdash_\varnothing \forall x.\bot| wasn’t provable and so the rule says nothing. On the other hand, that rule does require |\vdash_{\{y\}} \bigwedge_{i \in \varnothing}\bot| to be provable, and it is. Of course, it no longer corresponds to |\forall x.\bot| with this set of free variables. The putative corresponding formula would be |\bigwedge_{i \in \{y\}}\bot| which is indeed refutable.

With the setup above, we can’t get the elimination rule for |\bigwedge| to correspond to the elimination rule for |\forall|, because there isn’t a singular set of terms. However, a more common if less clean approach is to allow all free variables all the time, i.e. to fix a single countably infinite set of variables once and for all. This would “resolve” this problem.

The differences in the introduction rules are more stark. The rules are:

\[\frac{\Gamma \vdash \varphi \quad x\textrm{ not free in }\Gamma}{\Gamma \vdash \forall x.\varphi}\forall I \qquad \frac{\left\{\Gamma \vdash \varphi_i \right\}_{i \in I}}{\Gamma \vdash \bigwedge_{i \in I}\varphi_i}{\wedge}I\]

Again, the most blatant difference is that (when |I| is infinite) |{\wedge}I| corresponds to an infinitely large derivation. Again, the uniformity aspects show through. |\forall I| requires a single derivation that will handle all terms, whereas |{\wedge}I| allows a different derivation for each |i \in I|.

We don’t run into the same issue as in the semantic view with needing to turn elements of the domain into terms/formulas. Given a formula |\varphi| with free variable |x|, we can easily make a formula |\varphi_t| for every term |t|, namely |\varphi_t = \varphi[x \mapsto t]|. We won’t have the issue that leads to |\omega|-inconsistency because |\forall x.\varphi| is derivable from |\bigwedge_{t \in \mathsf{Term}(V)}\varphi[x \mapsto t]|. Of course, the reason this is true is because one of the terms in |\mathsf{Term}(V)| will be a variable not occurring in |\Gamma| allowing us to derive the premise of |\forall I|. On the other hand, if we choose |I = \mathsf{Term}(\varnothing)|, i.e. only consider closed terms, which is what the |\omega| rule in arithmetic is doing, then we definitely can get |\omega|-inconsistency-like situations. Most notably, in the case of theories, like ZFC, which have no closed terms.

Constructive View

A constructive perspective allows us to accentuate the contrast between universal quantification and infinitary conjunction even more as well as bring more clarity to the notion of uniformity.

We’ll start with the BHK interpretation of Intuitionistic logic and specifically a realizabilty interpretation. For this, we’ll allow infinitary conjunction only for |I = \mathbb N|.

I’ll write |n\textbf{ realizes }\varphi| for the statement that the natural number |n| realizes the formula |\varphi|. As in the linked articles, we’ll need a computable pairing function which computably encodes a pair of natural numbers as a natural number. I’ll just write this using normal pairing notation, i.e. |(n,m)|. We’ll also need Gödel numbering to computably map a natural number |n| to a computable function |f_n|.

\[\begin{align} (n_0, n_1)\textbf{ realizes }\varphi_1 \land \varphi_2 \quad & \textrm{if and only if} \quad n_0\textbf{ realizes }\varphi_0\textrm{ and } n_1\textbf{ realizes }\varphi_1 \\ n\textbf{ realizes }\forall x.\varphi \quad & \textrm{if and only if}\quad \textrm{for all }m, f_n(m)\textbf{ realizes }\varphi[x \mapsto m] \\ (k, n_k)\textbf{ realizes }\varphi_1 \lor \varphi_2 \quad & \textrm{if and only if} \quad k \in \{0, 1\}\textrm{ and }n_k\textbf{ realizes }\varphi_k \\ n\textbf{ realizes }\neg\varphi \quad & \textrm{if and only if} \quad\textrm{there is no }m\textrm{ such that }m\textbf{ realizes }\varphi \end{align}\]

I included disjunction and negation in the above so I could talk about the Law of the Excluded Middle. Via the above interpretation, given any formula |\varphi| with free variable |x|, the meaning of |\forall x.\varphi\lor\neg\varphi| would be a computable function which for each natural number |m| produces a bit indicating whether or not |\varphi[x \mapsto m]| holds. The Law of Excluded Middle holding would thus mean every such formula is computationally decidable which we know isn’t the case. For example, choose |\varphi| as the formula which asserts that the |x|-th Turing machine halts.

This example illustrates the uniformity constraint. Assuming a traditional, classical meta-language, e.g. ZFC, then it is the case that |(\varphi\lor\neg\varphi)[x \mapsto m]| is realized for each |m| in the case where |\varphi| is asserting the halting of the |x|-th Turing machine4. But this interpretation of universal quantification requires not only that the quantified formula holds for all naturals, but also that we can computably find this out.

It’s clear that trying to formulate a notion of infinitary conjunction with regards to realizability would require using something other than natural numbers as realizers if we just directly generalize the finite conjunction case. For example, we might use potentially infinite sequences of natural numbers as realizers. Regardless, the discussion of the previous example makes it clear an interpretation of infinitary conjunction can’t be done in standard computability5, while, obviously, universal quantification can.

Categorical View

The categorical semantics of universal quantification and conjunction are quite different which also suggests that they are not related, at least not in some straightforward way.

One way to get to categorical semantics is to restate traditional, set-theoretic semantics in categorical terms. Traditionally, the semantics of a formula is a subset of some product of the domain set, one for each free variable. Categorically, that suggests we want finite products and the categorical semantics of a formula should be a subobject of a product of some object representing the domain.

Conjunction is traditionally represented via intersection of subsets, and categorically we form the intersection of subobjects via pulling back. So to support finite conjunctions, we need our category to additionally have finite pullbacks of monomorphisms. Infinitary conjunctions simply require infinitely wide pullbacks of monomorphisms. However, we can start to see some cracks here. What does it mean for a pullback to be infinitely wide? It means the obvious thing; namely, that we have an infinite set of monomorphisms sharing a codomain, and we’ll take the limit of this diagram. The key here, though, is “set”. Regardless of whatever the objects of our semantic category are, the infinitary conjunctions are indexed by a set.

To talk about the categorical semantics of universal quantification, we need to bring to the foreground some structure that we have been leaving – and traditionally accounts do leave – in the background. Before, I said the semantics of a formula, |\varphi|, depends on the free variables in that formula, e.g. if |D| is our domain object, then the semantics of a formula with three free variables would be a subobject of |\prod_{v \in \mathsf{fv}(\varphi)}D \cong D\times D \times D| which I’ll continue to write as |D^{\mathsf{fv}(\varphi)}| though now it will be interpreted as a product rather than a function space. For |\mathbf{Set}|, this makes no difference. It would be more accurate to say that a formula can be given semantics in any product of the domain object indexed by any superset of the free variables. This is just to say that a formula doesn’t need to use every free variable that is available. Nevertheless, even if it is induced by the same formula, a subobject of |D^{\mathsf{fv}(\varphi)}| is a different subobject than a subobject of |D^{\mathsf{fv}(\varphi) \cup \{u\}}| where |u| is a variable not free in |\varphi|, so we need a way of relating the semantics of formulas considered with respect to different sets of free variables.

To do this, we will formulate a category of contexts and index our semantics by it. Fix a category |\mathcal C| and an object |D| of |\mathcal C|. Our category of contexts, |\mathsf{Ctx}|, will be the full subcategory of |\mathcal C| with objects of the form |D^S| where |S| is a finite subset of |V|, a fixed set of variables. We’ll assume these products exist, though typically we’ll just assume that |\mathcal C| has all finite products. From here, we use the |\mathsf{Sub}| functor. |\mathsf{Sub} : \mathsf{Ctx}^{op} \to \mathbf{Pos}| maps an object of |\mathsf{Ctx}| to the poset of its subobjects as objects of |\mathcal C|6. Now an arrow |f : D^{\{x,y,z,w\}} \to D^{\{x,y,z\}}| would induce a monotonic function |\mathsf{Sub}(f) : \mathsf{Sub}(D^{\{x,y,z\}}) \to \mathsf{Sub}(D^{\{x,y,z,w\}})|. This is defined for each subobject by pulling back a representative monomorphism of that subobject along |f|. Arrows of |\mathsf{Ctx}| are the semantic analogues of substitutions, and |\mathsf{Sub}(f)| applies these “substitutions” to the semantics of formulas.

Universal quantification is then characterized as the (indexed) right adjoint (Galois connection in this context) of |\mathsf{Sub}(\pi^x)| where |\pi^x : D^S \to D^{S \setminus \{x\}}| is just projection. The indexed nature of this adjoint leads to Beck-Chevalley conditions reflecting the fact universal quantification should respect substitution. |\mathsf{Sub}(\pi^x)| corresponds to adding |x| as a new, unused free variable to a formula. Let |U| be a subobject of |D^{S \setminus \{x\}}| and |V| a subobject of |D^S|. Furthermore, write |U \sqsubseteq U’| to indicate that |U| is a subobject of the subobject |U’|, i.e. that the monos that represent |U| factor through the monos that represent |U’|. The adjunction then states: \[\mathsf{Sub}(\pi^x)(U) \sqsubseteq V\quad \textrm{if and only if}\quad U \sqsubseteq \forall_x(V)\] The |\implies| direction is a fairly direct semantic analogue of the |\forall I| rule: \[\frac{\Gamma \vdash \varphi\quad x\textrm{ not free in }\Gamma}{\Gamma \vdash \forall x.\varphi}\] Indeed, it is easy to show that the converse of this rule is derivable with |\forall E| validating the semantic “if and only if”. To be clear, the full adjunction is natural in |U| and |V| and indexed, effectively, in |S|.

Incidentally, we’d also want the semantics of infinite conjunctions to respect substitution, so they too have a Beck-Chevalley condition they satisfy and give rise to an indexed right adjoint.

It’s hard to even compare the categorical semantics of infinitary conjunction and universal quantification, let alone conflate them, even when |\mathcal C = \mathbf{Set}|. This isn’t too surprising as these semantics work just fine for constructive logics where, as illustrated earlier, these can be semantically distinct. As mentioned, both of these constructs can be described by indexed right adjoints. However, they are adjoints between very different indexed categories. If |\mathcal M| is our indexed category (above it was |\mathsf{Sub}|), then we’ll have |I|-indexed products if |\Delta_{\mathcal M} : \mathcal M \to [DI, -] \circ \mathcal M| has an indexed right adjoint where |D : \mathbf{Set} \to \mathbf{cat}| is the discrete (small) category functor. For |\mathcal M| to have universal quantification, we need an indexed right adjoint to an indexed functor |\mathcal M \circ \mathsf{cod} \circ \iota \to \mathcal M \circ \mathsf{dom} \circ \iota| where |\iota : s(\mathsf{Ctx}) \hookrightarrow \mathsf{Ctx}^{\to}| is the full subcategory of the arrow category |\mathsf{Ctx}^{\to}| consisting of just the projections.

Conclusion

My hope is that the preceding makes it abundantly clear that viewing universal quantification as some kind of special “infinite conjunction” is not sensible even approximately. To do so is to seriously misunderstand universal quantification. Most discussions “equating” them involve significant conflations of syntax and semantics where a specific choice of domain is fixed and elements of that specific domain are used as terms.

A secondary goal was to illustrate an aspect of logic from a variety of perspectives and illustrate some of the concerns in meta-logical reasoning. For example, quantifiers and connectives are syntactical concepts and thus can’t depend on the details of the semantic domain. As another example, better perspectives on quantifiers and connectives are more robust to weakening the logic. I’d say this is especially true when going from classical to constructive logic. Structural proof theory and categorical semantics are good at formulating logical concepts modularly so that they still make sense in very weak logics.

Unfortunately, the traditional trend towards minimalism strongly pushes in the other direction leading to the exploiting of every symmetry and coincidence a stronger logic (namely classical logic) provides producing definitions that don’t survive even mild weakening of the logic7. The attempt to identify universal quantification with infinite conjunction here takes that impulse too far and doesn’t even work in classical logic as demonstrated. While there’s certainly value in recognizing redundancy, I personally find minimizing logical assumptions far more important and valuable than minimizing (primitive) logical connectives.


  1. “Universal statements are true if they are true for every individual in the world. They can be thought of as an infinite conjunction,” from some random AI lecture notes. You can find many others.↩︎

  2. The domain doesn’t even need to be a set.↩︎

  3. For example, we may formulate our syntax in a second-order arithmetic identifying our syntax’s meta-theoretic sets with unary predicates, while our semantics is in ZFC. Just from cardinality concerns, we know that there’s no way of injectively mapping every ZFC set to a set of natural numbers.↩︎

  4. It’s probably worth pointing out that not only will this classical meta-language not tell us whether it’s |\varphi[x \mapsto m]| or |\neg\varphi[x \mapsto m]| that holds for every specific |m|, but it’s easy to show (assuming consistency of ZFC) that |\varphi[x \mapsto m]| is independent of ZFC for specific values of |m|. For example, it’s easy to make a Turing machine that halts if and only if it finds a contradiction in the theory of ZFC.↩︎

  5. Interestingly, for some models of computation, e.g. ones based on Turing machines, infinitary disjunction, or, specifically, |\mathbb N|-ary disjunction is not problematic. Given an infinite sequence of halting Turing machines, we can interleave their execution such that every Turing machine in the sequence will halt at some finite time. Accordingly, extending the definition of disjunction in realizability to the |\mathbb N|-ary case does not run into any of the issues that |\mathbb N|-ary conjunction has and is completely unproblematic. We just let |k| be an arbitrary natural instead of just |\{0, 1\}|.↩︎

  6. This is a place we could generalize the categorical semantics further. There’s no reason we need to consider this particular functor. We could consider other functors from |\mathsf{Ctx}^{op} \to \mathbf{Pos}|, i.e. other indexed |(0,1)|-categories. This setup is called a hyperdoctrine↩︎

  7. The most obvious example of this is defining quantifiers and connectives in terms of other connectives particularly when negation is involved. A less obvious example is the overwhelming focus on |\mathbf 2|-valued semantics when classical logic naturally allows arbitrary Boolean-algebra-valued semantics.↩︎

January 03, 2024 06:00 AM

January 01, 2024

Monday Morning Haskell

How to Write “Hello World” in Haskell

In this article we're going to write the easiest program we can in the Haskell programming language. We're going to write a simple example program that prints "Hello World!" to the console. It's such a simple program that we can do it in one line! But it's still the first thing you should do when starting a new programming language. Even with such a simple program there are several details we can learn about writing a Haskell program. Here's a quick table of contents if you want to jump around!

Now let's get started!

Writing Haskell "Hello World"

To write our "Haskell Hello World" program, we just need to open a file named 'HelloWorld.hs' in our code editor and write the following line:

main = putStrLn "Hello World!"

This is all the code you need! With just this one line, there's still another way you could write it. You could use the function 'print' instead of 'putStrLn':

main = print "Hello World!"

These programs will both accomplish our goal, but their behavior is slightly different! But to explore this, we first need to run our program!

The Simplest Way to Run the Code

Hopefully you've already installed the Haskell language tools on your machine. The old way to do this was through Haskell Platform, but now you should use GHCup. You can read our Startup Guide for more instructions on that! But assuming you've installed everything, the simplest way to run your program is to use the 'runghc' command on your file:

>> runghc HelloWorld.hs

With the first version of our code using 'putStrLn', we'll see this printed to our terminal:

Hello World!

If we use 'print' instead, we'll get this output:

"Hello World!"

In the second example, there are quotation marks! To understand why this is, we need to understand a little more about types, which are extremely important in Haskell code.

Functional Programming and Types

Haskell is a functional programming language with a strong, static type system. Even something as simple as our "Hello World" program is comprised of expressions, and each of these expressions has a type. For that matter, our whole program has a type!

In fact, every Haskell program has the same type: 'IO ()'. The IO type signifies any expression which can perform Input/Output activities, like printing to the terminal and reading user input. Most functions you write in Haskell won't need to do these tasks. But since we're printing, we need the IO signifier. The second part of the type is the empty tuple, '()'. This is also referred to as the "unit type". When used following 'IO', it is similar to having a 'void' return value in other programming languages.

Now, our 'main' expression signifies our whole program, and we can explicitly declare it to have this type by putting a type signature above it in our code. We give the expression name, two colons, and then the type:

main :: IO ()
main = putStrLn "Hello World!"

Our program will run the same with the type signature. We didn't need to put it there, because GHC, the Haskell compiler, can usually infer the types of expressions. With more complicated programs, it can get stuck without explicit type signatures, but we don't have to worry about that right now.

Requirements of an Executable Haskell Program

Now if we gave any other type to our main function, we won't be able to run our program! Our file is supposed to be an entry point - the root of an executable program. And Haskell has several requirements for such files.

