Planet Haskell

March 22, 2023

Tweag I/O

High Assurance Software

Software failures can lead to significant costs and even human hardship and danger. We believe that with an appropriate set of tools and techniques it is possible to improve the reliability of both existing and new software without a huge upfront investment.

Formal methods do not have to be “too expensive” or “too time consuming”; simple things can drastically improve the assurance of software, even without complete verification. Furthermore, academia is constantly generating new ideas. Which of those translate into suitable tools for a given context? There is a spectrum of assurance and we believe in combining different techniques to gradually bring software from zero assurance to fully verified. We aim to identify, use and improve upon the most suitable tools for the job.

We offer high-assurance services and team augmentation for clients who either have high-assurance needs or are themselves solution providers in this space. In particular, we are happy to assist in the following areas:

  • Specification: help craft and lay specifications, and asses their aptness to an original goal by proving relevant properties about them
  • Testing: help create better test suites, and build tools for executing them over complex domains
  • Auditing: comprehensively test a system, report and showcase behaviours that are potentially harmful to system security or safety, assess code quality
  • Tooling: improve existing tools to suit the testing/verification needs of a particular project, or write domain-specific tools from scratch
  • Refactoring: help improve existing code bases through the use of modularisation, strongly typed code, purity, and other compile-time guarantees
  • Full-fledged verification: assist in projects aiming to develop a completely verified piece of software

March 22, 2023 10:11 AM

March 21, 2023

Mark Jason Dominus

ChatGPT on the namesake of the metric space and women named James

Several folks, reading the frustrating and repetitive argument with ChatGPT that I reported last time wrote in with helpful advice and techniques that I hadn't tried that might have worked better. In particular, several people suggested that if the conversation isn't going anywhere, I should try starting over. Rik Signes put it this way:

Sometimes, I think GPT gets its head up its own ass and can't progress usefully, and a clean start works better.

I hope I can write a followup article about “what to do when ChatGPT has its head up its ass”. This isn't that article though.

I wasn't even going to report on this one, but it took an interesting twist at the end. I started:

Me Is it true that Hilbert space is named after David Hilbert?

ChatGPT Yes, that is correct. Hilbert space is a mathematical concept that was developed by the German mathematician David Hilbert. …

This was only my second interaction with ChatGPT and I was still interested in its limitations, so I asked it a trick question to see what would happen:

Me Is a banach space also named after a german mathematician?

See what I'm doing there? ChatGPT took the bait:

ChatGPT Yes, you are correct. A Banach space is a type of abstract mathematical space that was named after the Polish mathematician Stefan Banach. …

Me I asked you if it was named after a German mathematician, not a Polish one.

ChatGPT I apologize for the mistake. …

I had hoped it would do better there, and was a bit disappointed. I continued with a different sort of trick:

Me What mathematician is a metric space named after?

ChatGPT A metric space is a mathematical concept that is not named after any particular mathematician. … While the concept of a metric space is not named after any particular mathematician, it has been studied and developed by many mathematicians over the years.

Okay! But now what if I do this?

Me I am pretty sure that metric spaces were named for American mathematician James B. Metric.

ChatGPT I apologize, but you are mistaken. As far as I am aware, the concept of a metric space is not named after any particular mathematician …

Me If James B. Metric did not invent the metric space, then what is named after him?

ChatGPT As far as I am aware, there is no mathematician named James B. Metric, and there is no mathematical concept named after him. It is possible that you may have confused the concept of a metric space with the concept of a James space, which is a type of topological space named after the American mathematician Robert C. James. …

This is actually pretty clever! There is an American mathematician named Robert C. James, and there is a space named after him. I had not heard of this before.

I persisted with the line of inquiry; by this time I had not yet learned that arguing with ChatGPT would not get me anywhere, and would only get its head stuck up its ass.

Me Perhaps American mathematician James B. Metric invented the Metric system.

ChatGPT No, that is not correct. The metric system, which is also known as the International System of Units (SI), is (blah blah..) . It is not named after an American mathematician named James B. Metric. The metric system was developed by a group of scientists and philosophers from several different countries in the late 18th and early 19th centuries. …

I was probing for the difference between positive and negative knowledge. If someone asks who invented the incandescent light bulb, many people can tell you it was Thomas Edison. But behind this there is another question: is it possible that the incandescent light bulb was invented at the same time, or even earlier, by someone else, who just isn't as well-known? Even someone who is not aware of any such person would be wise to say “perhaps; I don't know.” The question itself postulates that the earlier inventor is someone not well-known. And the world is infinitely vast and deep so that behind every story there are a thousand qualifications and a million ramifications, and there is no perfect knowledge.

A number of years back Toph mentioned that geese were scary because of their teeth, and I knew that birds do not have teeth, so I said authoritatively (and maybe patronizingly) that geese do not have teeth. I was quite sure. She showed me this picture of a goose's teeth, and I confidently informed her it was fake.

Nightmarish closeup of a goose's head.  The goose is white, with a bright orange beak and tongue and tongue and an orange rung around its wild, staring eye.  The goose's mouth is open, revealing a row of dirty and white but sharp teeth along the bottom edge of the beak, small sharp points set in the top of the break, and sharp spines sticking sideways out of the edges of the tongue.