These files must have an expression named 'main'. This expression must have the type 'IO ()'. Finally, if we put a module name on our code, that module name should be Main. Module names go at the top of our file, prefaced by "module", and followed by the word "where". Here's how we can explicitly declare the name of our module:

module Main where

main :: IO ()
main = putStrLn "Hello World!"

Like the type signature on our function 'main', GHC could infer the module name as well. But let's try giving it a different module name:

module HelloWorld where

main :: IO ()
main = putStrLn "Hello World!"

For most Haskell modules you write, using the file name (minus the '.hs' extension) IS how you want to name the module. But runnable entry point modules are different. If we use the 'runghc' command on this code, it will still work. However, if we get into more specific behaviors of GHC, we'll see that Haskell treats our file differently if we don't use 'Main'.

Using the GHC Compiler

Instead of using 'runghc', a command designed mainly for one-off files like this, let's try to compile our code more directly using the Haskell compiler. Suppose we have used HelloWorld as the module name. What files does it produce when we compile it with the 'ghc' command?

>> ghc HelloWorld.hs
[1 of 1] Compiling HelloWorld       ( HelloWorld.hs, HelloWorld.o )
>> ls
HelloWorld.hi HelloWorld.hs HelloWorld.o

This produces two output files beside our source module. The '.hi' file is an interface file. The '.o' file is an object file. Unfortunately, neither of these are runnable! So let's try changing our module name back to Main.

module Main where

main :: IO ()
main = putStrLn "Hello World!"

Now we'll go back to the command line and run it again:

>> ghc HelloWorld.hs
[1 of 2] Compiling Main       ( HelloWorld.hs, HelloWorld.o )
[2 of 2] Linking HelloWorld
>> ls 
HelloWorld HelloWorld.hi HelloWorld.hs HelloWorld.o

This time, things are different! We now have two compilation steps. The first says 'Compiling Main', referring to our code module. The second says 'Linking HelloWorld'. This refers to the creation of the 'HelloWorld' file, which is executable code! (On Windows, this file will be called 'HelloWorld.exe'). We can "run" this file on the command line now, and our program will run!

>> ./HelloWorld
Hello World!

Using GHCI - The Haskell Interpreter

Now there's another simple way for us to run our code. We can also use the GHC Interpreter, known as GHCI. We open it with the command 'ghci' on our command line terminal. This brings us a prompt where we can enter Haskell expressions. We can also load code from our modules, using the ':load' command. Let's load our hello world program and run its 'main' function.

>> ghci
GHCI, version 9.4.7: https://www.haskell.org/ghc/   :? for help
ghci> :load HelloWorld
[1 of 2] Compiling Main          ( HelloWorld.hs, interpreted )
ghci> main
Hello World!

If we wanted, we could also just run our "Hello World" code in the interpreter itself:

ghci> putStrLn "Hello World!"
Hello World!

It's also possible to assign our string to a value and then use it in another expression:

ghci> let myString = "Hello World!"
ghci> putStrLn myString
Hello World!

A Closer Look at Our Types

A very useful function of GHCI is that it can tell us the types of our expressions. We just have to use the ':type' command, or ':t' for short. We have two expressions in our Haskell program: 'putStrLn', and "Hello World!". Let's look at their types. We'll start with "Hello World!":

ghci> :type "Hello World!"
"Hello World!" :: String

The type of "Hello World!" itself is a 'String'. This is the name given for a list of characters. We can look at the type of an individual character as well:

ghci> :type 'H'
'H' :: Char

What about 'putStrLn'?

ghci> :t putStrLn
putStrLn :: String -> IO ()

The type for 'putStrLn' looks like 'String -> IO ()'. Any type with an arrow in it ('->') is a function. It takes a 'String' as an input and it returns a value of type 'IO ()', which we've discussed. In order to apply a function, we place its argument next to it in our code. This is very different from other programming languages, where you usually need parentheses to apply a function on arguments. Once we apply a function, the type of the resulting expression is just whatever is on the right side of the arrow. So applying our string to the function 'putStrLn', we get 'IO ()' as the resulting type!

ghci> :t putStrLn "Hello World!"
putStrLn "Hello World!" :: IO ()

Compilation Errors

For a different example, let's see what happens if we try to use an integer with 'putStrLn':

ghci> putStrLn 5
No instance for (Num String) arising from the literal '5'

The 'putStrLn' function only works with values of the 'String' type, while 5 has a type more like 'Int'. So we can't use these expressions together.

A Quick Look At Type Classes

However, this is where 'print' comes in. Let's look at its type signature:

ghci> :t print
print :: Show a => a -> IO ()

Unlike 'putStrLn', the 'print' function takes a more generic input. A "type class" is a general category describing a behavior. Many different types can perform the behavior. One such class is 'Show'. The behavior is that Show-able items can be converted to strings for printing. The 'Int' type is part of this type class, so we can use 'print' with it!

ghci> print 5
5

When use 'show' on a string, Haskell adds quotation marks to the string. This is why it looks different to use 'print' instead of 'putStrLn' in our initial program:

ghci> print "Hello World!"
"Hello World!"

Echo - Another Example Program

Our Haskell "Hello World" program is the most basic example of a program we can write. It only showed one side of the input/output equation. Here's an "echo" program, which first waits for the user to enter some text on the command line and then prints that line back out:

main :: IO ()
main = do
  input <- getLine
  putStrLn input

Let's quickly check the type of 'getLine':

ghci> :t getLine
getLine :: IO String

We can see that 'getLine' is an IO action returning a string. When we use the backwards arrow '<-' in our code, this means we unwrap the IO value and get the result on the left side. So the type of 'input' in our code is just 'String', meaning we can then use it with 'putStrLn'! Then we use the 'do' keyword to string together two consecutive IO actions. Here's what it looks like to run the program. The first line is us entering input, the second line is our program repeating it back to us!

>> runghc Echo.hs
I'm entering input!
I'm entering input!

A Complete Introduction to the Haskell Programming Language

Our Haskell "Hello World" program is the most basic thing you can do with the language. But if you want a comprehensive look at the syntax and every fundamental concept of Haskell, you should take our beginners course, Haskell From Scratch.

You'll get several hours of video lectures, plus a lot of hands-on experience with 100+ exercise problems with automated testing.

All-in-all, you'll only need 10-15 hours to work through all the material, so within a couple weeks you'll be ready for action! Read more about the course here!

by James Bowen at January 01, 2024 04:00 PM

December 26, 2023

Sandy Maguire

FRP in Yampa: Part 4: Routing

In the last post, we investigated the switch combinator, and saw how it can give us the ability to work with “state machine�-sorts of things in our functionally reactive programs.

Today we turn our attention towards game objects—that is, independently operating entities inside of the game, capable of behaving on their own and communicating with one another. I originally learned of this technique from the paper The Yampa Arcade, but haven’t looked at it in a few years, so any shortcomings here are my own.

Nevertheless, the material presented here does in fact work—I’ve actually shipped a game using this exact technique!

Game Objects🔗

Before we dive into the Yampa, it’s worth taking some time to think about what it is we’re actually trying to accomplish. There are a series of constraints necessary to get everything working, and we’ll learn a lot about the problem domain by solving those constraints simultaneously.

The problem: we’d like several Objects running around, which we’d like to program independently, but which behave compositionally. There are going to be a lot of moving pieces here—not only in our game, but also in our solution—so let’s take a moment to define a type synonym for ourselves:

type Object = SF ObjectInput ObjectOutput

Of course, we haven’t yet defined ObjectInput or ObjectOutput, but that’s OK! They will be subject to a boatload of constraints, so we’ll sort them out as we go. At the very least, we will need the ability for an Object to render itself, so we can add a Render field:

data ObjectOutput = ObjectOutput
  { oo_render :: Render
  , ...
  }

We would like Objects to be able to interact with one another. The usual functional approach to this problem is to use message passing—that is, Objects can send values of some message type to one another. Those messages could be things like “I shot you!� or “teleport to me,� or any sort of crazy game-specific behavior you’d like.

In order to do this, we’ll need some sort of Name for each Object. The exact structure of this type depends on your game. For the purposes of this post we’ll leave the thing abstract:

data Name = ...

We’ll also need a Message type, which again we leave abstract:

data Message = ...

Sending messages is clearly an output of the Object, so we will add them to ObjectOutput:

data ObjectOutput = ObjectOutput
  { oo_render :: Render
  , oo_outbox :: [(Name, Message)]
  , ...
  }

There are actions we’d like to perform in the world which are not messages we want to send to anyone; particularly things like “kill my Object� or “start a new Object.� These two are particularly important, but you could imagine updating global game state or something else here.

data Command
  = Die
  | Spawn Name ObjectState Object
  | ...

Commands are also outputs:

data ObjectOutput = ObjectOutput
  { oo_render   :: Render
  , oo_outbox   :: [(Name, Message)]
  , oo_commands :: [Command]
  , ...
  }

Finally, it’s often helpful to have some common pieces of state that belong to all Objects—things like their current position, and hot boxes, and anything else that might make sense to track in your game. We’ll leave this abstract:

data ObjecState = ...

data ObjectOutput = ObjectOutput
  { oo_render   :: Render
  , oo_outbox   :: [(Name, Message)]
  , oo_commands :: [Command]
  , oo_state    :: ObjectState
  }

Let’s turn our attention now to the input side. It’s pretty clear we’re going to want incoming messages, and our current state:

data ObjectInput = ObjectInput
  { oi_inbox :: [(Name, Message)]
  , oi_state :: ObjectState
  }

What’s more interesting, however, than knowing our own state is knowing everyone’s state. Once we have that, we can re-derive oi_state if we know our own Name. Thus, instead:

data ObjectInput = ObjectInput
  { oi_inbox    :: [(Name, Message)]
  , oi_me       :: Name
  , oi_everyone :: Map Name ObjectState
  }

oi_state :: ObjectInput -> ObjectState
oi_state oi
    = fromMaybe (error "impossible!")
    $ Data.Map.lookup (oi_me oi)
    $ oi_everyone oi

Parallel Switching🔗

Armed with our input and output types, we need now figure out how to implement any of this. The relevant combinator is Yampa’s pSwitch, with the ridiculous type:

pSwitch
  :: Functor col
  => (forall sf. gi -> col sf -> col (li, sf))
  -> col (SF li o)
  -> SF (gi, col o) (Event e)
  -> (col (SF li o) -> e -> SF gi (col o))
  -> SF gi (col o)

Yes, there are five type variables here (six, if you include the rank-2 type.) In order, they are:

  1. col: the data structure we’d like to store everything in
  2. gi: the global input, fed to the eventual signal
  3. li: the local input, fed to each object
  4. o: the output of each object signal
  5. e: the type we will use to articulate desired changes to the world

Big scary types like these are an excellent opportunity to turn on -XTypeApplications, and explicitly fill out the type parameters. From our work earlier, we know the types of li and o—they ought to be ObjectInput and ObjectOutput:

pSwitch @_
        @_
        @ObjectInput
        @ObjectOutput
        @_
  :: Functor col
  => (forall sf. gi -> col sf -> col (ObjectInput, sf))
  -> col (SF ObjectInput ObjectOutput)
  -> SF (gi, col ObjectOutput) (Event e)
  -> (col (SF ObjectInput ObjectOutput) -> e -> SF gi (col ObjectOutput))
  -> SF gi (col ObjectOutput)

It’s a little clearer what’s going on here. We can split it up by its four parameters:

  1. The first (value) parameter is this rank-2 function which is responsible for splitting the global input into a local input for each object.
  2. The second parameter is the collection of starting objects.
  3. The third parameter extracts the desired changes from the collection of outputs
  4. The final parameter applies the desired changes, resulting in a new signal of collections.

We are left with a few decisions, the big ones are: what should col be, and what should e be? My answer for the first is:

data ObjectMap a = ObjectMap
  { om_objects  :: Map Name (ObjectState, a)
  , om_messages :: MonoidalMap Name [(Name, Message)]
  }
  deriving stock Functor

which not only conveniently associates names with their corresponding objects and states, but also keeps track of the messages which haven’t yet been delivered. We’ll investigate this further momentarily.

For maximum switching power, we can therefore make our event type be ObjectMap Object -> ObjectMap Object. Filling all the types in, we get:

pSwitch @ObjectMap
        @_
        @ObjectInput
        @ObjectOutput
        @(ObjectMap Object -> ObjectMap Object)
  :: (forall sf. gi -> ObjectMap sf -> ObjectMap (ObjectInput, sf))
  -> ObjectMap Object
  -> SF (gi, ObjectMap ObjectOutput)
        (Event (ObjectMap Object -> ObjectMap Object))
  -> ( ObjectMap Object
    -> (ObjectMap Object -> ObjectMap Object)
    -> SF gi (ObjectMap ObjectOutput)
     )
  -> SF gi (ObjectMap ObjectOutput)

which is something that feels almost reasonable. Let’s write a function that calls pSwitch at these types. Thankfully, we can immediately fill in two of these parameters:

router
    :: ObjectMap Object
    -> SF gi (ObjectMap ObjectOutput)
router objs =
  pSwitch @ObjectMap
          @_
          @ObjectInput
          @ObjectOutput
          @(ObjectMap Object -> ObjectMap Object)
    _
    objs
    _
    (\om f -> router' $ (f om) { om_messages = mempty })

We are left with two holes: one which constructs ObjectInputs, the other which destructs ObjectOutputs. The first is simple enough:

routeInput :: gi -> ObjectMap sf -> ObjectMap (ObjectInput, sf)
routeInput gi om@(ObjectMap objs msgs) = om
  { om_objects = flip Data.Map.mapWithKey objs $ \name (_, sf) ->
      (, sf) $ ObjectInput
        { oi_inbox    = fromMaybe mempty $ Data.MonoidalMap.lookup name msgs
        , oi_me       = name
        , oi_everyone = fmap fst objs
        }
  }

Writing decodeOutput is a little more work—we need to accumulate every change that ObjectOutput might want to enact:

decodeOutput :: Name -> ObjectOutput -> Endo (ObjectMap Object)
decodeOutput from (ObjectOutput _ msgs cmds _) = mconcat
  [ flip foldMap msgs $ uncurry $ send from
  , flip foldMap cmds $ decodeCommand from
  ]

send :: Name -> Name -> Message -> Endo (ObjectMap Object)
send from to msg
  = Endo $ #om_messages <>~ Data.MonoidalMap.singleton to [(from, msg)]

decodeCommand :: Name -> Command -> Endo (ObjectMap Object)
decodeCommand _ (Spawn name st obj)
  = Endo $ #om_objects . at name ?~ (st, obj)
decodeCommand who Die
  = Endo $ #om_objects %~ Data.Map.delete who

There’s quite a lot going on here. Rather than dealing with ObjectMap Object -> ObjectMap Object directly, we instead work with Endo (ObjectMap Object) which gives us a nice monoid for combining endomorphisms. Then by exploiting mconcat and foldMap, we can split up all of the work of building the total transformation into pieces. Then send handles sending a message from one object to another, while also decodeCommand transforms each Command into an endomap.

We can tie everything together:

router
    :: ObjectMap Object
    -> SF gi (ObjectMap ObjectOutput)
router objs =
  pSwitch @ObjectMap
          @_
          @ObjectInput
          @ObjectOutput
          @(ObjectMap Object -> ObjectMap Object)
    routeInput
    objs
    (arr $ Event
         . appEndo
         . foldMap (uncurry decodeOutput)
         . Data.Map.assocs
         . om_objects
         . snd
         )
    (\om f -> router' $ (f om) { om_messages = mempty })

Notice that we’ve again done the monoid trick to run decodeOutput on every output in the ObjectMap. If you’re not already on the monoid bandwagon, hopefully this point will help to change your mind about that!

So our router is finally done! Except not quite. For some reason I don’t understand, pSwitch is capable of immediately switching if the Event you generate for decodeOutput immediately fires. This makes sense, but means Yampa will happily get itself into an infinite loop. The solution is to delay the event by an infinitesimal amount:

router
    :: ObjectMap Object
    -> SF gi (ObjectMap ObjectOutput)
router objs =
  pSwitch @ObjectMap
          @_
          @ObjectInput
          @ObjectOutput
          @(ObjectMap Object -> ObjectMap Object)
    routeInput
    objs
    ((arr $ Event
         . appEndo
         . foldMap (uncurry decodeOutput)
         . Data.Map.assocs
         . om_objects
         . snd
         ) >>> notYet)
    (\om f -> router' $ (f om) { om_messages = mempty })

There’s probably a more elegant solution to this problem, and if you know it, please do get in touch!

Wrapping Up🔗

Today we saw how to use the pSwitch combinator in order to build a router capable of managing independent objects, implementing message passing between them in the process.

You should now have enough knowledge of Yampa to get real tasks done, although if I’m feeling inspired, I might write one more post on integrating a Yampa stream into your main function, and doing all the annoying boilerplate like setting up a game window. Maybe! Watch this space for updates!

December 26, 2023 12:00 AM

December 24, 2023

Sandy Maguire

FRP in Yampa: Part 3: Switching

Yesterday we looked at arrowized FRP in Yampa, and saw how it the proc notation is to arrows as do is for monads. While these syntaxes don’t give you any new power, notation nevertheless matters and helps us better structure our programs.

So far all of our programs have consisted of a single signal function. We’ve sketched out how to build a lobotomized version of the Snake game, but real games have things like title screens and option menus as well as the actual gameplay component. If you were determined, you could probably figure out how to build these missing components with what we’ve seen so far, but it wouldn’t be fun.

Instead, we turn our attention to switches.

Switching🔗

Yampa’s SF type isn’t monadic, but the switch combinator gets you surprisingly close:

switch :: SF i (o, Event e) -> (e -> SF i o) -> SF i o

The idea is that you run the first SF until the outputted Event produces an event, at which point you take its value and use it to generate a new SF, which you subsequently run.

As an example, let’s build a little coproduct type for the choices we might make on the menu screen:

data MenuOption = Start | Options

Our menu screen is now an SF that outputs the things we’d like to draw on the screen (a Render), as well as an Event MenuOption corresponding to an event for when we actually make a selection:

menuScreen :: SF () (Render, Event MenuOption)
menuScreen = ...

As before, we have our main Snake game, and now a new screen for the options:

mainGame :: SF () Render
mainGame = ...

optionsScreen :: SF () Render
optionsScreen = ...

We can tie it all together by switching from menuScreen to the appropriate next SF:

program :: SF () Render
program = switch menuScreen $ \case
  Start   -> mainGame
  Options -> optionsScreen

Again, you can kind of squint to get the picture, but things get a little gnarlier when you actually get into the gritty details here. For example, in a real game, you might go back to the menu screen after the game ends, and you’d certainly go back after setting up the appropriate options. If we wanted to encode those rules, we’d need to fiddle with some types.

Let’s add Event ()s to mainGame and optionScreen, corresponding to when the player has died and when the options have been set, respectively:

mainGame :: SF () (Render, Event ())
optionsScreen :: SF () (Render, Event ())

With a creative amount of switching, it’s possible to encode everything we’d like:

program :: SF () Render
program = switch menuScreen $ \case
  Start   -> switch mainGame      $ const program
  Options -> switch optionsScreen $ const program

Of course, we can use switch for much more than just modeling state machines—the following example uses it as a combinator to do something for a while:

timed :: Time -> SF i o -> SF i o
timed dur s1 s2 =
  switch
    (proc i -> do
      o  <- s1 -< i
      ev <- after dur () -< ()
      returnA -< (o, ev)
    ) $ const s2

or, more interestingly, a combinator which interpolates a function:

interpolate :: Time -> (Time -> a) -> SF (i, a) o -> SF i o -> SF i o
interpolate dur f interp final =
  switch
    (proc i -> do
      t  <- time -< ()
      o  <- s1 -< (i, t / dur)
      ev <- after dur () -< ()
      returnA -< (o, ev)
    ) $ const final

The parameter f here will be called with values of time from 0 to 1, linearly increasing until dur. This is the sort of combinator that is extremely useful for animating objects, where you’d like to tween from a known starting point to a know ending point.

Making a Real Monad🔗

Most of what I know about Yampa I learned by reverse-engineering Alex Stuart’s excellent game Peoplemon (source here). As you might expect, it’s a fun parody on Pokemon.

One night while desperately trying to work out how he programmed up the menu-based battle system in Peoplemon, I came across the mysteriously named Lightarrow.hs, which makes the following improvement over the switching technique above.

He sticks the whole thing into the Cont monad:

newtype Cont r a = Cont { runCont :: (a -> r) -> r }

I think this is the first and only time I’ve seen a use for Cont in the wild, that doesn’t stem directly from trying to CPS everything in order to make your program go faster from fusion. It’s so COOL to see a real world opportunity to throw Cont at a problem!

Anyway. This type is known as Swont, which I’ve always assumed was something like “signal continuation� but your guess is as good as mine:

newtype Swont i o a = Swont { unSwont :: Cont (SF i o) a }
  deriving newtype (Functor, Applicative, Monad)

We can lift any SF i (b, Event c) into a Swont via swont:

swont :: SF i (o, Event e) -> Swont i o e
swont = Swont . cont . switch

and we can lower the whole thing again by way of switchSwont:

switchSwont :: Swont i o e -> (e -> SF i o) -> SF i o
switchSwont sw end = runCont (unSwont sw) end

What’s really nice about Swont is that it is a genuine, bona-fide monad. This gives us a really lovely notation for programming sequential things like state machines or battle animations—stuff that consists of needing to switch between disparate things with discrete reasons to change.

We can use Swont to encode our above state machine in a much more familiar way:

foreverSwont :: Swont i o e -> SF i o
foreverSwont sw = switchSwont (forever sw) $ error "impossible"

program :: SF () Render
program = foreverSwont $ do
  menuScreen >>= \case
    Start   -> mainGame
    Options -> optionsScreen

Not bad at all!

Wrapping Up🔗

Today we looked at Yampa’s switch combinator, seen how it can be used to string disparate signals together, and seen how wrapping the whole thing in a continuation monad can make the whole thing tolerable to work with.

In tomorrow’s post, we’ll look at writing object routers in Yampa—essentially, the main data structure for tracking lots of game objects, and allowing them to communicate with one another. Until then, I hope you’re having a very special Christmas weekend.

December 24, 2023 12:00 AM

December 22, 2023

Joachim Breitner

The Haskell Interlude Podcast

It was pointed out to me that I have not blogged about this, so better now than never:

Since 2021 I am – together with four other hosts – producing a regular podcast about Haskell, the Haskell Interlude. Roughly every two weeks two of us interview someone from the Haskell Community, and we chat for approximately an hour about how they came to Haskell, what they are doing with it, why they are doing it and what else is on their mind. Sometimes we talk to very famous people, like Simon Peyton Jones, and sometimes to people who maybe should be famous, but aren’t quite yet.

For most episodes we also have a transcript, so you can read the interviews instead, if you prefer, and you should find the podcast on most podcast apps as well. I do not know how reliable these statistics are, but supposedly we regularly have around 1300 listeners. We don’t get much feedback, however, so if you like the show, or dislike it, or have feedback, let us know (for example on the Haskell Disourse, which has a thread for each episode).

At the time of writing, we released 40 episodes. For the benefit of my (likely hypothetical) fans, or those who want to train an AI voice model for nefarious purposes, here is the list of episodes co-hosted by me:

Can’t decide where to start? The one with Ryan Trinkle might be my favorite.

Thanks to the Haskell Foundation and its sponsors for supporting this podcast (hosting, editing, transscription).

by Joachim Breitner (mail@joachim-breitner.de) at December 22, 2023 09:04 AM

Derek Elkins

What is the coproduct of two groups?

Introduction

The purpose of this article is to answer the question: what is the coproduct of two groups? The approach, however, will be somewhat absurd. Instead of simply presenting a construction and proving that it satisfies the appropriate universal property, I want to find the general answer and simply instantiate it for the case of groups.

Specifically, this will be a path through the theory of Lawvere theories and their models with the goal of motivating some of the theory around it in pursuit of the answer to this relatively simple question.

If you really just want to know the answer to the title question, then the construction is usually called the free product and is described on the linked Wikipedia page.

Groups as Models of a Lawvere Theory

A group is a model of an equational theory. This means a group is described by a set equipped with a collection of operations that must satisfy some equations. So we’d have a set, |G|, and operations |\mathtt{e} : () \to G|, |\mathtt{i} : G \to G|, and |\mathtt{m} : G \times G \to G|. These operations satisfy the equations, \[ \begin{align} \mathtt{m}(\mathtt{m}(x, y), z) = \mathtt{m}(x, \mathtt{m}(y, z)) \\ \mathtt{m}(\mathtt{e}(), x) = x = \mathtt{m}(x, \mathtt{e}()) \\ \mathtt{m}(\mathtt{i}(x), x) = \mathtt{e}() = \mathtt{m}(x, \mathtt{i}(x)) \end{align} \] universally quantified over |x|, |y|, and |z|.

These equations can easily be represented by commutative diagrams, i.e. equations of compositions of arrows, in any category with finite products of an object, |G|, with itself. For example, the left inverse law becomes: \[ \mathtt{m} \circ (\mathtt{i} \times id_G) = \mathtt{e} \circ {!}_G \] where |{!}_G : G \to 1| is the unique arrow into the terminal object corresponding to the |0|-ary product of copies of |G|.

One nice thing about this categorical description is that we can now talk about a group object in any category with finite products. Even better, we can make this pattern describing what a group is first-class. The (Lawvere) theory of a group is a (small) category, |\mathcal{T}_{\mathbf{Grp}}| whose objects are an object |\mathsf{G}| and all its powers, |\mathsf{G}^n|, where |\mathsf{G}^0 = 1| and |\mathsf{G}^{n+1} = \mathsf{G} \times \mathsf{G}^n|. The arrows consist of the relevant projection and tupling operations, the three arrows above, |\mathsf{m} : \mathsf{G}^2 \to \mathsf{G}^1|, |\mathsf{i} : \mathsf{G}^1 \to \mathsf{G}^1|, |\mathsf{e} : \mathsf{G}^0 \to \mathsf{G}^1|, and all composites that could be made with these arrows. See my previous article for a more explicit description of this, but it should be fairly intuitive.

An actual group is then, simply, a finite-product-preserving functor |\mathcal{T}_{\mathbf{Grp}} \to \mathbf{Set}|. It must be finite-product-preserving so the image of |\mathsf{m}| actually gets sent to a binary function and not some function with some arbitrary domain. The category, |\mathbf{Grp}|, of groups and group homomorphisms is equivalent to the category |\mathbf{Mod}_{\mathcal{T}_{\mathbf{Grp}}}| which is defined to be the full subcategory of the category of functors from |\mathcal{T}_{\mathbf{Grp}} \to \mathbf{Set}| consisting of the functors which preserve finite products. While we’ll not explore it more here, we could use any category with finite products as the target, not just |\mathbf{Set}|. For example, we’ll show that |\mathbf{Grp}| has finite products, and in fact all limits and colimits, so we can talk about the models of the theory of groups in the category of groups. This turns out to be equivalent to the category of Abelian groups via the well-known Eckmann-Hilton argument.

A Bit of Organization

First, a construction that will become even more useful later. Given any category, |\mathcal{C}|, we define |\mathcal{C}^{\times}|, or, more precisely, an inclusion |\sigma : \mathcal{C} \hookrightarrow \mathcal{C}^{\times}| to be the free category-with-finite-products generated from |\mathcal{C}|. Its universal property is: given any functor |F : \mathcal{C} \to \mathcal{E}| into a category-with-finite-products |\mathcal E|, there exists a unique finite-product-preserving functor |\bar{F} : \mathcal{C}^{\times} \to \mathcal E| such that |F = \bar{F} \circ \sigma|.

An explicit construction of |\mathcal{C}^{\times}| is the following. Its objects consist of (finite) lists of objects of |\mathcal{C}| with concatenation as the categorical product and the empty list as the terminal object. The arrows are tuples with a component for each object in the codomain list. Each component is a pair of an index into the domain list and an arrow from the corresponding object in the domain list to the object in the codomain list for this component. For example, the arrow |[A, B] \to [B, A]| would be |((1, id_B), (0, id_A))|. Identity and composition is straightforward. |\sigma| then maps each object to a singleton list and each arrow |f| to |((0, f))|.

Like most free constructions, this construction completely ignores any finite products the original category may have had. In particular, we want the category |\mathcal{T}_{\mathbf{Set}} = \mathbf{1}^{\times}|, called the theory of a set. The fact that the one object of the category |\mathbf{1}| is terminal has nothing to do with its image via |\sigma| which is not the terminal object.

We now define the general notion of a (Lawvere) theory as a small category with finite products, |\mathcal{T}|, equipped with a finite-product-preserving, identity-on-objects functor |\mathcal{T}_{\mathbf{Set}} \to \mathcal{T}|. A morphism of (Lawvere) theories is a finite-product-preserving functor that preserves these inclusions a la: \[ \xymatrix { & \mathcal{T}_{\mathbf{Set}} \ar[dl] \ar[dr] & \\ \mathcal{T}_1 \ar[rr] & & \mathcal{T}_2 } \]

The identity-on-objects aspect of the inclusion of |\mathcal{T}_{\mathbf{Set}}| along with finite-product-preservation ensures that the only objects in |\mathcal{T}| are powers of a single object which we’ll generically call |\mathsf{G}|. This is sometimes called the “generic object”, though the term “generic object” has other meanings in category theory.

A model of a theory (in |\mathbf{Set}|) is then simply a finite-product-preserving functor into |\mathbf{Set}|. |\mathbf{Mod}_{\mathcal{T}}| is the full subcategory of functors from |\mathcal{T} \to \mathbf{Set}| which preserve finite products. The morphisms of models are simply the natural transformations. As an exercise, you should show that for a natural transformation |\tau : M \to N| where |M| and |N| are two models of the same theory, |\tau_{\mathsf{G}^n} = \tau_{\mathsf{G}}^n|.

The Easy Categorical Constructions

This relatively simple definition of model already gives us a large swathe of results. An easy result in basic category theory is that (co)limits in functor categories are computed pointwise whenever the corresponding (co)limits exist in the codomain category. In our case, |\mathbf{Set}| has all (co)limits, so all categories of |\mathbf{Set}|-valued functors have all (co)limits and they are computed pointwise.

However, the (co)limit of finite-product-preserving functors into |\mathbf{Set}| may not be finite-product-preserving, so we don’t immediately get that |\mathbf{Mod}_{\mathcal{T}}| has all (co)limits (and they are computed pointwise). That said, finite products are limits and limits commute with each other, so we do get that |\mathbf{Mod}_{\mathcal{T}}| has all limits and they are computed pointwise. Similarly, sifted colimits, which are colimits that commute with finite products in |\mathbf{Set}| also exist and are computed pointwise in |\mathbf{Mod}_{\mathcal{T}}|. Sifted colimits include the better known filtered colimits which commute with all finite limits.

I’ll not elaborate on sifted colimits. We’re here for (finite) coproducts, and, as you’ve probably already guessed, coproducts are not sifted colimits.

When the Coproduct of Groups is Easy

There is one class of groups whose coproduct is easy to compute for general reasons: the free groups. The free group construction, like most “free constructions”, is a left adjoint and left adjoints preserve colimits, so the coproduct of two free groups is just the free group on the coproduct, i.e. disjoint union, of their generating sets. We haven’t defined the free group yet, though.

Normally, the free group construction would be defined as left adjoint to the underlying set functor. We have a very straightforward way to define the underlying set functor. Define |U : \mathbf{Mod}_{\mathcal T} \to \mathbf{Set}| as |U(M) = M(\mathsf{G}^1)| and |U(\tau) = \tau_{\mathsf{G}^1}|. Identifying |\mathsf{G}^1| with the functor |\mathsf G : \mathbf{1} \to \mathcal{T}| we have |U(M) = M \circ \mathsf{G}| giving a functor |\mathbf{1} \to \mathbf{Set}| which we identify with a set. The left adjoint to precomposition by |\mathsf{G}| is the left Kan extension along |\mathsf{G}|.

We then compute |F(S) = \mathrm{Lan}_{\mathsf{G}}(S) \cong \int^{{*} : \mathbf{1}} \mathcal{T}(\mathsf{G}({*}), {-}) \times S({*}) \cong \mathcal{T}(\mathsf{G}^1, {-}) \times S|. This is the left Kan extension and does form an adjunction but not with the category of models because the functor produced by |F(S)| does not preserve finite products. We should have |F(S)(\mathsf{G}^n) \cong F(S)(\mathsf{G})^n|, but substituting in the definition of |F(S)| clearly does not satisfy this. For example, consider |F(\varnothing)(\mathsf{G}^0)|.

We can and will show that the left Kan extension of a functor into |\mathbf{Set}| preserves finite products when the original functor did. Once we have that result we can correct our definition of the free construction. We simply replace |S : \mathbf{1} \to \mathbf{Set}| with a functor that does preserve finite products, namely |\bar{S} : \mathbf{1}^{\times} \to \mathbf{Set}|. Of course, |\mathbf{1}^{\times}| is exactly our definition of |\mathcal{T}_{\mathbf{Set}}|. We see now that a model of |\mathcal{T}_{\mathbf{Set}}| is the same thing as having a set, hence the name. Indeed, we have an equivalence of categories between |\mathbf{Set}| and |\mathbf{Mod}_{\mathcal{T}_{\mathbf{Set}}}|. (More generally, this theory is called “the theory of an object” as we may consider models in categories other than |\mathbf{Set}|, and we’ll still have this relation.)