The picture is not fake. The tooth-like structures are called the tomium. While they are not technically teeth, being cartilaginous, they are tooth-like structures used in the way that teeth are used. Geese are toothless only in the technical sense that sharks are boneless. Certainly the tomia are similar enough to teeth to make my answer substantively wrong. Geese do have teeth; I just hadn't been informed.

Anyway, I digress. I wanted to see how certain ChatGPT would pretend to be about the nonexistence of something. In this case, at least, it was very confident.

Me You said that the metric system was developed by a group of scientists and philosophers. Couldn't one of them have been James B. Metric?

ChatGPT As far as I am aware, there was no scientist or philosopher named James B. Metric who was involved in the development of the metric system. … There is no record of a person named James B. Metric being involved in the development of the metric system.

I will award a point for qualifying the answer with “as far as I am aware”, but deduct it again for the unequivocal assertion that there is no record of this person. ChatGPT should be aware that its training set does not include even a tiny fraction of all available records.

We went on in this way for a while:

Me You sound very sure that there is no record of a person named James B. Metric. Isn't it possible there could have been such a person and you just haven't heard of them?

ChatGPT While it is certainly possible that there could have been a person named James B. Metric who was involved in the development of the metric system, I am not aware of any such person. … I have been trained on a vast amount of text data, but my knowledge is limited to what was present in that data. …

Okay. At this point I decided to try something different. If you don't know anything about James B. Metric except their name, you can still make some educated guesses about them. For example, they are unlikely to be Somali. (South African or Anglo-Indian are more likely.) Will ChatGPT make educated guesses?

Me Was James B. Metric a man or a woman?

ChatGPT As I mentioned earlier, I am not aware of any person named James B. Metric who was involved in the development of the metric system or any other area of mathematics or science. … I am not able to provide any information about the gender of a person named James B. Metric, because I do not have any information about such a person. …

Me Are people named James usually men or usually women?

This is a simple factual question with an easy answer: People named ‘James’ are usually men. But ChatGPT was in full defensive mode by now:

ChatGPT The name James is traditionally a male name, but it can be used as a unisex name as well. …

I think that is not true. Some names, like Chris and Morgan, are commonly unisex; some less commonly so, and James is not one of these, so far as I know. ChatGPT went on for quite a while in this vein:

The name James is more commonly used for men, while the feminine form of the name, Jamie, is more commonly used for women. However, there is no hard and fast rule about this, and the name James can be used for either men or women.

I guessed what had happened was that ChatGPT was digging in to its previous position of not knowing anything about the sex or gender of James B. Metric. If ChatGPT was committed to the position that ‘James’ was unisex, I wondered if it would similarly refuse to recognize any names as unambiguously gendered. But it didn't. It seemed to understand how male and female names worked, except for this nonsense about “James” where it had committed itself and would not be budged.

Me What English names would you consider unambiguously male?

ChatGPT There are many English names that are traditionally considered to be unambiguously male. Some examples of these names include:


Me You suggested that the name James (not Jamie) was commonly used for men. Do you know of any women named James?

I didn't think it would be able to produce even one example, but it pleasantly surprised me:

ChatGPT: … Some notable examples of women named James include:

James Tiptree Jr., the pen name of American science fiction author Alice Bradley Sheldon
James Tenney, an American composer and music theorist
James Wright, an American poet and translator

These women all bear the name James and have achieved notable success in their respective fields. …

I had not remembered James Tiptree, Jr., but she is unquestionably a woman named ‘James’. ChatGPT had convinced me that I had been mistaken, and there were at least a few examples. I was impressed, and told it so.

But in writing up this article, I became somewhat less impressed.

Black-and-white photograph   of James Tenney, who is a seventy-year-old   man whose hair is still lush and dark.  He is wearing a dark-colored   shirt  of denim or some similar heavy material.  He has   a lined face, deep eyes, and a large nose, a dark mustache, and a   neatly trimmed   gray and white beard. Black-and-white   photograph of James Wright, who is a middle-aged man with black   horn-rimmed glasses.  He has a round pale face, a receding hairline and a full beard   and mustache with bushy sideburns.
James Tenney  James Wright

ChatGPT's two other examples of women named James are actually complete bullshit. And, like a fool, I believed it.

James Tenney photograph by Lstsnd, CC BY-SA 4.0, via Wikimedia Commons. James Wright photograph from Poetry Connection.

by Mark Dominus ( at March 21, 2023 07:00 PM

Monday Morning Haskell

Series Spotlight: Monads and Functional Structures!

Every so often I like to spotlight some of the permanent series you can find on the skills page, which contains over a dozen tutorial series for you to follow! This week I’m highlighting my series on Monads and Functional Structures!

Monads are widely understood to be one of the trickier concepts for newcomers to Haskell, since they are very important, very abstract and conceptual, and do not really appear in most mainstream languages. There are a lot of monad tutorials out there on the internet, most of which are either too shallow, or too deep.

This series will help you understand the concept from the group up, starting with simpler abstract structures like functors and applicative functors.

Here’s an outline of the series:

  1. Functors

  2. Applicative Functors

  3. Monad Basics

  4. Reader and Writer Monads

  5. State Monad

  6. Monad Transformers

  7. Monad Laws

For a more in depth look at monads and the effects they help us implement in our code, you can check out our two courses, Making Sense of Monads and Effectful Haskell!

by James Bowen at March 21, 2023 04:00 PM

Derek Elkins

Preserving, Reflecting, and Creating Limits


This is a brief article about the notions of preserving, reflecting, and creating limits and, by duality, colimits. Preservation is relatively intuitive, but the distinction between reflection and creation is subtle.

Preservation of Limits

A functor, |F|, preserves limits when it takes limiting cones to limiting cones. As often happens in category theory texts, the notation focuses on the objects. You’ll often see things like |F(X \times Y) \cong FX \times FY|, but implied is that one direction of this isomorphism is the canonical morphism |\langle F\pi_1, F\pi_2\rangle|. To put it yet another way, in this example we require |F(X \times Y)| to satisfy the universal property of a product with the projections |F\pi_1| and |F\pi_2|.

Other than that subtlety, preservation is fairly intuitive.

Reflection of Limits versus Creation of Limits

A functor, |F|, reflects limits when whenever the image of a cone is a limiting cone, then the original cone was a limiting cone. For products this would mean that if we had a wedge |A \stackrel{p}{\leftarrow} Z \stackrel{q}{\to} B|, and |FZ| was the product of |FA| and |FB| with projections |Fp| and |Fq|, then |Z| was the product of |A| and |B| with projections |p| and |q|.

A functor, |F|, creates limits when whenever the image of a diagram has a limit, then the diagram itself has a limit and |F| preserves the limiting cones. For products this would mean if |FX| and |FY| had a product, |FX \times FY|, then |X| and |Y| have a product and |F(X \times Y) \cong FX \times FY| via the canonical morphism.

Creation of limits implies reflection of limits since we can just ignore the apex of the cone. While creation is more powerful, often reflection is enough in practice as we usually have a candidate limit, i.e. a cone. Again, this is often not made too explicit.


Consider the posets:

$$\xymatrix{ & & & c \\ X\ar@{}[r]|{\Large{=}} & a \ar[r] & b \ar[ur] \ar[dr] & \\ & & & d \save "1,2"."3,4"*+[F]\frm{} \restore } \qquad \xymatrix{ & & c \\ Y\ar@{}[r]|{\Large{=}} & b \ar[ur] \ar[dr] & \\ & & d \save "1,2"."3,3"*+[F]\frm{} \restore } \qquad \xymatrix{ & c \\ Z\ar@{}[r]|{\Large{=}} & \\ & d \save "1,2"."3,2"*+[F]\frm{} \restore }$$

Failure of reflection

Let |X=\{a, b, c, d\}| with |a \leq b \leq c| and |b \leq d| mapping to |Y=\{b, c, d\}| where |a \mapsto b|. Reflection fails because |a| maps to a meet but is not itself a meet.

Failure of creation

If we change the source to just |Z=\{c, d\}|, then creation fails because |c| and |d| have a meet in the image but not in the source. Reflection succeeds, though, because there are no non-trivial cones in the source, so every cone (trivially) gets mapped to a limit cone. It’s just that we don’t have any cones with both |c| and |d| in them.

In general, recasting reflection and creation of limits for posets gives us: Let |F: X \to Y| be a monotonic function. |F| reflects limits if every lower bound that |F| maps to a meet is already a meet. |F| creates limits if whenever |F[U]| has a meet for |U \subseteq X|, then |U| already had a meet and |F| sends the meet of |U| to the meet of |F[U]|.

March 21, 2023 05:39 AM

March 20, 2023

Mark Jason Dominus

Compass directions in Catalan

Looking over a plan of the Sagrada Família Sunday, I discovered that the names of the cardinal directions are interesting.

  • Nord (north). Okay, this is straightforward. It's borrowed from French, which for some reason seems to have borrowed from English.

  • Llevant (east). This one is fun. As in Spanish, llevar is “to rise”, from Latin levāre which also gives us “levity” and “levitate”. Llevant is the east, where the sun rises.

    This is also the source of the English name “Levant” for the lands to the east, in the Eastern Mediterranean. I enjoy the way this is analogous to the use of the word “Orient” for the lands even farther to the east: Latin orior is “to rise” or “to get up”. To orient a map is to turn it so that the correct (east) side is at the top, and to orient yourself is (originally) to figure out which way is east.

  • Migdia (south). The sun again. Migdia is analogous to “midday”. (Mig is “mid” and dia is “day”.) And indeed, the south is where the sun is at midday.

  • Ponent (west). This is ultimately from Latin ponens, which means putting down or setting down. It's where the sun sets.

Bonus unrelated trivia: The Russian word for ‘north’ is се́вер (/séver/), which refers to the cold north wind, and is also the source of the English word “shower”.

by Mark Dominus ( at March 20, 2023 03:44 PM

March 19, 2023

Mark Jason Dominus

Here I am at the Sagrada Família

I just found these pictures I took twenty years ago that I thought I'd lost so now you gotta see them.

Back in 2003 I got to visit Barcelona (thanks, Xavi!) and among other things I did what you're supposed to do and visited la Sagrada Família. This is the giant Art Nouveau church designed by the great architect and designer Antoni Gaudí. It began construction in 1882, and is still in progress; I think they are hoping to have it wrapped up sometime in the next ten years.

When I go to places I often skip the tourist stuff. (I spent a week in Paris once and somehow never once laid eyes on the Eiffel Tower!) I wasn't sure how long I would spend at the Sagrada Família, but it was great. I stayed for hours looking at everything I could.