The correct definition of |F| is |F(S) = \mathrm{Lan}_{\iota}(\bar S) \cong \int^{\mathsf{G}^n:\mathcal{T}_{\mathbf{Set}}} \mathcal{T}(\iota(\mathsf{G}^n), {-}) \times \bar{S}(\mathsf{G}^n) \cong \int^{\mathsf{G}^n:\mathcal{T}_{\mathbf{Set}}} \mathcal{T}(\iota(\mathsf{G}^n), {-}) \times S^n| where |\iota : \mathcal{T}_{\mathbf{Set}} \to \mathcal{T}| is the inclusion we give as part of the definition of a theory. We can also see |\iota| as |\bar{\mathsf{G}}|.

We can start to see the term algebra in this definition. An element of |F(S)| is a choice of |n|, an |n|-tuple of elements of |S|, and a (potentially compound) |n|-ary operation. We can think of an element of |\mathcal{T}(\mathsf{G}^n, {-})| as a term with |n| free variables which we’ll label with the elements of |S^n| in |F(S)|. The equivalence relation in the explicit construction of the coend allows us to swap projections and tupling morphisms from the term to the tuple of labels. For example, it equates a unary term paired with one label with a binary term paired with two labels but where the binary term immediately discards one of its inputs. Essentially, if you are given a unary term and two labels, you can either discard one of the labels or you can make the unary term binary by precomposing with a projection. Similarly for tupling.

It’s still not obvious this definition produces a functor which preserves finite products. As a lemma to help in the proof of that fact, we have a bit of coend calculus.

Lemma 1: Let |F \dashv U : \mathcal{D} \to \mathcal{C}| and |H : \mathcal D^{op} \times \mathcal{C} \to \mathcal{E}|. Then, |\int^C H(FC, C) \cong \int^D H(D, UD)| when one, and thus both, exist. Proof: \[ \begin{align} \mathcal{E}\left(\int^C H(FC, C), {-}\right) & \cong \int_C \mathcal{E}(H(FC, C), {-}) \tag{continuity} \\ & \cong \int_C \int_D [\mathcal{D}(FC, D), \mathcal{E}(H(D, C), {-})] \tag{Yoneda} \\ & \cong \int_C \int_D [\mathcal{C}(C, UD), \mathcal{E}(H(D, C), {-})] \tag{adjunction} \\ & \cong \int_D \int_C [\mathcal{C}(C, UD), \mathcal{E}(H(D, C), {-})] \tag{Fubini} \\ & \cong \int_D \mathcal{E}(H(D, UD), {-}) \tag{Yoneda} \\ & \cong \mathcal{E}\left(\int^D H(D, UD), {-}\right) \tag{continuity} \\ & \square \end{align} \]

Using the adjunction |\Delta \dashv \times : \mathcal{C} \times \mathcal{C}\to \mathcal{C}| gives the following corollary.

Corollary 2: For any |H : \mathcal{C}^{op} \times \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{E}|, \[\int^{C} H(C, C, C) \cong \int^{C_1}\int^{C_2} H(C_1, C_2, C_1 \times C_2)\] when both exists. This allows us to combine two (co)ends into one.

Now our theorem.

Theorem 3: Let |F : \mathcal{T}_1 \to \mathbf{Set}| and |J : \mathcal{T}_1 \to \mathcal{T}_2| where |\mathcal{T}_1| and |\mathcal{T}_2| have finite products. Then |\mathrm{Lan}_J(F)| preserves finite products if |F| does.

Proof: \[ \begin{flalign} \mathrm{Lan}_J(F)(X \times Y) & \cong \int^A \mathcal{T}_2(J(A), X \times Y) \times F(A) \tag{coend formula for left Kan extension} \\ & \cong \int^A \mathcal{T}_2(J(A), X) \times \mathcal{T}_2(J(A), Y) \times F(A) \tag{continuity} \\ & \cong \int^{A_1}\int^{A_2}\mathcal{T}_2(J(A_1), X) \times \mathcal{T}_2(J(A_2), Y) \times F(A_1 \times A_2) \tag{Corollary 2} \\ & \cong \int^{A_1}\int^{A_2}\mathcal{T}_2(J(A_1), X) \times \mathcal{T}_2(J(A_2), Y) \times F(A_1) \times F(A_2) \tag{finite product preservation} \\ & \cong \left(\int^{A_1}\mathcal{T}_2(J(A_1), X) \times F(A_1) \right) \times \left(\int^{A_2}\mathcal{T}_2(J(A_2), Y) \times F(A_2)\right) \tag{commutativity and cocontinuity of $\times$} \\ & \cong \mathrm{Lan}_J(F)(X) \times \mathrm{Lan}_J(F)(Y) \tag{coend formula for left Kan extension} \\ & \square \end{flalign} \]

The Coproduct of Groups

To get general coproducts (and all colimits), we’ll show that |\mathbf{Mod}_{\mathcal{T}}| is a reflective subcategory of |[\mathcal{T}, \mathbf{Set}]|. Write |\iota : \mathbf{Mod}_{\mathcal{T}} \hookrightarrow [\mathcal{T}, \mathbf{Set}]|. If we had a functor |R| such that |R \dashv \iota|, then we have |R \circ \iota = Id| which allows us to quickly produce colimits in the subcategory via |\int^I D(I) \cong R\int^I \iota D(I)|. It’s easy to verify that |R\int^I \iota D(I)| has the appropriate universal property to be |\int^I D(I)|.

We’ll compute |R| by composing two adjunctions. First, we have |\bar{({-})} \dashv \iota({-}) \circ \sigma : \mathbf{Mod}_{\mathcal{T}^{\times}} \to [\mathcal T, \mathbf{Set}]|. This is essentially the universal property of |\mathcal{T}^{\times}|. When |\mathcal{T}| has finite products, which, of course, we’re assuming, then we can use the universal property of |\mathcal{T}^{\times}| to factor |Id_{\mathcal{T}}| into |Id = \bar{Id} \circ \sigma|. The second adjunction is then |\mathrm{Lan}_{\bar{Id}} \dashv {-} \circ \bar{Id} : \mathbf{Mod}_{\mathcal{T}} \to \mathbf{Mod}_{\mathcal{T}^{\times}}|. The left adjoint sends finite-product-preserving functors to finite-product-preserving functors via Theorem 3. The right adjoint is the composition of finite-product-preserving functors.

The composite of the left adjoints is |\iota({-} \circ \bar{Id}) \circ \sigma = \iota({-}) \circ \bar{Id} \circ \sigma = \iota({-})|. The composite of the right adjoint is \[ \begin{align} R(F) & = \mathrm{Lan}_{\bar{Id}}(\bar{F}) \\ & \cong \int^X \mathcal{T}(\bar{Id}(X), {-}) \times \bar{F}(X) \\ & \cong \int^X \mathcal{T}\left(\prod_{i=1}^{\lvert X\rvert} X_i, {-}\right) \times \prod_{i=1}^{\lvert X \rvert} F(X_i) \end{align} \] where we view the list |X : \mathcal{T}^{\times}| as a |\lvert X\rvert|-tuple with components |X_i|.

This construction of the reflector, |R|, is quite similar to the free construction. The main difference is that here we factor |Id| via |\mathcal{T}^{\times}| where there we factored |\mathsf{G} : \mathbf{1} \to \mathcal{T}| via |\mathbf{1}^{\times} = \mathcal{T}_{\mathbf{Set}}|.

Let’s now explicitly describe the coproducts via |R|. As a warm-up, we’ll consider the initial object, i.e. nullary coproducts. We consider |R(\Delta 0)|. Because |0 \times S = 0|, the only case in the coend that isn’t |0| is when |\lvert X \rvert = 0| so the underlying set of the coend reduces to |\mathcal{T}(\mathsf{G}^0, \mathsf{G}^1)|, i.e. the nullary terms. For groups, this is just the unit element. For bounded lattices, it would be the two element set consisting of the top and bottom elements. For lattices without bounds, it would be the empty set. Of course, |R(\Delta 0)| matches |F(0)|, i.e. the free model on |0|.

Next, we consider two models |G| and |H|. First, we compute to the coproduct of |G| and |H| as (plain) functors which is just computed pointwise, i.e. |(G+H)(\mathsf{G}^n) = G(\mathsf{G}^n)+H(\mathsf{G}^n) \cong G(\mathsf{G^1})^n + H(\mathsf{G^1})^n|. Considering the case where |X_i = \mathsf{G}^1| for all |i| and where |\lvert X \rvert = n|, which subsumes all the other cases, we see we have a term with |n| free variables each labelled by either an element of |G| or an element of |H|. If we normalized the term into a list of variables representing a product of variables, then we’d have a essentially a word as described on the Wikipedia page for the free product. If we then only considered quotienting by the equivalences induced by projection and tupling, we’d have the free group on the disjoint union of the underlying sets of the |G| and |H|. However, for |R|, we quotient also by the action of the other operations. The lists of objects with |X_i \neq \mathsf{G}^1| come in here to support equating non-unary ops. For example, a pair of the binary term |\mathsf{m}| and the 2-tuple of elements |(g_1, g_2)| for |g_1, g_2 \in U(G)|, will be equated with the pair of the unary term |id| and the 1-tuple of elements |(g)| where |g = g_1 g_2| in |G|. Similarly for |H| and the other operations (and terms generally). Ultimately, the quotient identifies every element with an element that consists of a pair of a term that is a fully right associated set of multiplications ending in a unit where each variable is labelled with an element from |U(G)| or |U(H)| in an alternating fashion. These are the reduced words in the Wikipedia article.

This, perhaps combined with a more explicit spelling out of the equivalence relation, should make it clear that this construction does actually correspond to the usual free product construction. The name “free product” is also made a bit clearer, as we are essentially building the free group on the disjoint union of the underlying sets of the inputs, and then quotienting that to get the result. While there are some categorical treatments of normalization, the normalization arguments used above were not guided by the category theory. The (underlying sets of the) models produced by the above |F| and |R| functors big equivalence classes of “terms”. The above constructions provide no guidance for finding “good” representatives of those equivalence classes.

Conclusions

This was, of course, a very complex and round-about way of answering the title question. Obviously the real goal was illustrating these ideas and illustrating how “abstract” categorical reasoning can lead to relatively “concrete” results. Of course, these concrete constructions are derived from other concrete constructions, usually concrete constructions of limits and colimits in |\mathbf{Set}|. That said, category theory allows you to get a lot from a small collection of relatively simple concrete constructions. Essentially, category theory is like a programming language with a small set of primitives. You can write “abstract” programs in terms of that language, but once you provide an “implementation” for those primitives, all those “abstract” programs can be made concrete.

I picked (finite) coproducts, in particular, as they are where a bunch of complexity suddenly arises when studying algebraic objects categorically, but (finite) coproducts are still fairly simple.

For Lawvere theories, one thing to note is that the Lawvere theory is independent of the presentation. Any presentation of the axioms of a group would give rise to the same Lawvere theory. Of course, to explicitly describe the category would end up requiring a presentation of the category anyway. Beyond Lawvere theories are algebraic theories and algebraic categories, and further into essentially algebraic theories and categories. These extend to the multi-sorted case and then into the finite limit preserving case. The theory of categories, for example, cannot be presented as a Lawvere theory but is an essentially algebraic theory. There’s much more that can be said even about specifically Lawvere theories, both from a theoretical perspective, starting with monadicity, and from practical perspectives like algebraic effects.

Familiarity with the properties of functor categories, and especially categories of (co)presheaves was behind many of these results, and many that I only mentioned in passing. It is always useful to learn more about categories of presheaves. That said, most of the theory works in an enriched context and often without too many assumptions. The fact that all we need to talk about models is for the codomains of the functors to have finite products allows quite broad application. We can talk about algebraic objects almost anywhere. For example, sheaves of rings, groups, etc. can equivalently be described as models of the theories of rings, groups, etc. in sheaves of sets.

Kan extensions unsurprisingly played a large role, as they almost always do when you’re talking about (co)presheaves. One of the motivations for me to make this article was a happy confluence of things I was reading leading to a nice, coend calculus way of describing and proving finite-product-preservation for free models.

Thinking about what exactly was going on around finite-product-preservation was fairly interesting. The incorrect definition of the free model functor could be corrected in a different (though, of course, ultimately equivalent) way. The key is to remember that the coend formula for the left Kan extension is generally a copower and not a cartesian product. The copower for |\mathbf{Set}|-valued functors is different from the copower for finite-product-preserving |\mathbf{Set}|-valued functors. For a category with (arbitrary) coproducts, the copower corresponds to the coproduct of a constant family. We get, |F(S) \cong \coprod_{S} \mathcal T(\mathsf{G}^1, {-})| as is immediately evident from |F| being a left adjoint and a set |S| being the coproduct of |1| |S|-many times. For the purposes of this article, this would have been less than satisfying as figuring out what coproducts were was the nominal point.

That said, it isn’t completely unsatisfying as this defines the free model in terms of a coproduct of, specifically, representables and those are more tractable. In particular, an easy and neat exercise is to work out what |\mathcal{T}(\mathsf{G}^n, {-}) + \mathcal{T}(\mathsf{G}^m, {-})| is. Just use Yoneda and work out what must be true of the mapping out property, and remember that the object you’re mapping into preserves finite products. Once you have finite coproducts described, you can get all the rest via filtered colimits, and since those commute with finite products that gives us arbitrary coproducts.

December 22, 2023 02:47 AM

Sandy Maguire

FRP in Yampa: Part 2: Arrowized FRP

In the last part, we got a feel for how FRP can help us with real-time programming tasks, especially when contrasted against implicit models of time. However, the interface we looked at yesterday left much to be desired—stringing together long signal functions felt clunky, and since SFs don’t form a monad, we couldn’t alleviate the problem with do-notation.

So today we’ll look at one of Haskell’s lesser-known features—arrow notation—and learn how it can help structure bigger reactive programs.

Arrows🔗

What an awful, overloaded word we’ve found ourselves with. Being Haskell programmers, we’re all very familiar with the everyday function arrow (->), which you should think of as a special case of a more general notion of arrow.

Notice how both function arrows (i -> o) and signal functions (SF i o) have two type parameters—one for the input side of things, and another for the output side. And indeed, we should think of these as sides of the computation, where we are transforming an i into an o.

For our purposes today, we’ll want to be very precise when we differentiate between functions-as-data and functions-as-ways-of-building things. In order to do so, we will give give ourselves a little type synonym to help differentiate:

type Fn i o = i -> o

And henceforth, we will use the Fn synonym to refer to functions we’re manipulating, reserving (->) to talk about combinators for building those functions.

For example, our favorite identity function is a Fn:

id :: Fn a a

We usually give the constant function the type a -> b -> a, but my claim is that it ought to be:

const :: a -> Fn b a

The subtle thing I’m trying to point out is that there is a (conceptual) difference between the functions we want to operate on at runtime (called Fns), and the combinators we use to build those functions (called (->).)

Like I said, it’s a bit hard to point to in Haskell, because one of the great successes of functional programming has been to blur this distinction.

Anyway, let’s return to our discussion of arrows. Both functions and SFs admit a notion of composition, which allow us to line up the output of one arrow with the input of another, fusing the two into a single computation. The types they have are:

  • (.) :: Fn b c -> Fn a b -> Fn a c
  • (<<<) :: SF b c -> SF a b -> SF a c

Despite our intimate familiarity with functions, this pattern of types with both an input and an output is quite uncommon in Haskell. Due to the immense mindshare that the monad meme takes up, we usually think about computation in terms of monads, and it can be hard to remember that not all computation is monadic (nor applicative.)

Monadic values are of the shape M o, with only a single type parameter that corresponds (roughly) with the output of the computation. That is to say, all of the interesting computational structure of a monad exists only in its output, and never in its input—in fact, we can’t even talk about the input to a monad. What we do instead is cheat; we take the input side of the computation directly from the function arrow.

If we expand out the types of (<*>) and flip (>>=), using our Fn notation from above, they get the types:

  • (<*>) :: M (Fn i o) -> Fn (M i) (M o)
  • flip (>>=) :: Fn i (M o) -> Fn (M i) (M o)

which makes it much clearer that the relevant interactions here are some sort of distributivity of our monad over the regular, everyday function arrows. In other words, that monads are cheating by getting their “inputs� from functions.

What the Hell?🔗

Enough philosophy. What the hell are arrows? The example that really made it stick for me is in the domain of digital circuits. A digital circuit is some piece of silicon with wire glued to it, that moves electrons from one side to the other—with the trick being that the eventual endpoint of the electrons depends on their original positions. With enough squinting, you can see the whole thing as a type Circuit i o, where i corresponds to which wires we chose to put a high voltage on, and o is which wires have a high voltage at the end of the computation. With a little more squinting, it’s not too hard to reconceptualize these wires as bits, which we can again reconceptualize as encodings of particular types.