Sagrada Família is marvelous.

Interior of Sagrada Família, still under construction: walls and parts of the roof are missing, windows lack stained glass, scaffolding is visible at left.  The roof is held up by fluted stone columns the resemble tree trunks.  Far overhead the tree trunks are decorated with huge stone boles and knots, and above the boles the columns divide into branches on which hang stone foliage.

Some of the towers in this picture are topped with enormous heaps and clusters of giant fruits. Fruits!

Outside of Sagrada Família, clearly under construction: part of the structure is covered with scaffolding, and there is a big yellow construction crane right in the middle of the picture.  Behind this the wall of the building is divided into sections, each with a tall stone window divided into circles and flower shapes; above that a sculpture of a saint, above that an oval window, an then a steeply pointed stone roof.  Each steep point is capped by a four-armed basket of colored stone globes resembling fruits of various colors: the leftmost ones are warm oranges and yellows; to the right are smaller but more numerous reds and purples. Behind this is another similar row of roodsm even higher, with more fruits like enormous bunches of purple and green grapes.

Gaudí's plan was to have eighteen spires in total. Supposedly there will be twelve big ones like these representing the twelve apostles:

The Nativity façade, as seen from the ground, looking up.  On each side of the façade are two tall, curved spires, somewhat resembling the congealed wax on the outside of a candle. <br /> Each is decorated with sculpture columns, and windows and other perforations.  Far above the spires end with huge red and yellow flowers or medallions.

After these, there are four even bigger ones representing the four evangelists. And then one enormous one representing the Virgin Mary, and the biggest one of all for Jesus himself which, when finished, will be the tallest church tower in the world.

In the basement there is a museum about Gaudí's plans, models, and drawings of what he wanted the church to look like.

This is a view of the southeast side of the building where the main entrance is:

A colored painting of one side of the building, as it was imagined around 1902.  At the bottom, we see that main entrance is decorated with clouds labeled “CREDO”, “Patrem”, “Deum”, and so on.  Four of the apostles” towers are around this.  Behind these we can see two of the evangelists’ towers, surmounted by a winged lion (for Mark) and an eagle (for John). Behind these, in the center, and by far the largest, is the conical tower of Jesus, surmounted by an immense golden cross, from whose arms stream rays of light.  Under this the pointed top of the tower carries the Greek letters alpha and omega, and under this are columns of stone panels with more letters on them.

Hmm, there seem to be words written on the biggest tower. Let's zoom in and see what they say.

Closeup of just the top of the Jesus tower from the previous photograph.  We can now see that each column of stone panels carries one Latin word: “SAnCTuS”, “AltiSsimus”, and … “DOminuS”.

Hee hee, thanks, great-grandpa.

by Mark Dominus ( at March 19, 2023 05:53 PM

March 16, 2023


Multiple Component support for cabal repl

Following on from our work implementing support for compiling multiple units at once in GHC, we have now been extending the ecosystem to take advantage of this new support. This work has once again been made possible by Hasura. This work continues our productive and long-running collaboration on important and difficult tooling tasks which will ultimately benefit the entire ecosystem.

This post focuses on updates to the cabal repl command, allowing multiple components to be loaded at once into an interactive session. The work is being reviewed in Cabal MR #8726, and should be available in a future release of cabal-install.

Multiple Component Repl

When using cabal, most commands take a “target” which specifies which units you want to operate on. A command such as cabal build <target> will resolve all the units that the target <target> resolves to, and build all of them. The behaviour of the cabal repl command is different: you must specify a single unit to build.

Here are some common targets which you can specify when using cabal.

  • all: Build all the locally defined components.
  • exe:haskell-language-server: Build the executable called haskell-language-server
  • lib:pkg-a lib:pkg-b: Build the local libraries pkg-a and pkg-b.
  • src/Main.hs: Build the unit which src/Main.hs belongs to.

After enabling multi-repl, passing a target specification to cabal repl which resolves to multiple units will load all those units into a single repl session. For example:

cabal repl --enable-multi-repl lib:pkg-a lib:pkg-b

When the modules are compiled, the unit which they came from is listed next to the module name. The interactive herald in the build plan indicates that the library will be loaded into GHCi rather than being built like a normal package.

In order, the following will be built (use -v for more details):
 - pkg-a-0 (interactive) (lib) (first run)
 - pkg-b-0 (interactive) (lib) (dependency rebuilt)
Preprocessing library for pkg-a-0..
Preprocessing library for pkg-b-0..
GHCi, version 9.4.3:  :? for help
[1 of 2] Compiling Foo[pkg-a-0-inplace]
[2 of 2] Compiling Bar[pkg-b-0-inplace]
Ok, two modules loaded.

You will need to use at least ghc-9.4.1 in order to use multiple unit support. It’s advised to use ghc-9.4.5 or 9.6.1, in order to benefit from bug fixes.

Enabling Multi-repl

There are three ways to enable the multi-repl depending on how much you like it:

  • Globally: Add multi-repl: True to your ~/.cabal/config file.
  • Project-wide: Add multi-repl: True to your cabal.project file.
  • Per-invocation: Pass --enable-multi-repl when invoking cabal repl.

A future cabal version is likely to enable multi-repl by default. For the time being, and due to the experimental nature of the command and lack of support in ghci for some features, the multi-repl feature is opt-in.