The point I was trying to make earlier about the distinction between (->) and Fn makes much more sense in this context; just replace Fn with Circuit. Here it makes much more sense to think about the identity circuit:

id :: Circuit a a

which is probably just a bundle of wires, and the constant circuit:

const :: o -> Circuit i o

which lets you pick some particular o value (at design time), and then make a circuit that is disconnected from its input wires and merely holds the chosen o value over its output wires.

Anyway. The important thing about digital circuits is that you have infinite flexibility when you are designing them, but once they’re manufactured, they stay that way. If you chose to wire the frobulator directly to the zanzigurgulator, those two components are, and always will be, wired together. In perpetuity.

Of course, you can do some amount of dynamic reconfiguring of a circuit, by conditionally choosing which wires you consider to be “relevant� right now, but those wires are going to have signals on them whether you’re interested in them or not.

In other words, there is a strict phase distinction between the components on the board and the data they carry at runtime.

And this is what arrows are all about.

Arrows are about computations whose internal structure must remain constant. You’ve got all the flexibility in the world when you’re designing them, but you can’t reconfigure anything at runtime.

Arrow Notation🔗

Yesterday’s post ended with the following code, written directly with the arrow combinators.

onPress :: (Controller -> Bool) -> a -> SF () (Event a)
onPress field a = fmap (fmap (const a)) $ fmap field controller >>> edge

arrowEvents :: Num a => SF () (Event (V2 a))
arrowEvents =
  (\u d l r -> asum [u, d, l r])
    <$> onPress ctrl_up    (V2 0 (-1))
    <*> onPress ctrl_down  (V2 0 1)
    <*> onPress ctrl_left  (V2 (-1) 0)
    <*> onPress ctrl_right (V2 1    0)

snakeDirection :: SF () (V2 Float)
snakeDirection = arrowEvents >>> hold (V2 0 1)

snakePosition :: SF () (V2 Float)
snakePosition = snakeDirection >>> integral

While technically you can get anything done in this style, it’s a lot like writing all of your monadic code directly in terms of (>>=). Possible certainly, but indisputably clunky.

Instead, let’s rewrite it with arrow notation:

{-# LANGUAGE Arrows #-}

snakePosition :: SF () (V2 Float)
snakePosition = proc i -> do
  u <- onPress ctrl_up    $ V2 0 (-1) -< i
  d <- onPress ctrl_down  $ V2 0 1    -< i
  l <- onPress ctrl_left  $ V2 (-1) 0 -< i
  r <- onPress ctrl_right $ V2 1    0 -< i

  dir <- hold $ V2 0 1 -< asum [u, d, l r]
  pos <- integral -< dir

  returnA -< pos

Much tidier, no? Reading arrow notation takes a little getting used to, but there are really only two things you need to understand. The first is that proc i -> do introduces an arrow computation, much like the do keyword introduces a monadic computation. Here, the input to the entire arrow is bound to i, but you can put any legal Haskell pattern you want there.

The other thing to know about arrow notation is that <- and -< are two halves of the same syntax. The notation here is:

  output <- arrow -< input

where arrow is of type SF i o, and input is any normal everyday Haskell value of type i. At the end of the day, you bind the result to output, whose type is obviously o.

The mnemonic for this whole thing is that you’re shooting an arrow (of bow and arrow fame) from the input to the output. And the name of the arrow is written on the shaft. It makes more sense if you play around with the whitespace:

  output   <-arrow-<   input

More importantly, the name of that arrow can be any valid Haskell expression, including one with infix operators. Thus, we should parse:

  u <- onPress ctrl_up $ V2 0 (-1) -< i

as

  u <- (onPress ctrl_up $ V2 0 (-1)) -< i

What’s likely to bite you as you get familiar with arrow notation is that the computations (the bits between <- and -<) exist in a completely different phase/namespace than the inputs and outputs. That means the following program is illegal:

  proc (i, j) -> do
    x <- blah  -< i
    y <- bar x -< j
    ...

because x simply isn’t in scope in the expression bar x. It’s the equivalent of designing a circuit board with n capacitors on it, where n will be determined by an input voltage supplied by the end-user. Completely nonsensical!

Wrapping Up🔗

That’s all for today, folks. The day caught me by surprise, so we’ll be back tomorrow to talk about building state machines in Yampa—something extremely important for making real video games.

December 22, 2023 12:00 AM

December 21, 2023

Sandy Maguire

FRP in Yampa: Part 1

I’ve been writing some Haskell lately, for the first time in a year, and it’s a total blast! In particular, school is out for the holidays, so I had some spare time, and thought I’d waste it by making a video game. In Haskell.

It’s always more fun to make video games with other people, but the few people I pitched it to all had the same response—“I don’t know how to do that.� So it seemed like a good opportunity to dust off the old blog and write about how to make a video game in Haskell, using arrowized FRP.

What the hell does that mean? Get ready to FIND OUT!

FRP?🔗

FRP is short for functional reactive programming, originally invented by Conal Elliott. The library we’ll be using today is called Yampa, which is certainly inspired by Elliott’s work, but my guess is it’s insufficiently true to the core idea for him to be excited about it.

Nevertheless, even an imperfect implementation of the idea is still orders of magnitude for making real-time applications than doing everything by hand. And to this extent, Yampa is an excellent library.

So what exactly is FRP? The core idea is that we want to talk about functions that are continuous in time, which give rise to extremely useful combinators-over-time. Real-time programs written as FRP are much easier to reason about, and significantly more expressive than you’d manage otherwise.

A Point of Contrast🔗

It’s informative to compare what writing a video game looks like under an imperative style. The idea is that you have your game loop (a fancy name for “infinite loop�) running:

void main() {
  setup();

  while (true) {
    float delta_time = waitForNextFrame();
    updateGame(delta_time);
    renderFrame();
  }
}

and this is kind of fine and manages to get the job done. But it’s inelegant for a few reasons. The biggest problem is that we are not actually modeling time here; we’re just running the game discretely, and time happens as a side effect of things changing. There’s this delta_time variable which counts how long it’s been since you last updated the game, which is to say it corresponds to “how much work you need to do this frame.�

What goes wrong is when updateGame or renderFrame takes too long to run; in that case, you might get spikes in how long it’s been since you last updated. Procedurally-written games compensate by interpolating everything a little further on the next frame, which gives the player the perception that they’re actually experiencing time.

But things can break down. If your last frame took too long, you need to simulate physics a little more this frame. In practice this usually means that you integrate your velocity a little more than usual—which really means your positions will teleport a little further than usual. This is a common bug in games, where it’s often easy to clip through obstacles when the frame-rate is too low.

The other problem with modeling your time only incidentally is that it makes it really annoying to actually do anything. For example, when you read from the controller you will only get whether the buttons are down or up, but you won’t get whether the button was just pressed this frame. If you want to know that you’ll have to compute it yourself:

bool last_a_button = false;

void updateGame(float delta_time) {
  controller ctrls = getControllerState();

  if (ctrls.a_button && !last_a_button) {
    // handle a press
  }

  last_a_button = ctrls.a_button;
}

It’s tedious, but it gets the job done. Another common pain point is when you want to do something five seconds in the future:

float timer;

void updateGame(float delta_time) {
  timer -= delta_time;

  if (getWantsToStartTimer()) {
    timer = 5.0;
  }

  // ...

  if (timer <= 0) {
    // handle timer finishing
  }
}

Again, nothing you can’t tackle, but in aggregate, this all becomes very weighty. Not being able to model time explicitly is a real pain, and everywhere you go you need to simulate it by diddling state changes.

Signal Functions🔗

If you’ve ever written a video game, it probably looked a lot like the examples from the previous section. That’s the sort of thing we’d like to abstract over, and work at a much higher level of detail than.

Here comes FRP to the rescue.

The core building block in Yampa is the “signal function�, written as SF i o. You can think of this as a transformer of signals of i into signals of o, where a signal is a function Time -> a. Unwrapping all of this, an SF i o is a function (Time -> i) -> (Time -> o).

That’s everything you need to know about what SFs are. I don’t know how they’re implemented, and I don’t need to, because the abstraction doesn’t leak. Being a haskell programmer, you’re probably looking at SF i o and thinking “that thing is clearly a Functor/Applicative/Monad.� Two out of three—it’s a functor and an applicative, but not a monad. We’ll come back to this momentarily.

The trick to working in FRP is to think of continuous streams of values over time. Thus, we can think about the player’s controller as an SF:

controller :: SF () Controller

which is to say, a continuous stream of Controller values. By marking the input side of the SF as a unit, it means we don’t need to provide anything in order to get this value, which makes sense since the controller state is obviously at the very periphery of our program.

Since SF is a functor, we can get the state of the A button by fmapping it:

aState :: SF () Bool
aState = fmap a_button controller

which isn’t very surprising. But what’s more interesting are the SF-operating primitives that Yampa gives us. For example, there’s delay:

delay :: Time -> a -> SF a a

which delays a signal by the given time, using the a parameter as the value for the initial value of the stream. Thus, we can get the value of the A button two seconds ago via:

aStateTwoSecondsAgo :: SF () Bool
aStateTwoSecondsAgo = aState >>> delay 2 False

where (>>>) :: SF a b -> SF b c -> SF a c is composition of SFs, analogous to function composition.

Already we can see the benefit of this approach. While it’s not clear exactly why we might want to look at the state of the controller two seconds ago, it’s also non-obvious how you’d go about implementing such a thing procedurally with a game loop.

One last signal function we might be interested for the time being is integral, which allows us to compute the integral of a stream:

integral :: Fractional a => SF a a

Events🔗

SFs are transformers of continuous signals, but often we want to talk about discrete moments in time. For this, we’ve got the Event type, which is isomorphic to Maybe:

data Event a
  = Event a
  | NoEvent

The interpretation you should have for an Event is that it’s a discrete piece of data arriving at a specific moment in time. In our earlier discussion of things you want to do in games, we’ve already seen two examples of events: when a timer expires, and when the player presses the A button. Under Yampa, the first is particularly easy to code up, by way of the after combinator:

after :: Time -> b -> SF a (Event b)

If we want to trigger a timer after 5 seconds, it’s just after 5 () :: SF a (Event ()), and we can listen to the output of this stream for an Event () value in order to know when the timer has elapsed.

Similarly, when we’re interested in the player pressing a button, what we’re really interested is in the edges of their button signal. We can get this functionality by way of the edge signal function:

edge :: SF Bool (Event ())

which generates an event whenever the input boolean goes from false to true.

Of course, just being able to generate events isn’t very useful if we don’t have any means of subsequently eliminating them. A simple means of eliminating events is via hold:

hold :: a -> SF (Event a) a

The hold function takes a stream of events, and holds onto the most recent value it received.

Making a Game of Snake🔗

We’ve already seen enough of FRP in order to make most of the old classic, Snake. In Snake, you are a snake who slithers around in a square, with a constant velocity, continuing in the direction you’re going until the player asks you to turn.

Begin with a Controller, and an SF to read it:

data Controller = Controller
  { ctrl_up    :: Bool
  , ctrl_down  :: Bool
  , ctrl_left  :: Bool
  , ctrl_right :: Bool
  }

controller :: SF () Controller
controller = ...

We can then write a little helper function to determine when a button has been pressed—tagging it with a particular value of our choice:

onPress :: (Controller -> Bool) -> a -> SF () (Event a)
onPress field a = fmap (fmap (const a)) $ fmap field controller >>> edge

Next, we can sum up an onPress for each direction on the controller, mapping them into direction vectors:

arrowEvents :: Num a => SF () (Event (V2 a))
arrowEvents =
  (\u d l r -> asum [u, d, l r])
    <$> onPress ctrl_up    (V2 0 (-1))
    <*> onPress ctrl_down  (V2 0 1)
    <*> onPress ctrl_left  (V2 (-1) 0)
    <*> onPress ctrl_right (V2 1    0)

Above, the use of asum allows us to combine these four events into one, meaning that if the player presses two directions at exactly the same moment, we will prefer up over down, and down over left, etc.

By holding onto the most recent arrow event, we can get the current direction our snake is facing:

snakeDirection :: SF () (V2 Float)
snakeDirection = arrowEvents >>> hold (V2 0 1)

which we can then integrate in order to have the snake move around:

snakePosition :: SF () (V2 Float)
snakePosition = snakeDirection >>> integral

Not too shabby at all! This particular snake will move at a rate of 1 unit per second, but we could make him faster by scaling up snakeDirection before taking its integral.

Wrapping Up🔗

Hopefully I’ve given you a taste of how FRP can radically simplify the implementation of real-time applications. Tomorrow we’ll look into arrowized FRP, and get a sense of how to build bigger, more interesting programs.

December 21, 2023 12:00 AM

December 09, 2023

Magnus Therning

Getting Amazonka S3 to work with localstack

I'm writing this in case someone else is getting strange errors when trying to use amazonka-s3 with localstack. It took me rather too long finding the answer and neither the errors I got from Amazonka nor from localstack were very helpful.

The code I started with for setting up the connection looked like this

main = do
  awsEnv <- AWS.overrideService localEndpoint <$> AWS.newEnv AWS.discover
  -- do S3 stuff
  where
    localEndpoint = AWS.setEndpoint False "localhost" 4566

A few years ago, when I last wrote some Haskell to talk to S3 this was enough1, but now I got some strange errors.

It turns out there are different ways to address buckets and the default, which is used by AWS itself, isn't used by localstack. The documentation of S3AddressingStyle has more details.

So to get it to work I had to change the S3 addressing style as well and ended up with this code instead

main = do
  awsEnv <- AWS.overrideService (s3AddrStyle . localEndpoint) <$> AWS.newEnv AWS.discover
  -- do S3 stuff
  where
    localEndpoint = AWS.setEndpoint False "localhost" 4566
    s3AddrStyle svc = svc {AWS.s3AddressingStyle = AWS.S3AddressingStylePath}

Footnotes:

1

That was before version 2.0 of Amazonka, so it did look slightly different, but overriding the endpoint was all that was needed.

December 09, 2023 04:23 PM

December 04, 2023

Matthew Sackman

Golang Bebop serialisation codec

I’ve been paying some attention to serialisation formats since 2014 or so. I used Cap’n Proto when I was writing GoshawkDB which dates from around 2015, and just before that I think I’d been using Protobuf. Two years ago, when working on WalkTogether, was the first time I’d used Bebop.

There are dozens of different serialisation formats, and a very wide range of trade-offs to each one. I tend to favour formats where you define the schema in a separate language and file, and from there use tools to generate suitable code for different languages. I think I was very resistant to this approach when I first came across it, but without it, it means you have no single source of truth as to what the protocol or its types are, and manually updating different implementations in different languages quickly becomes a disaster.

JSON has its place: it benefits from being mildly human-readable on the wire, and if you really need something vaguely self-describing then it’s probably the obvious choice. Tooling support for JSON is great, especially in browsers, though JSON Schema is pretty awful. But JSON doesn’t scale, and the awful efficiency of the encoding is frustrating. About a year ago I found myself dealing with 100GB+ JSON files. That wasn’t a lot of fun, though I would be surprised if even the most efficient serialisation format would get that below 70GB and so you’d still be writing your own tooling to deal with streaming and processing files at that size.

Bebop brings nothing new to the table. It doesn’t say anything about memory allocation like Cap’n Proto, and it doesn’t have built-in mechanisms to deal with versioning and schema evolution (though you can build support for that yourself by using its unions). It doesn’t have good documentation – certainly nothing that I would consider acceptable as a specification. There are also some statements there that at best are questionable. For example when talking about opcodes, it says:

All the compiler does is check that no opcode is used twice…

Well that’s impossible unless you’re doing compile-the-world. But there’s no specification of how compilation should proceed. So in my implementation it’s a runtime check (via Go’s init() funcs) that no opcode is used twice.

There is an existing Go binding which I’ve used before. It’s fast and it seems to work just fine. Nevertheless, I wasn’t super keen on a few aspects: the parser is hand rolled; the code generation is done by appending strings together; and the generated code isn’t all that nice. None of these are critical issues really, but nevertheless I decided to see whether I could address these points, and here is the result.

For the parser, I’m using Pigeon which I’ve used before. Given that upstream doesn’t have a specification of the Bebop language, I propose my PEG grammar as a specification (of sorts). That grammar is fairly relaxed about where you need semicolons, new lines, that sort of thing.

I’m using Go’s stdlib templates to drive the code generation. In truth the code to drive it is not super pretty: it would probably benefit from a bunch of refactoring and tidying. The generated code though is fairly nice: I don’t think it’s terribly far off the sort of code I’d write if I was rolling it by hand. There’s a []byte-based APIs (Marshal/Unmarshal) which makes sense when you have something else doing framing (for example maybe you’re sending over websocket and so your transport is message oriented already, or you’re reading in from a key-value store so the value’s length is already known), and there’s an io.Reader/io.Writer-based API too for when your transport is a stream (e.g. plain TCP).

December 04, 2023 04:01 PM

November 30, 2023

Ken T Takusagawa

[iuigljdm] and (False,True)

some more silliness resulting from the Foldable Traversable Proposal (FTP) in Haskell:

*Main> and (False,True)
True
*Main> and [False,True]
False
*Main> uncurry (&&) (False,True)
False
*Main> snd (False,True)
True

"and" may be called on a tuple because tuples are instances of Foldable.  this is similar to length (1,2) == 1.

"and" == "snd" was discovered by accident.  I had accidentally typed "and" instead of "snd" (A and S are adjacent on a QWERTY keyboard), calling it on an argument of type (a,Bool).  despite the typo, seemingly substituting functions of completely different types, the program compiled and ran successfully.  I think "and" and "snd" always give the same answer for inputs of type (a,Bool).

by Unknown (noreply@blogger.com) at November 30, 2023 10:38 PM

November 27, 2023

Monday Morning Haskell

Black Friday Sale: Last Day!

We've come to Cyber Monday, marking the last day of our Black Friday sale! Today is your last chance to get big discounts on all of our courses. You can get 20% by using the code BFSOLVE23 at checkout. Or you can subscribe to our mailing list to receive a 30% discount code. You must use these codes by the end of the day in order to get the discount!

Here's a final runthrough of the courses we have available, including our newest course, Solve.hs!

Solve.hs

We just released the first part of our newest course last week! These two detailed modules dive into the fundamentals of problem solving in Haskell. You'll get to rewrite the list type and most of its API from scratch, teaching you all the different ways you can write "loop" code in Haskell. Then you'll get an in-depth look at how data structures work in Haskell, including the quick process to learn a data structure from start to finish!

Course Page

Normal Price: $89 Sale Price: $71.20 Subscriber Price: $62.30

Haskell From Scratch

This is our extensive, 7-module beginners course. You'll get a complete introduction to Haskell's syntax and core concepts, including things like monads and tricky type conversions.

Course Page

Normal Price: $99 Sale Price: $79.20 Subscriber Price: $69.30

Practical Haskell

Practical Haskell is designed to break the idea that "Haskell is only an academic language". In our longest and most detailed course, you'll learn the ins and outs of communicating with a database in Haskell, building a web server, and connecting that server to a functional frontend page. You'll also learn about the flexibility that comes with Haskell's effect systems, as well as best practices for testing your code, including tricky test cases like IO based functions!

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Normal Price: $149 Sale Price: $119.20 Subscriber Price: $104.30

Making Sense of Monads

The first of our shorter, more targeted courses, Making Sense of Monads will teach you how to navigate monads, one of Haskell's defining concepts. This idea is a bit tricky at first but also quite important for unleashing Haskell's full power. The course is well suited to beginners who know all the basic syntax but want more conceptual practice.

Note that Making Sense of Monads is bundled with Haskell From Scratch. So if you buy the full beginners course, you'll get this in-depth look at monads for free!

Course Page

Normal Price: $29 Sale Price: $23.20 Subscriber Price: $20.30

Effectful Haskell

If Making Sense of Monads is best for teaching the basics of monads, Effectful Haskell will show you how to maximize the potential of this idea. You'll develop a more complete idea of what we mean by "effects" in your code. You'll see a variety of ways to incorporate them into your code and learn some interesting ideas about effect substitution!

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Normal Price: $39 Sale Price: $31.20 Subscriber Price: $27.30

Haskell Brain

Last, but not least, Haskell Brain will teach you how to perform machine learning tasks in Haskell with TensorFlow. There's a lot of steps involved in linking these two technologies. So while machine learning is a valuable skill to have in today's world, understanding the ways we can link software together is almost as valuable!

Course Page

Normal Price: $39 Sale Price: $31.20 Subscriber Price: $27.30

Conclusion

So don't miss out on this special offer! You can use the code BFSOLVE23 for 20% off, or you can subscribe to our mailing list to get a code for 30% off! This offer ends tonight, so don't wait!

by James Bowen at November 27, 2023 03:30 PM

November 24, 2023

Monday Morning Haskell

Spotlight: Quick, Focused Haskell Courses

A couple days ago I gave a brief spotlight on the longer, more in-depth courses I've written. The newest of these is Solve.hs, with its focus on problem solving, and the original two I wrote were Haskell From Scratch and Practical Haskell.

After my first two courses, I transitioned towards writing a few shorter courses. These are designed to teach vital concepts in a shorter period of time. They all consist of just a single module and have a shorter total lecture time (1.5 to 2 hours each). You can finish any of them in a concentrated 1-2 week effort. Today I'll give a brief summary of each of these, listed from most abstract to most practical, and easiest to hardest.

Remember, all of these are on sale at 20% off using the code BFSOLVE23 at checkout! You can also subscribe to our mailing list to get an even bigger discount, at 30% off!

Making Sense of Monads

This is for those of you who have been writing Haskell long enough that you've got the hang of the syntax, but you still struggle a bit to understand monads. You might look at parts of Modules 4 and 5 from Haskell From Scratch and think they look useful, but you don't think you need the rest of the course.

Making Sense of Monads really "zooms in" on Module 5. It goes deeper in understanding all of the simpler structures that help us understand monads, and it gives a sizable amount of practice with writing monadic code. You'll also get a crash course on parsing (a common use of monadic operations), and write two fairly complex parsers. So it's a great option if you want a shorter but more concentrated approach on some of the basics!

Effectful Haskell

Effectful Haskell takes a lot of the core ideas and concepts in Making Sense of Monads and goes one step beyond into the more practical realm of applying monadic effects in a program. You'll learn more abstractly what an effect is, but then also the different ways to incorporate polymorphic effects into your Haskell program. You'll see how to use monads and monad classes to swap effectful behaviors in your program, and why this is useful.

This course culminates in a similar (but smaller) project to Practical Haskell, where you'll deploy an effectful web server to Heroku.

Haskell Brain

This course is the hardest and most practically-oriented of this series. You will take on the challenge of incorporating TensorFlow and machine learning into Haskell. This is easier said than done, because TensorFlow has many dependencies beyond the normal packages you can simply pick up on Hackage. So you'll gain valuable experience going through this installation process, and then we'll run through some of the main information you need to know when it comes to creating tensors in Haskell, and building moderately complex models.

Conclusion

So while these courses are shorter, they still pack a decent amount of material! And with the subscriber discount, you can get each of them for less than $30! This offer will only last until Monday though, so make up your mind quickly!

by James Bowen at November 24, 2023 03:30 PM

November 19, 2023

Magnus Therning

Making Emacs without terminal emulator a little more usable

After reading Andrey Listopadov's You don't need a terminal emulator (mentioned at Irreal too) I decided to give up on using Emacs as a terminal for my shell. In my experience Emacs simply isn't a very good terminal to run a shell in anyway. I removed the almost completely unused shell-pop from my configuration and the keybinding with a binding to async-shell-command. I'm keeping terminal-here in my config for the time being though.

I realised projectile didn't have a function for running it in the root of a project, so I wrote one heavily based on project-async-shell-command.

(defun mep-projectile-async-shell-command ()
  "Run `async-shell-command' in the current project's root directory."
  (declare (interactive-only async-shell-command))
  (interactive)
  (let ((default-directory (projectile-project-root)))
    (call-interactively #'async-shell-command)))

I quickly found that the completion offered by Emacs for shell-command and async-shell-command is far from as sophisticated as what I'm used to from Z shell. After a bit of searching I found emacs-bash-completion. Bash isn't my shell of choice, partly because I've found the completion to not be as good as in Z shell, but it's an improvement over what stock Emacs offers. The instructions in the repo was good, but had to be adjusted slightly:

(use-package bash-completion
  :straight (:host github
             :repo "szermatt/emacs-bash-completion")
  :config
  (add-hook 'shell-dynamic-complete-functions 'bash-completion-dynamic-complete))

I just wish I'll find a package offering completions reaching Z shell levels.

November 19, 2023 07:50 AM

November 16, 2023

Magnus Therning

Using the golang mode shipped with Emacs

A few weeks ago I wanted to try out tree-sitter and switched a few of the modes I use for coding to their -ts-mode variants. Based on the excellent How to Get Started with Tree-Sitter I added bits like this to the setup I have for coding modes:1

(use-package X-mode
  :init
  (add-to-list 'treesit-language-source-alist '(X "https://github.com/tree-sitter/tree-sitter-X"))
  ;; (treesit-install-language-grammar 'X)
  (add-to-list 'major-mode-remap-alist '(X-mode . X-ts-mode))
  ;; ...
  )

I then manually evaluated the expression that's commented out to download and compile the tree-sitter grammar. It's a rather small change, it works, and I can switch over language by language. I swapped a couple of languages to the tree-sitter modes like this, including golang. The only mode that I noticed changes in was golang, in particular my adding of gofmt-before-save to before-save-hook had stopped having any effect.

What I hadn't realised was that the go-mode I was using didn't ship with Emacs and that when I switched to go-ts-mode I switched to one that was. It turns out that gofmt-before-save is hard-wired to work only in go-mode, something others have noticed.

I don't feel like waiting for go-mode to fix that though, especially not when there's a perfectly fine golang mode shipping with Emacs now, and not when emacs-reformatter make it so easy to define formatters (as I've written about before).

My golang setup, sans keybindings, now looks like this:2

(use-package go-ts-mode
  :hook
  (go-ts-mode . lsp-deferred)
  (go-ts-mode . go-format-on-save-mode)
  :init
  (add-to-list 'treesit-language-source-alist '(go "https://github.com/tree-sitter/tree-sitter-go"))
  (add-to-list 'treesit-language-source-alist '(gomod "https://github.com/camdencheek/tree-sitter-go-mod"))
  ;; (dolist (lang '(go gomod)) (treesit-install-language-grammar lang))
  (add-to-list 'auto-mode-alist '("\\.go\\'" . go-ts-mode))
  (add-to-list 'auto-mode-alist '("/go\\.mod\\'" . go-mod-ts-mode))
  :config
  (reformatter-define go-format
    :program "goimports"
    :args '("/dev/stdin"))
  :general
  ;; ...
  )

So far I'm happy with the built-in go-ts-mode and I've got to say that using a minor mode for the format-on-save functionality is more elegant than adding a function to before-save-hook (something that go-mode may get through this PR).

Footnotes:

1

There were a few more things that I needed to modify. As the tree-sitter modes are completely separate from the non-tree-sitter modes things like hooks and keybindings in the modes' keymaps.

2

The full file is here.

November 16, 2023 06:02 AM

November 10, 2023

GHC Developer Blog

GHC 9.4.8 is now available

GHC 9.4.8 is now available

Zubin Duggal - 2023-11-10

The GHC developers are happy to announce the availability of GHC 9.4.8. Binary distributions, source distributions, and documentation are available on the release page.

This release is primarily a bugfix release addressing a few issues found in the 9.4 series. These include:

  • A fix for a recompilation checking bug where GHC may miss changes in transitive dependencies when deciding to relink a program (#23724).
  • A fix for a code generator bug on AArch64 platforms resulting in invalid conditional jumps (#23746).
  • Support for -split-sections on Windows.
  • Enabling -split-sections for various Linux and Windows binary distributions, enabling GHC to produce smaller binaries on these platforms.
  • And a few other fixes

A full accounting of changes can be found in the release notes. As some of the fixed issues do affect correctness users are encouraged to upgrade promptly.

We would like to thank Microsoft Azure, GitHub, IOG, the Zw3rk stake pool, Well-Typed, Tweag I/O, Serokell, Equinix, SimSpace, Haskell Foundation, and other anonymous contributors whose on-going financial and in-kind support has facilitated GHC maintenance and release management over the years. Finally, this release would not have been possible without the hundreds of open-source contributors whose work comprise this release.

As always, do give this release a try and open a ticket if you see anything amiss.

Enjoy!

-Bryan

by ghc-devs at November 10, 2023 12:00 AM

November 07, 2023

Chris Reade

Graphs, Kites and Darts – and Theorems

We continue our exploration of properties of Penrose’s aperiodic tilings with kites and darts using Haskell and Haskell Diagrams.

In this blog we discuss some interesting properties we have discovered concerning the \small\texttt{decompose}, \small\texttt{compose}, and \small\texttt{force} operations along with some proofs.

Index

  1. Quick Recap (including operations \small\texttt{compose}, \small\texttt{force}, \small\texttt{decompose} on Tgraphs)
  2. Composition Problems and a Compose Force Theorem (composition is not a simple inverse to decomposition)
  3. Perfect Composition Theorem (establishing relationships between \small\texttt{compose}, \small\texttt{force}, \small\texttt{decompose})
  4. Multiple Compositions (extending the Compose Force theorem for multiple compositions)
  5. Proof of the Compose Force Theorem (showing \small\texttt{compose} is total on forced Tgraphs)

1. Quick Recap

Haskell diagrams allowed us to render finite patches of tiles easily as discussed in Diagrams for Penrose tiles. Following a suggestion of Stephen Huggett, we found that the description and manipulation of such tilings is greatly enhanced by using planar graphs. In Graphs, Kites and Darts we introduced a specialised planar graph representation for finite tilings of kites and darts which we called Tgraphs (tile graphs). These enabled us to implement operations that use neighbouring tile information and in particular operations \small\texttt{decompose}, \small\texttt{compose}, and \small\texttt{force}.

For ease of reference, we reproduce the half-tiles we are working with here.

Figure 1: Half-tile faces
Figure 1: Half-tile faces

Figure 1 shows the right-dart (RD), left-dart (LD), left-kite (LK) and right-kite (RK) half-tiles. Each has a join edge (shown dotted) and a short edge and a long edge. The origin vertex is shown red in each case. The vertex at the opposite end of the join edge from the origin we call the opp vertex, and the remaining vertex we call the wing vertex.

If the short edges have unit length then the long edges have length \phi (the golden ratio) and all angles are multiples of 36^{\circ} (a tenth turn) with kite halves having  two 2s and a 1, and dart halves having a 3 and two 1s. This geometry of the tiles is abstracted away from at the graph representation level but used when checking validity of tile additions and by the drawing functions.

There are rules for how the tiles can be put together to make a legal tiling (see e.g. Diagrams for Penrose tiles). We defined a Tgraph (in Graphs, Kites and Darts) as a list of such half-tiles which are constrained to form a legal tiling but must also be connected with no crossing boundaries (see below).

As a simple example consider kingGraph (2 kites and 3 darts round a king vertex). We represent each half-tile as a TileFace with three vertex numbers, then apply makeTgraph to the list of ten Tilefaces. The function makeTgraph :: [TileFace] -> Tgraph performs the necessary checks to ensure the result is a valid Tgraph.

kingGraph :: Tgraph
kingGraph = makeTgraph 
  [LD (1,2,3),RD (1,11,2),LD (1,4,5),RD (1,3,4),LD (1,10,11)
  ,RD (1,9,10),LK (9,1,7),RK (9,7,8),RK (5,7,1),LK (5,6,7)
  ]

To view the Tgraph we simply form a diagram (in this case 2 diagrams horizontally separated by 1 unit)

  hsep 1 [labelled drawj kingGraph, draw kingGraph]

and the result is shown in figure 2 with labels and dashed join edges (left) and without labels and join edges (right).

Figure 2: kingGraph with labels and dashed join edges (left) and without (right).
Figure 2: kingGraph with labels and dashed join edges (left) and without (right).

The boundary of the Tgraph consists of the edges of half-tiles which are not shared with another half-tile, so they go round untiled/external regions. The no crossing boundary constraint (equivalently, locally tile-connected) means that a boundary vertex has exactly two incident boundary edges and therefore has a single external angle in the tiling. This ensures we can always locally determine the relative angles of tiles at a vertex. We say a collection of half-tiles is a valid Tgraph if it constitutes a legal tiling but also satisfies the connectedness and no crossing boundaries constraints.

Our key operations on Tgraphs are \small\texttt{decompose}, \small\texttt{force}, and \small\texttt{compose} which are illustrated in figure 3.

Figure 3: decompose, force, and compose
Figure 3: decompose, force, and compose

Figure 3 shows the kingGraph with its decomposition above it (left), the result of forcing the kingGraph (right) and the composition of the forced kingGraph (bottom right).

Decompose

An important property of Penrose dart and kite tilings is that it is possible to divide the half-tile faces of a tiling into smaller half-tile faces, to form a new (smaller scale) tiling.

Figure 4: Decomposition of (left) half-tiles
Figure 4: Decomposition of (left) half-tiles

Figure 4 illustrates the decomposition of a left-dart (top row) and a left-kite (bottom row). With our Tgraph representation we simply introduce new vertices for dart and kite long edges and kite join edges and then form the new faces using these. This does not involve any geometry, because that is taken care of by drawing operations.