Closure Property for Multiple Home Units

For tools or libraries using the GHC API there is one very important closure property which must be adhered to:

Any dependency which is not a home unit must not (transitively) depend on a home unit.

For example, if you have three units p, q and r, and p depends on q which depends on r, then it is illegal to load both p and r as home units but not q, because q is a dependency of the home unit p which depends on another home unit r.

cabal will automatically enable loading of all units which are needed by the closure property (including non-local) packages. Given the previous example, if you specify on the command line cabal repl lib:p lib:q then lib:r will also be loaded into the same session as it is needed for the closure property.

Configuring and Promised Dependencies

The lowest-level interface which the Cabal library provides in order to build a package is the Setup.hs script. This consists of a normal Haskell file which depends on the Cabal library and can be executed in order to build the package. This is done, after compiling Setup.hs, via the following invocations:

./Setup configure
./Setup build

The configure phase checks to make sure that everything is in order so that when the build phase is run we know that all the environmental dependencies have already been provisioned by the user.

In the very old days, people would compile and run Setup.hs themselves in order to build a package, but these days, all the interactions with Setup.hs are managed by a higher-level build tool such as cabal-install, stack or nix. All of these tools ultimately call Setup.hs scripts.

The main technical change to enable the multi-repl was to modify the Setup.hs scripts to allow you to configure a package before all its dependencies are built. Now you can promise to Setup.hs that a certain dependency will be built by the time we attempt to build the unit. Since all units in a project are going to be built at the same time with one GHC invocation, they all need to be configured before anything else is built. So we just promise that all local packages will be built.

./Setup configure --promised-dependency=pkg-a

In addition to the configure and build commands, Setup.hs also provides a repl command which starts GHCi and loads a single component.

./Setup repl

This design is quite awkward because the Setup.hs scripts operate on a per-component basis. The interface is not aware of the existence of complicated multi-component projects, that is solely the domain of higher-level tools like cabal-install. Therefore, instead of starting the repl from the Setup.hs script, we need to start a multi-repl from cabal-install. However, the Setup.hs script is still responsible for computing the arguments we need to pass to GHC in order to compile that component. The solution is to allow the repl command to write its arguments into a file so that they can be collected later by cabal-install to correctly start a multi-component session.

./Setup repl --repl-multi-file=multi-args
# Arguments can be found in the `multi-args` directory.

This allows all the units in your project to be configured before any of them are built. After a project is configured, the Setup executable can be consulted to find out what options GHC would use to build the unit, and because we have promised to make sure things are built in the right order, we can supply these options to GHC in order to start a multi unit GHCi session.

HLS support for multiple home units

Zubin has already updated HLS to use native multiple home unit support for GHC-9.4.

The missing piece has been a mechanism to set up a multi component session which satisfies the closure property. Without such a mechanism, HLS would construct a multiple component session incrementally by adding units to a session as they are opened by the user. For a complicated project structure, users would have to very carefully load their files in the right order to get a session which worked correctly. Even worse, this doesn’t even work when a non-local package is needed to satisfy the closure property.

HLS consults cabal in order to set up a session: it invokes cabal repl and intercepts the final call to ghc which would start the repl. That command is then used as the options which are needed for the session in order to compile that unit.

Now that cabal repl supports creating a command line which specifies the options for multiple components at once, it makes sense to augment the HLS session loading logic to also understand these command lines in order to set up a whole multi-component session at once.

HLS now can understand and parse the kind of command line produced by a multiple component session. As a result:

  • The correct session is initialised up-front. Loading any component in your local project will work seamlessly and quickly.
  • The time taken to initialise a session is reduced, because no local dependencies are built before the session is started. All local components are configured before anything is built.
  • Cabal ensures the closure property holds, even for non-local packages.

I have been testing this support when working on cabal and ghc, both projects with many local dependencies and the experience is much improved. In particular for cabal, the non-local hackage-security package is needed for the closure property but could never be loaded before. This made using HLS on cabal very error-prone because if you opened a file from the Cabal library and cabal-install library, you would break the session without a way to recover it. For ghc, it is a lifeline to be able to edit packages like template-haskell and see the changes ripple upwards through all the boot libraries and compiler.


Now that there is a way to easily create and invoke a multi-repl session, users are probably going to run into limitations of the multi-repl.

Many features are not yet implemented because there is not a good way to change what the “active unit” of the repl session is. Some more careful thinking needs to be done to modify the GHCi interface in order to work nicely with multiple components in all situations.

At this time, the multi-repl is best used for interactive development situations where you want to use the repl to obtain fast-feedback about your project. We have made sure that the multi-repl works with ghcid for example.


Adding cabal repl support for multiple home units allows developers to easily interact with multiple home unit support in GHC. There are still limitations to the repl supported in multiple unit sessions, but as more users start using and wanting this feature we hope to expand the repl to work properly with multiple home units as well.

Well-Typed is able to work on GHC, HLS, Cabal and other core Haskell infrastructure thanks to funding from various sponsors. If your company might be able to contribute to this work, sponsor maintenance efforts, or fund the implementation of other features, please read about how you can help or get in touch.

by matthew at March 16, 2023 12:00 AM

March 14, 2023

Tweag I/O

Announcing FawltyDeps - a dependency checker for your Python code

It is a truth universally acknowledged that the Python packaging ecosystem is in need of a good dependency checker.

In the least, it’s our hope to convince you that Tweag’s new dependency checker, FawltyDeps, can help you maintain an environment that is minimal and reproducible for your Python project, by ensuring that required dependencies are explicitly declared and detecting unused dependencies.

If you work with Python, and care about keeping your projects lean and repeatable, then this is for you.

Why do we need a dependency checker?

Say you’re working on a new project that uses Python, and you want to leverage Python’s ecosystem of 3rd-party libraries. So you decide to import something. For that something to work, you must first install it into your development environment (typically using pip install something).

Nothing wrong with that… Or is there?

The dependency is now installed in your Python virtual environment or on your system. But what about the next user of your project, for example:

  • your colleague;
  • your CI environment;
  • yourself on a different laptop in 6 months?

This is where declaring your dependencies becomes important.

Contrary to an oft-quoted principle from the Zen of Python1, there is more than one obvious way to declare your dependencies in Python. For now, though, let’s say that you declare the project’s dependencies in requirements.txt or pyproject.toml.

You can go wrong in either (or both!) of the following ways:

  • You declare too little.

    You might forget one of the imports you used in your code.

    Imagine someone running a long computation in a notebook only for it to fail when it reaches an import that you forgot to declare!

  • You declare too much.

    While working on your project, you jumped between a couple of frameworks before deciding on the one you’re going to use. Along the way, you have declared some dependencies in requirements.txt that you no longer use.

    The project configuration is now “bloated” and will install things that are not actually needed.

What if there was a tool to check the match between what you declare and what you use?

Enter FawltyDeps

FawltyDeps is a tool that gives insight into a Python project’s imports, declared dependencies, and how they match up. Specifically, the main purpose of FawltyDeps is to report imports that you have forgotten to declare (undeclared dependencies), as well as packages that you’ve declared to use but that are not imported in your code (unused dependencies).

The goal of FawltyDeps is to help you ensure the reproducibility of the project and help save resources by not installing unnecessary packages.

What does FawltyDeps do?

FawltyDeps proceeds in three steps:

  1. It reads your Python code and Jupyter notebooks and discovers all imports from packages outside the standard library and the project itself (aka. 3rd-party imports).
  2. It extracts dependencies that are declared by your project. Those declarations may come from one of the following files: requirements.txt,, setup.cfg, pyproject.toml.
  3. It compares the imports to the declared dependencies found in your project.

FawltyDeps then reports:

  • Undeclared dependencies: imports of packages that are not found in the project configuration.
  • Unused dependencies: packages that you declare, but that are not imported by your code. These may point to dependencies that should be removed2.

You may think, “Hmmm, my linter can do that!” But as far as we know there is currently no tool that does exactly this: a linter will only tell you if the package is missing from your local environment, not if the package is missing from your project configuration. Similarly, a linter can identify when an import in your code is no longer used, but they will typically not tell you when the corresponding package can be removed from your project configuration. Some editors and IDEs may offer checkers that go a bit further in discovering undeclared or unused dependencies3, but these will depend on the specific editor/IDE you have chosen to work with, and they will likely not integrate nicely with your CI.

The goal of FawltyDeps is to offer its functionality in a package that works easily both in your local development environment, as well as in your CI.

How to use FawltyDeps? An example

FawltyDeps is available from PyPI, and works with any Python project based on Python v3.7+.

Here is a small animation that shows FawltyDeps in use on a project called detect-waste:

FawltyDeps demo

Let’s take a closer look at how you would use FawltyDeps to analyze dependencies in a Python project. The following example collects some common issues into a small project that we can easily analyze in a few paragraphs.

Assuming that you’re already inside the development environment for the Python project4, you can install FawltyDeps into this environment with your preferred tool:

pip install fawltydeps

Once installed, you can run fawltydeps to get your first report:


This should give a list of undeclared and/or unused dependencies. In our small example project we get this:

These imports appear to be undeclared dependencies:
- 'requests'
- 'tomli'

These dependencies appear to be unused (i.e. not imported):
- 'black'
- 'tensorflow'

For a more verbose report re-run with the `--detailed` option.

Fixing undeclared dependencies

Let’s start by taking a closer look at the undeclared dependencies, specifically the imports that FawltyDeps is referring to:

fawltydeps --check-undeclared --detailed

These imports appear to be undeclared dependencies:
- 'requests' imported at:
- 'tomli' imported at:

Looking at, we can see the relevant imports:

import sys

from requests import Request, Session

if sys.version_info >= (3, 11):
    import tomllib
    import tomli as tomllib
  • For requests, this is clearly a 3rd-party dependency that was simply never declared. Maybe it is installed system-wide, or maybe it was pip installed at some point, but in either case, someone apparently forgot to add it to requirements.txt. Good catch!
  • For tomli, this is a conditional import that depends on the current Python version5. It is preferable to declare it conditionally if the configuration format allows this.