Force

Figure 5 illustrates the rules used by our \small\texttt{force} operation (we omit a mirror-reflected version of each rule).

Figure 5: Force rules
Figure 5: Force rules

In each case the yellow half-tile is added in the presence of the other half-tiles shown. The yellow half-tile is forced because, by the legal tiling rules, there is no choice for adding a different half-tile on the edge where the yellow tile is added.

We call a Tgraph correct if it represents a tiling which can be continued infinitely to cover the whole plane without getting stuck, and incorrect otherwise. Forcing involves adding half-tiles by the illustrated rules round the boundary until either no more rules apply (in which case the result is a forced Tgraph) or a stuck tiling is encountered (in which case an incorrect Tgraph error is raised). Hence \small\texttt{force} is a partial function but total on correct Tgraphs.

Compose: This is discussed in the next section.

2. Composition Problems and a Theorem

Compose Choices

For an infinite tiling, composition is a simple inverse to decomposition. However, for a finite tiling with boundary, composition is not so straight forward. Firstly, we may need to leave half-tiles out of a composition because the necessary parts of a composed half-tile are missing. For example, a half-dart with a boundary short edge or a whole kite with both short edges on the boundary must necessarily be excluded from a composition. Secondly, on the boundary, there can sometimes be a problem of choosing whether a half-dart should compose to become a half-dart or a half-kite. This choice in composing only arises when there is a half-dart with its wing on the boundary but insufficient local information to determine whether it should be part of a larger half-dart or a larger half-kite.

In the literature (see for example 1 and 2) there is an often repeated method for composing (also called inflating). This method always make the kite choice when there is a choice. Whilst this is a sound method for an unbounded tiling (where there will be no choice), we show that this is an unsound method for finite tilings as follows.

Clearly composing should preserve correctness. However, figure 6 (left) shows a correct Tgraph which is a forced queen, but the kite-favouring composition of the forced queen produces the incorrect Tgraph shown in figure 6 (centre). Applying our \small\texttt{force} function to this reveals a stuck tiling and reports an incorrect Tgraph.

Figure 6: An erroneous and a safe composition
Figure 6: An erroneous and a safe composition

Our algorithm (discussed in Graphs, Kites and Darts) detects dart wings on the boundary where there is a choice and classifies them as unknowns. Our composition refrains from making a choice by not composing a half dart with an unknown wing vertex. The rightmost Tgraph in figure 6 shows the result of our composition of the forced queen with the half-tile faces left out of the composition (the remainder faces) shown in green. This avoidance of making a choice (when there is a choice) guarantees our composition preserves correctness.

Compose is a Partial Function

A different composition problem can arise when we consider Tgraphs that are not decompositions of Tgraphs. In general, \small\texttt{compose} is a partial function on Tgraphs.

Figure 7: Composition may fail to produce a Tgraph
Figure 7: Composition may fail to produce a Tgraph

Figure 7 shows a Tgraph (left) with its sucessful composition (centre) and the half-tile faces that would result from a second composition (right) which do not form a valid Tgraph because of a crossing boundary (at vertex 6). Thus composition of a Tgraph may fail to produce a Tgraph when the resulting faces are disconnected or have a crossing boundary.

However, we claim that \small\texttt{compose} is a total function on forced Tgraphs.

Compose Force Theorem

Theorem: Composition of a forced Tgraph produces a valid Tgraph.

We postpone the proof (outline) for this theorem to section 5. Meanwhile we use the result to establish relationships between \small\texttt{compose}, \small\texttt{force}, and \small\texttt{decompose} in the next section.

3. Perfect Composition Theorem

In Graphs, Kites and Darts we produced a diagram showing relationships between multiple decompositions of a dart and the forced versions of these Tgraphs. We reproduce this here along with a similar diagram for multiple decompositions of a kite.

Figure 8: Commuting Diagrams
Figure 8: Commuting Diagrams

In figure 8 we show separate (apparently) commuting diagrams for the dart and for the kite. The bottom rows show the decompositions, the middle rows show the result of forcing the decompositions, and the top rows illustrate how the compositions of the forced Tgraphs work by showing both the composed faces (black edges) and the remainder faces (green edges) which are removed in the composition. The diagrams are examples of some commutativity relationships concerning \small\texttt{force}, \small\texttt{compose} and \small\texttt{decompose} which we will prove.

It should be noted that these diagrams break down if we consider only half-tiles as the starting points (bottom right of each diagram). The decomposition of a half-tile does not recompose to its original, but produces an empty composition. So we do not even have g = (\small\texttt{compose} \cdot \small\texttt{decompose}) \ g in these cases. Forcing the decomposition also results in an empty composition. Clearly there is something special about the depicted cases and it is not merely that they are wholetile complete because the decompositions are not wholetile complete. [Wholetile complete means there are no join edges on the boundary, so every half-tile has its other half.]

Below we have captured the properties that are sufficient for the diagrams to commute as in figure 8. In the proofs we use a partial ordering on Tgraphs (modulo vertex relabelling) which we define next.

Partial ordering of Tgraphs

If g_0 and g_1 are both valid Tgraphs and g_0 consists of a subset of the (half-tile) faces of g_1 we have

\displaystyle g_0 \subseteq g_1

which gives us a partial order on Tgraphs. Often, though, g_0 is only isomorphic to a subset of the faces of g_1, requiring a vertex relabelling to become a subset. In that case we write

\displaystyle g_0 \sqsubseteq g_1

which is also a partial ordering and induces an equivalence of Tgraphs defined by

\displaystyle g_0 \equiv g_1 \text{ if and only if } g_0 \sqsubseteq g_1 \text{ and } g_1 \sqsubseteq g_0

in which case g_0 and g_1 are isomorphic as Tgraphs.

Both \small\texttt{compose} and \small\texttt{decompose} are monotonic with respect to \sqsubseteq meaning:

\displaystyle g_0 \sqsubseteq g_1 \text{ implies } \small\texttt{compose} \ g_0 \sqsubseteq \small\texttt{compose} \ g_1 \text{ and } \small\texttt{decompose} \ g_0 \sqsubseteq \small\texttt{decompose} \ g_1

We also have \small\texttt{force} is monotonic, but only when restricted to correct Tgraphs. Also, when restricted to correct Tgraphs, we have \small\texttt{force} is non decreasing because it only adds faces:

\displaystyle g \sqsubseteq \small\texttt{force} \ g

and \small\texttt{force} is idempotent (forcing a forced correct Tgraph leaves it the same):

\displaystyle (\small\texttt{force} \cdot \small\texttt{force}) \ g \equiv \small\texttt{force} \ g

Composing perfectly and perfect compositions

Definition: A Tgraph g composes perfectly if all faces of g are composable (i.e there are no remainder faces of g when composing).

We note that the composed faces must be a valid Tgraph (connected with no crossing boundaries) if all faces are included in the composition because g has those properties. Clearly, if g composes perfectly then

\displaystyle (\small\texttt{decompose} \cdot \small\texttt{compose}) \ g \equiv g

In general, for arbitrary g where the composition is defined, we only have

\displaystyle (\small\texttt{decompose} \cdot \small\texttt{compose}) \ g \sqsubseteq g

Definition: A Tgraph g' is a perfect composition if \small\texttt{decompose} \ g' composes perfectly.

Clearly if g' is a perfect composition then

\displaystyle (\small\texttt{compose} \cdot \small\texttt{decompose}) \ g' \equiv g'

(We could use equality here because any new vertex labels introduced by \small\texttt{decompose} will be removed by \small\texttt{compose}). In general, for arbitrary g',

\displaystyle (\small\texttt{compose} \cdot \small\texttt{decompose}) \ g' \sqsubseteq g'

Lemma 1: g' is a perfect composition if and only if g' has the following 2 properties:

  1. every half-kite with a boundary join has either a half-dart or a whole kite on the short edge, and
  2. every half-dart with a boundary join has a half-kite on the short edge,

(Proof outline:) Firstly note that unknowns in g (= \small\texttt{decompose} \ g') can only come from boundary joins in g'. The properties 1 and 2 guarantee that g has no unknowns. Since every face of g has come from a decomposed face in g', there can be no faces in g that will not recompose, so g will compose perfectly to g'. Conversely, if g' is a perfect composition, its decomposition g can have no unknowns. This implies boundary joins in g' must satisfy properties 1 and 2. \square

(Note: a perfect composition g' may have unknowns even though its decomposition g has none.)

It is easy to see two special cases:

  1. If g' is wholetile complete then g' is a perfect composition.Proof: Wholetile complete implies no boundary joins which implies properties 1 and 2 in lemma 1 which implies g' is a perfect composition. \square
  2. If g' is a decomposition then g' is a perfect composition.Proof: If g' is a decomposition, then every half-dart has a half-kite on the short edge which implies property 2 of lemma 1. Also, any half-kite with a boundary join in g' must have come from a decomposed half-dart since a decomposed half-kite produces a whole kite with no boundary kite join. So the half-kite must have a half-dart on the short edge which implies property 1 of lemma 1. The two properties imply g' is a perfect composition. \square

We note that these two special cases cover all the Tgraphs in the bottom rows of the diagrams in figure 8. So the Tgraphs in each bottom row are perfect compositions, and furthermore, they all compose perfectly except for the rightmost Tgraphs which have empty compositions.

In the following results we make the assumption that a Tgraph is correct, which guarantees that when \small\texttt{force} is applied, it terminates with a correct Tgraph. We also note that \small\texttt{decompose} preserves correctness as does \small\texttt{compose} (provided the composition is defined).

Lemma 2: If g_f is a forced, correct Tgraph then

\displaystyle (\small\texttt{compose} \cdot \small\texttt{force} \cdot \small\texttt{decompose}) \ g_f \equiv g_f

(Proof outline:) The proof uses a case analysis of boundary and internal vertices of g_f. For internal vertices we just check there is no change at the vertex after (\small\texttt{compose} \cdot \small\texttt{force} \cdot \small\texttt{decompose}) using figure 11 (plus an extra case for the forced star). For boundary vertices we check local contexts similar to those depicted in figure 10 (but including empty composition cases). This reveals there is no local change of the boundary at any boundary vertex, and since this is true for all boundary vertices, there can be no global change. (We omit the full details). \square

Lemma 3: If g' is a perfect composition and a correct Tgraph, then

\displaystyle \small\texttt{force} \ g' \sqsubseteq (\small\texttt{compose} \cdot \small\texttt{force} \cdot \small\texttt{decompose}) \ g'

(Proof outline:) The proof is by analysis of each possible force rule applicable on a boundary edge of g' and checking local contexts to establish that (i) the result of applying (\small\texttt{compose} \cdot \small\texttt{force} \cdot \small\texttt{decompose}) to the local context must include the added half-tile, and (ii) if the added half tile has a new boundary join, then the result must include both halves of the new half-tile. The two properties of perfect compositions mentioned in lemma 1 are critical for the proof. However, since the result of adding a single half-tile may break the condition of the Tgraph being a pefect composition, we need to arrange that half-tiles are completed first then each subsequent half-tile addition is paired with its wholetile completion. This ensures the perfect composition condition holds at each step for a proof by induction. [A separate proof is needed to show that the ordering of applying force rules makes no difference to a final correct Tgraph (apart from vertex relabelling)]. \square

Lemma 4 If g composes perfectly and is a correct Tgraph then

\displaystyle \small\texttt{force} \ g \equiv (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{force} \cdot \small\texttt{compose})\ g

Proof: Assume g composes perfectly and is a correct Tgraph. Since \small\texttt{force} is non-decreasing (with respect to \sqsubseteq on correct Tgraphs)

\displaystyle \small\texttt{compose} \ g \sqsubseteq (\small\texttt{force} \cdot \small\texttt{compose}) \ g

and since \small\texttt{decompose} is monotonic

\displaystyle (\small\texttt{decompose} \cdot \small\texttt{compose}) \ g \sqsubseteq (\small\texttt{decompose} \cdot \small\texttt{force} \cdot \small\texttt{compose}) \ g

Since g composes perfectly, the left hand side is just g, so

\displaystyle g \sqsubseteq (\small\texttt{decompose} \cdot \small\texttt{force} \cdot \small\texttt{compose}) \ g

and since \small\texttt{force} is monotonic (with respect to \sqsubseteq on correct Tgraphs)

\displaystyle (*) \ \ \ \ \ \small\texttt{force} \ g \sqsubseteq (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{force} \cdot \small\texttt{compose}) \ g

For the opposite direction, we substitute \small\texttt{compose} \ g for g' in lemma 3 to get

\displaystyle (\small\texttt{force} \cdot \small\texttt{compose}) \ g \sqsubseteq (\small\texttt{compose} \cdot \small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{compose}) \ g

Then, since (\small\texttt{decompose} \cdot \small\texttt{compose}) \ g \equiv g, we have

\displaystyle (\small\texttt{force} \cdot \small\texttt{compose}) \ g \sqsubseteq (\small\texttt{compose} \cdot \small\texttt{force}) \ g

Apply \small\texttt{decompose} to both sides (using monotonicity)

\displaystyle (\small\texttt{decompose} \cdot \small\texttt{force} \cdot \small\texttt{compose}) \ g \sqsubseteq (\small\texttt{decompose} \cdot \small\texttt{compose} \cdot \small\texttt{force}) \ g

For any g'' for which the composition is defined we have (\small\texttt{decompose} \cdot \small\texttt{compose})\ g'' \sqsubseteq g'' so we get

\displaystyle (\small\texttt{decompose} \cdot \small\texttt{force} \cdot \small\texttt{compose}) \ g \sqsubseteq \small\texttt{force} \ g

Now apply \small\texttt{force} to both sides and note (\small\texttt{force} \cdot \small\texttt{force})\ g \equiv \small\texttt{force} \ g to get

\displaystyle (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{force} \cdot \small\texttt{compose}) \ g \sqsubseteq \small\texttt{force} \ g

Combining this with (*) above proves the required equivalence. \square

Theorem (Perfect Composition): If g composes perfectly and is a correct Tgraph then

\displaystyle (\small\texttt{compose} \cdot \small\texttt{force}) \ g \equiv (\small\texttt{force} \cdot \small\texttt{compose}) \ g

Proof: Assume g composes perfectly and is a correct Tgraph. By lemma 4 we have

\displaystyle \small\texttt{force} \ g \equiv (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{force} \cdot \small\texttt{compose})\ g

Applying \small\texttt{compose} to both sides, gives

\displaystyle (\small\texttt{compose} \cdot \small\texttt{force}) \ g \equiv (\small\texttt{compose} \cdot \small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{force} \cdot \small\texttt{compose})\ g

Now by lemma 2, with g_f = (\small\texttt{force} \cdot \small\texttt{compose}) \ g, the right hand side is equivalent to

\displaystyle (\small\texttt{force} \cdot \small\texttt{compose}) \ g

which establishes the result. \square

Corollaries (of the perfect composition theorem):

  1. If g' is a perfect composition and a correct Tgraph then
    \displaystyle \small\texttt{force} \ g' \equiv (\small\texttt{compose} \cdot \small\texttt{force} \cdot \small\texttt{decompose}) \ g'

    Proof: Let g' = \small\texttt{compose} \ g (so g \equiv \small\texttt{decompose} \ g') in the theorem. \square

    [This result generalises lemma 2 because any correct forced Tgraph g_f is necessarily wholetile complete and therefore a perfect composition, and \small\texttt{force} \ g_f \equiv g_f.]

  2. If g' is a perfect composition and a correct Tgraph then
    \displaystyle (\small\texttt{decompose} \cdot \small\texttt{force}) \ g' \sqsubseteq (\small\texttt{force} \cdot \small\texttt{decompose}) \ g'

    Proof: Apply \small\texttt{decompose} to both sides of the previous corollary and note that

    \displaystyle (\small\texttt{decompose} \cdot \small\texttt{compose}) \ g'' \sqsubseteq g'' \textit{ for any } g''

    provided the composition is defined, which it must be for a forced Tgraph by the Compose Force theorem. \square

  3. If g' is a perfect composition and a correct Tgraph then
    \displaystyle (\small\texttt{force} \cdot \small\texttt{decompose}) \ g' \equiv (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{force}) \ g'

    Proof: Apply \small\texttt{force} to both sides of the previous corollary noting \small\texttt{force} is monotonic and idempotent for correct Tgraphs

    \displaystyle (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{force}) \ g' \sqsubseteq (\small\texttt{force} \cdot \small\texttt{decompose}) \ g'

    From the fact that \small\texttt{force} is non decreasing and \small\texttt{decompose} and \small\texttt{force} are monotonic, we also have

    \displaystyle (\small\texttt{force} \cdot \small\texttt{decompose}) \ g' \sqsubseteq (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{force}) \ g'

    Hence combining these two sub-Tgraph results we have

    \displaystyle (\small\texttt{force} \cdot \small\texttt{decompose}) \ g' \equiv (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{force}) \ g'

    \square

It is important to point out that if g is a correct Tgraph and \small\texttt{compose} \ g is a perfect composition then this is not the same as g composes perfectly. It could be the case that g has more faces than (\small\texttt{decompose} \cdot \small\texttt{compose}) \ g and so g could have unknowns. In this case we can only prove that

\displaystyle (\small\texttt{force} \cdot \small\texttt{compose}) \ g \sqsubseteq (\small\texttt{compose} \cdot \small\texttt{force}) \ g

As an example where this is not an equivalence, choose g to be a star. Then its composition is the empty Tgraph (which is still a pefect composition) and so the left hand side is the empty Tgraph, but the right hand side is a sun.