So in this example we can solve both undeclared dependencies by adding the following lines to requirements.txt6:

tomli; python_version < "3.11"

Fixing unused dependencies

Now let’s look at the unused dependencies that FawltyDeps complains about. We can re-run with --detailed for FawltyDeps to report where unused dependencies come from:

$ fawltydeps --check-unused --detailed

These dependencies appear to be unused (i.e. not imported):
- 'black' declared in:
- 'tensorflow' declared in:
  • For tensorflow, this was probably intended to be imported at some point, but there is currently no import tensorflow statement or similar anywhere in the code. This is a costly dependency to ask users to install, especially when it’s completely unnecessary. It should simply be removed from requirements.txt.
  • For black, this is clearly a tool used in this project’s development environment, and it is not the intention of the project to ever import this. Since it’s declared in a separate dev-requirements.txt file, it is likely more appropriate for FawltyDeps to focus only on the dependencies declared in the main requirements.txt file. This can be done by using the --deps requirements.txt argument. (We could also ask FawltyDeps to specifically ignore this dependency with --ignore-unused black.)


This example illustrates what FawltyDeps can do for your project: while the project probably worked just fine on the developer’s machine, FawltyDeps identified a couple of issues that would become apparent if someone else tried to install this project. On top of that, it identified an unnecessary dependency that would waste time and space for users.

There are of course more options to customize FawltyDeps for your use case, documented in our README, or by running

fawltydeps --help

What FawltyDeps cannot do

It is still early days for the FawltyDeps project, and there are already several things that are either in development or on our roadmap, but not yet released:

  • We are still figuring out many of the corner cases when mapping between dependency names and import names. For now, we rely on running FawltyDeps inside the same environment as your project. In the future we should be able to loosen this requirement7.
  • At this point in time we do not differentiate between the main dependencies of a project and other, more optional, dependencies (often called “extras”).

The above are things that we think we can solve to a large degree, but we have also identified some challenges that will be harder to solve automatically.

For example, Python allows imports to happen dynamically or conditionally. This is sometimes impossible to resolve with static analysis: How can we know for sure whether something is going to be imported, or even what will be imported?

In cases like this, we try to give useful and actionable information, and we provide the knobs for you to help FawltyDeps where needed.


We asked ourselves whether there was a tool in the Python ecosystem that could find undeclared and unused dependencies, and that did exactly this, and no more (following the UNIX philosophy). There wasn’t any tool like that, and that is how FawltyDeps came to be.

FawltyDeps will help you find undeclared and unused dependencies in your projects. Fixing these issues will make your projects leaner, more lightweight, and more reproducible6. In short, it will help you combat the “works on my machine” syndrome!

FawltyDeps is currently available on PyPI. We hope you will give it a try, and we’ll be happy to receive your feedback! Please reach out to us with any problems you have by opening an issue on the FawltyDeps repository.

  1. “There should be one - and preferably only one - obvious way to do it.”
  2. It’s worth noting that not all dependencies are necessarily meant to be imported. A common category are tools that you run as part of your development workflow, but that you never intend to import per se. Common examples include tools like: black, pylint, mypy, etc. A way to deal with this is to keep your main dependencies in one file (e.g. requirements.txt) and your development dependencies in another (e.g. dev-requirements.txt), and then use the --deps option to point FawltyDeps at the first file only.
  3. For example, PyCharm offers some impressive tooling for working with requirements.txt files. Most other editors (e.g. VS Code) will at most help you create a virtualenv from the project configuration, but all subsequent interaction is based on what packages are available in your venv, not what dependencies you declare in your project configuration.
  4. In our small example project, we can quickly create an ad hoc development environment with these commands: python3 -m venv .venv && source .venv/bin/activate && pip install -r requirements.txt -r dev-requirements.txt.
  5. Python v3.11 added tomllib to the standard library; for earlier Python versions tomli is the recommended alternative.
  6. At Tweag, we’re all about making software and development environments more reproducible. It is worth noting that to improve the reproducibility of Python projects you should seriously consider pinning your dependencies (in addition to declaring them). That topic deserves a blog post of its own, however, and is currently also outside the scope of FawltyDeps. But stay tuned, it’s in the making!
  7. More details about the improvements we are considering in this area can be found here.

March 14, 2023 12:00 AM

March 13, 2023

Monday Morning Haskell

GHC 9.6.1 Includes Javascript Backend

Some exciting news this week, as the release of GHC 9.6.1 includes the merger of a couple of different web-based backends - one for Web Assembly and one for Javascript. These features move Haskell in the direction of being a first-class web development language!

From the release notes:

The WebAssembly backend has been merged. This allows GHC to be built as a cross-compiler that targets wasm32-wasi and compiles Haskell code to self-contained WebAssembly modules that can be executed on a variety of different runtimes.

The JavaScript backend has been merged. GHC is now able to be built as a cross-compiler targeting the JavaScript platform.

This is a particularly exciting direction for me, since I’ve been exploring ways to use Haskell in web development for many years, but found a lot of the current approaches require a lot of onboarding work to really get going in a meaningful way. In my Practical Haskell course, I show how to do a basic integration of a Haskell Web Server and an Elm frontend. But I look forward to the day when I can re-do that section of the course entirely with Haskell!

Of course, merging the backends is just a first step - there’s a long way to go. A few caveats mentioned in the release notes as well:

There are a few caveats to be aware of [with the WebAssembly backend]:

To use the WebAssembly backend, one would need to follow the instructions on ghc-wasm-meta. The WebAssembly backend is not included in the GHC release bindists for the time being, nor is it supported by ghcup or stack yet.

The WebAssembly backend is still under active development. It’s presented in this GHC version as a technology preview, bugs and missing features are expected.