Perfectly composing generators

The perfect composition theorem and lemmas and the three corollaries justify all the commuting implied by the diagrams in figure 8. However, one might ask more general questions like: Under what circumstances do we have (for a correct forced Tgraph g_f)

\displaystyle (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{compose}) \ g_f \equiv g_f

Definition A generator of a correct forced Tgraph g_f is any Tgraph g such that g \sqsubseteq g_f and \small\texttt{force} \ g \equiv g_f.

We can now state that

Corollary If a correct forced Tgraph g_f has a generator which composes perfectly, then

\displaystyle (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{compose}) \ g_f \equiv g_f

Proof: This follows directly from lemma 4 and the perfect composition theorem. \square

As an example where the required generator does not exist, consider the rightmost Tgraph of the middle row in figure 9. It is generated by the Tgraph directly below it, but it has no generator with a perfect composition. The Tgraph directly above it in the top row is the result of applying (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{compose}) which has lost the leftmost dart of the Tgraph.

Figure 9: A Tgraph without a perfectly composing generator
Figure 9: A Tgraph without a perfectly composing generator

We could summarise this section by saying that \small\texttt{compose} can lose information which cannot be recovered by a subsequent \small\texttt{force} and, similarly, \small\texttt{decompose} can lose information which cannot be recovered by a subsequent \small\texttt{force}. We have defined perfect compositions which are the Tgraphs that do not lose information when decomposed and Tgraphs which compose perfectly which are those that do not lose information when composed. Forcing does the same thing at each level of composition (that is it commutes with composition) provided information is not lost when composing.

4. Multiple Compositions

We know from the Compose Force theorem that the composition of a Tgraph that is forced is always a valid Tgraph. In this section we use this and the results from the last section to show that composing a forced, correct Tgraph produces a forced Tgraph.

First we note that:

Lemma 5: The composition of a forced, correct Tgraph is wholetile complete.

Proof: Let g' = \small\texttt{compose} \ g_f where g_f is a forced, correct Tgraph. A boundary join in g' implies there must be a boundary dart wing of the composable faces of g_f. (See for example figure 4 where this would be vertex 2 for the half dart case, and vertex 5 for the half-kite face). This dart wing cannot be an unknown as the half-dart is in the composable faces. However, a known dart wing must be either a large kite centre or a large dart base and therefore internal in the composable faces of g_f (because of the force rules) and therefore not on the boundary in g'. This is a contradiction showing that g' can have no boundary joins and is therefore wholetile complete. \square

Theorem: The composition of a forced, correct Tgraph is a forced Tgraph.

Proof: Let g' = \small\texttt{compose} \ g_f for some forced, correct Tgraph g_f, then g' is wholetile complete (by lemma 5) and therefore a perfect composition. Let g = \small\texttt{decompose} \ g', so g composes perfectly (g' \equiv \small\texttt{compose} \ g). By the perfect composition theorem we have

\displaystyle (**) \ \ \ \ \ (\small\texttt{compose} \cdot \small\texttt{force}) \ g \equiv (\small\texttt{force} \cdot \small\texttt{compose}) \ g \equiv \small\texttt{force} \ g'

We also have

\displaystyle g = \small\texttt{decompose} \ g' = (\small\texttt{decompose} \cdot \small\texttt{compose}) \ g_f \sqsubseteq g_f

Applying \small\texttt{force} to both sides, noting that \small\texttt{force} is monotonic and the identity on forced Tgraphs, we have

\displaystyle \small\texttt{force} \ g \sqsubseteq \small\texttt{force} \ g_f \equiv g_f

Applying \small\texttt{compose} to both sides, noting that \small\texttt{compose} is monotonic, we have

\displaystyle (\small\texttt{compose} \cdot \small\texttt{force}) \ g \sqsubseteq \small\texttt{compose} \ g_f \equiv g'

By (**) above, the left hand side is equivalent to \small\texttt{force} \ g' so we have

\displaystyle \small\texttt{force} \ g' \sqsubseteq g'

but since we also have (\small\texttt{force} being non-decreasing)

\displaystyle g' \sqsubseteq \small\texttt{force} \ g'

we have established that

\displaystyle g' \equiv \small\texttt{force} \ g'

which means g' is a forced Tgraph. \square

This result means that after forcing once we can repeatedly compose creating valid Tgraphs until we reach the empty Tgraph.

We can also use lemma 5 to establish the converse to a previous corollary:

Corollary If a correct forced Tgraph g_f satisfies:

\displaystyle (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{compose}) \ g_f \equiv g_f

then g_f has a generator which composes perfectly.

Proof: By lemma 5, \small\texttt{compose} \ g_f is wholetile complete and hence a perfect composition. This means that (\small\texttt{decompose} \cdot \small\texttt{compose}) \ g_f composes perfectly and it is also a generator for g_f because

\displaystyle (\small\texttt{force} \cdot \small\texttt{decompose} \cdot \small\texttt{compose}) \ g_f \equiv g_f

\square

5. Proof of the Compose Force theorem

Theorem (Compose Force): Composition of a forced Tgraph produces a valid Tgraph.

Proof: For any forced Tgraph we can construct the composed faces. For the result to be a valid Tgraph we need to show no crossing boundaries and connectedness for the composed faces. These are proved separately by case analysis below.

Proof of no crossing boundaries

Assume g_f is a forced Tgraph and that it has a non-empty set of composed faces (we can ignore cases where the composition is empty as the empty Tgraph is valid). Consider a vertex v in the composed faces of g_f and first take the case that v is on the boundary of g_f . We consider the possible local contexts for a vertex v on a forced Tgraph boundary and the nature of the composed faces at v in each case.

Figure 10: Forced Boundary Vertex Contexts
Figure 10: Forced Boundary Vertex Contexts

Figure 10 shows local contexts for a boundary vertex v in a forced Tgraph where the composition is non-empty. In each case v is shown as a red dot, and the composition is shown filled yellow. The cases for v are shown in rows: the first row is for dart origins, the second row is for kite origins, the next two rows are for kite wings, and the last two rows are for kite opps. The dart wing cases are a subset of the kite opp cases, so not repeated, and dart opp vertices are excluded because they cannot be on the boundary of a forced Tgraph. We only show left-hand versions, so there is a mirror symmetric set for right-hand versions.

It is easy to see that there are no crossing boundaries of the composed faces at v in each case. Since any boundary vertex of any forced Tgraph (with a non-empty composition) must match one of these local context cases around the vertex, we can conclude that a boundary vertex of g_f cannot become a crossing boundary in compose \ g_f.

Next take the case where v is an internal vertex of g_f .

Figure 11: Vertex types and their relationships
Figure 11: Vertex types and their relationships

Figure 11 shows relationships between the forced Tgraphs of the 7 (internal) vertex types (plus a kite at the top right). The red faces are those around the vertex type and the black faces are those produced by forcing (if any). Each forced Tgraph has its composition directly above with empty compositions for the top row. We note that a (forced) star, jack, king, and queen vertex remains an internal vertex in the respective composition so cannot become a crossing boundary vertex. A deuce vertex becomes the centre of a larger kite and is no longer present in the composition (top right). That leaves cases for the sun vertex and ace vertex (=fool vertex). The sun Tgraph (sunGraph) and fool Tgraph (fool) consist of just the red faces at the respective vertex (shown top left and top centre). These both have empty compositions when there is no surrounding context. We thus need to check possible forced local contexts for sunGraph and fool.

The fool case is simple and similar to a duece vertex in that it is never part of a composition. [To see this consider inverting the decomposition arrows shown in figure 4. In both cases we see the half-dart opp vertex (labelled 4 in figure 4) is removed].

For the sunGraph there are only 7 local forced context cases to consider where the sun vertex is on the boundary of the composition.

Figure 12: Forced Contexts for a sun vertex v where v is on the composition boundary
Figure 12: Forced Contexts for a sun vertex v where v is on the composition boundary

Six of these are shown in figure 12 (the missing one is just a mirror reflection of the fourth case). Again, the relevant vertex v is shown as a red dot and the composed faces are shown filled yellow, so it is easy to check that there is no crossing boundary of the composed faces at v in each case. Every forced Tgraph containing an internal sun vertex where the vertex is on the boundary of the composition must match one of the 7 cases locally round the vertex.

Thus no vertex from g_f can become a crossing boundary vertex in the composed faces and since the vertices of the composed faces are a subset of those of g_f, we can have no crossing boundary vertex in the composed faces.

Proof of Connectedness

Assume g_f is a forced Tgraph as before. We refer to the half-tile faces of g_f that get included in the composed faces as the composable faces and the rest as the remainder faces. We want to prove that the composable faces are connected as this will imply the composed faces are connected.

As before we can ignore cases where the set of composable faces is empty, and assume this is not the case. We study the nature of the remainder faces of g_f. Firstly, we note:

Lemma (remainder faces)

The remainder faces of g_f are made up entirely of groups of half-tiles which are either:

  1. Half-fools (= a half dart and both halves of the kite attached to its short edge) where the other half-fool is entirely composable faces, or
  2. Both halves of a kite with both short edges on the (g_f) boundary (so they are not part of a half-fool) where only the origin is in common with composable faces, or
  3. Whole fools with just the shared kite origin in common with composable faces.
Figure 13: Remainder face groups (cases 1,2, and 3)
Figure 13: Remainder face groups (cases 1,2, and 3)

These 3 cases of remainder face groups are shown in figure 13. In each case the border in common with composable faces is shown yellow and the red edges are necessarily on the boundary of g_f (the black boundary could be on the boundary of g_f or shared with another reamainder face group). [A mirror symmetric version for the first group is not shown.] Examples can be seen in e.g. figure 12 where the first Tgraph has four examples of case 1, and two of case 2, the second has six examples of case 1 and two of case 2, and the fifth Tgraph has an example of case 3 as well as four of case 1. [We omit the detailed proof of this lemma which reasons about what gets excluded in a composition after forcing. However, all the local context cases are included in figure 14 (left-hand versions), where we only show those contexts where there is a non-empty composition.]

We note from the (remainder faces) lemma that the common boundary of the group of remainder faces with the composable faces (shown yellow in figure 13) is just a single vertex in cases 2 and 3. In case 1, the common boundary is just a single edge of the composed faces which is made up of 2 adjacent edges of the composable faces that constitute the join of two half-fools.

This means each (remainder face) group shares boundary with exactly one connected component of the composable faces.

Next we establish that if two (remainder face) groups are connected they must share boundary with the same connected component of the composable faces. We need to consider how each (remainder face) group can be connected with a neighbouring such group. It is enough to consider forced contexts of boundary dart long edges (for cases 1 and 3) and boundary kite short edges (for case 2). The cases where the composition is non-empty all appear in figure 14 (left-hand versions) along with boundary kite long edges (middle two rows) which are not relevant here.

Figure 14: Forced contexts for boundary edges
Figure 14: Forced contexts for boundary edges

We note that, whenever one group of the remainder faces (half-fool, whole-kite, whole-fool) is connected to a neighbouring group of the remainder faces, the common boundary (shared edges and vertices) with the compososable faces is also connected, forming either 2 adjacent composed face boundary edges (= 4 adjacent edges of the composable faces), or a composed face boundary edge and one of its end vertices, or a single composed face boundary vertex.

It follows that any connected collection of the remainder face groups shares boundary with a unique connected component of the composable faces. Since the collection of composable and remainder faces together is connected (g_f is connected) the removal of the remainder faces cannot disconnect the composable faces. For this to happen, at least one connected collection of remainder face groups would have to be connected to more than one connected component of composable faces.

This establishes connectedness of any composition of a forced Tgraph, and this completes the proof of the Compose Force theorem. \square

References

[1] Martin Gardner (1977) MATHEMATICAL GAMES. Scientific American, 236(1), (pages 110 to 121). http://www.jstor.org/stable/24953856

[2] Grünbaum B., Shephard G.C. (1987) Tilings and Patterns. W. H. Freeman and Company, New York. ISBN 0-7167-1193-1 (Hardback) (pages 540 to 542).

by readerunner at November 07, 2023 01:55 PM

Donnacha Oisín Kidney

POPL Paper—Algebraic Effects Meet Hoare Logic in Cubical Agda

Posted on November 7, 2023
Tags:

New paper: “Algebraic Effects Meet Hoare Logic in Cubical Agda”, by myself, Zhixuan Yang, and Nicolas Wu, will be published at POPL 2024.

Zhixuan has a nice summary of it here.

The preprint is available here.

by Donnacha Oisín Kidney at November 07, 2023 12:00 AM

November 01, 2023

Joachim Breitner

Joining the Lean FRO

Tomorrow is going to be a new first day in a new job for me: I am joining the Lean FRO, and I’m excited.

What is Lean?

Lean is the new kid on the block of theorem provers.

It’s a pure functional programming language (like Haskell, with and on which I have worked a lot), but it’s dependently typed (which Haskell may be evolving to be as well, but rather slowly and carefully). It has a refreshing syntax, built on top of a rather good (I have been told, not an expert here) macro system.

As a dependently typed programming language, it is also a theorem prover, or proof assistant, and there exists already a lively community of mathematicians who started to formalize mathematics in a coherent library, creatively called mathlib.

What is a FRO?

A Focused Research Organization has the organizational form of a small start up (small team, little overhead, a few years of runway), but its goals and measure for success are not commercial, as funding is provided by donors (in the case of the Lean FRO, the Simons Foundation International, the Alfred P. Sloan Foundation, and Richard Merkin). This allows us to build something that we believe is a contribution for the greater good, even though it’s not (or not yet) commercially interesting enough and does not fit other forms of funding (such as research grants) well. This is a very comfortable situation to be in.

Why am I excited?

To me, working on Lean seems to be the perfect mix: I have been working on language implementation for about a decade now, and always with a preference for functional languages. Add to that my interest in theorem proving, where I have used Isabelle and Coq so far, and played with Agda and others. So technically, clearly up my alley.

Furthermore, the language isn’t too old, and plenty of interesting things are simply still to do, rather than tried before. The ecosystem is still evolving, so there is a good chance to have some impact.

On the other hand, the language isn’t too young either. It is no longer an open question whether we will have users: we have them already, they hang out on zulip, so if I improve something, there is likely someone going to be happy about it, which is great. And the community seems to be welcoming and full of nice people.

Finally, this library of mathematics that these users are building is itself an amazing artifact: Lots of math in a consistent, machine-readable, maintained, documented, checked form! With a little bit of optimism I can imagine this changing how math research and education will be done in the future. It could be for math what Wikipedia is for encyclopedic knowledge and OpenStreetMap for maps – and the thought of facilitating that excites me.

With this new job I find that when I am telling friends and colleagues about it, I do not hesitate or hedge when asked why I am doing this. This is a good sign.

What will I be doing?

We’ll see what main tasks I’ll get to tackle initially, but knowing myself, I expect I’ll get broadly involved.

To get up to speed I started playing around with a few things already, and for example created Loogle, a Mathlib search engine inspired by Haskell’s Hoogle, including a Zulip bot integration. This seems to be useful and quite well received, so I’ll continue maintaining that.

Expect more about this and other contributions here in the future.

by Joachim Breitner (mail@joachim-breitner.de) at November 01, 2023 08:47 PM

October 30, 2023

Sandy Maguire

Certainty by Construction: Done!

Happy days and happy news: it’s done.

Certainty by Construction

After a year of work, I’m thrilled to announce the completion my new book, Certainty by Construction.

Certainty by Construction is a book on doing mathematics and software design in the proof assistant Agda, which is the language Haskell wants to be when it grows up. The book is part Agda primer, introduction to abstract algebra, and algorithm design manual, with a healthy dose of philosophy mixed in to help build intuition.

If you’re the sort of person who would like to learn more math (including all the proof burden), and see how to apply it to writing real software, I think you’d groove on this book. If it sounds up your alley, I’d highly encourage you to give it a read.

I’m not much on social media these days, but if you are, I’d really appreciate a signal boost on this announcement! Thanks to everyone for their support and understanding over the last year. I love you all!

Go cop Certainty by Construction!

October 30, 2023 12:00 AM