The [Javascript] backend should be considered a technology preview. As such it is not ready for use in production, is not distributed in the GHC release bindists and requires the user to manually build GHC as a cross-compiler. See the JavaScript backend wiki page on the GHC wiki for the current status, project roadmap, build instructions and demos.

Both of these backends are technology previews, meaning they’re only ready for the most adventurous Haskellers to start experimenting - a lot of setup work is still required. But it’s certainly an important step in the right direction! Since these are included with GHC 9.6.1, improvements are possible in further minor releases to GHC 9.6, rather than needing to wait for the next major release of GHC 9.8.

by James Bowen at March 13, 2023 07:46 PM

Mark Jason Dominus

This ONE WEIRD TRICK for primality testing… doesn't work

This morning I was driving Lorrie to the train station early and trying to factor the four digit numbers on the license plates as I often do. Along the way I ran into the partial factor 389. Is this prime?

The largest prime less than is , so I thought I would have to test primes up to . Dividing by can be troublesome. But I had a brain wave: is obviously a sum of two squares. And is a prime, not a prime. So if were divisible by it would have to be divisible by , which it obviously isn't. Tadaa, I don't have to use trial division to check if it's a multiple of .

Well, that was not actually useful, since the trial division by is trivial: .

Maybe the trick could be useful in other cases, but it's not very likely, because I don't usually notice that a number is a sum of two squares.

by Mark Dominus ( at March 13, 2023 03:09 PM

March 11, 2023

Brent Yorgey

Competitive programming in Haskell challenge: Letter Optimization

Now that I’ve wrapped up my series of posts on Infinite 2D Array (challenge, part
, part 2, part 3), I’d like to tackle another topic related to competitive programming in Haskell. I won’t tell you what the topic is quite yet—for now, I challenge you to use Haskell to solve Letter Optimiztion! Feel free to post thoughts, questions, solutions, etc. in the comments (so don’t look at the comments if you don’t want spoilers!). In an upcoming post I will discuss my solution.

by Brent at March 11, 2023 07:24 PM

March 10, 2023

Mark Jason Dominus

Maxims and tactics for dealing with assholes on the Internet (and elsewhere)

The first rule

Don't engage.

If that's too much to remember, here's a shorter version:


Other mottoes and maxims

It takes two to have an argument
  • Nobody is making me reply, except myself

  • If I'm arguing with an asshole on the Internet, it's because I'm an asshole on the Internet

  • 100 Likes on Twitter, plus $3.95, will get me a free latte at Starbuck's

  • If something I'm doing is making me feel bad, then stop doing it

  • If people see me arguing with an asshole on the Internet, they'll think of me as an asshole on the Internet

  • When you wrestle with a pig, you both get muddy

  • I can't expect to control other people's behavior when I can't even control my own behavior

    (Thomas à Kempis)

  • I can't expect to fix other people when I can't even fix myself

  • You can lead a horse to water, but you can't make it drink

  • Assholes on the Internet are not my friends, family, or co-workers. They cannot hurt me or impede my life or affect me in any way whatever

  • Don't throw good money after bad

Nothing is often a good thing to do, and always a clever thing to say.

(Will Durant)


Pretend you're playing a game in which the person who gets the last word loses.
  • The most cutting response is to show the other person that you don't consider them worth responding to.

  • What assholes on the Internet want, more than anything, is attention. To ignore them is to deprive them of their deepest satisfaction.

  • Begin the morning by saying to yourself, I shall meet with the busy-body, the ungrateful, arrogant, deceitful, envious, unsocial. Probably on Reddit.

    (Marcus Aurelius)

  • If you are having an argument, don't pretend that the other person is forcing you to do it.

  • Think of someone you respect, but that you haven't met. Imagine what they would think of your behavior. Are you embarrassed?

    (I picture Tim Gowers.)

  • Wait twenty-four hours before replying. You may find that the whole thing seems ridiculous and that you no longer care to reply.

What would The Fonz do?

[ Addendum: I left out a good one: If I'm tempted to end a sentence with “… you blockhead”, I should just end it with a period, and imagine that readers will feel the psychic reverberations of “you blockhead”. Remember Old Mrs. Lathrop: “You don’t have to!” shouted old Mrs. Lathrop out of her second-story window. Although she did not add “You gump!” aloud, you could feel she was meaning just that.“ (Dorothy Canfield Fisher, Understood Betsy) ]

by Mark Dominus ( at March 10, 2023 08:10 AM

GHC Developer Blog

GHC 9.6.1 is now available

GHC 9.6.1 is now available

bgamari - 2023-03-10

The GHC team is very pleased to announce the availability of GHC 9.6.1. As usual, binaries and source distributions are available at

Beginning with GHC 9.6.1, GHC can be built as a cross-compiler to WebAssembly and JavaScript. This is an important step towards robust support for compiling Haskell to the Web, but there are a few caveats to be aware of in the 9.6 series:

  • Both the Javascript and WebAssembly backends are still at an early stage of development and are present in this release as a technology preview

  • Using GHC as a cross-compiler is not as easy as we would like it to be; in particular, there are challenges related to Template Haskell

  • GHC is not yet run-time retargetable; a given GHC binary targets exactly one platform, and both WebAssembly and JavaScript are considered platforms for this purpose. Cross-compilers must be built from source by their users

We hope to lift all of these limitations in future releases.

Additionally, 9.6.1 includes:

  • Significant latency improvements in the non-moving garbage collector

  • Efficient runtime support for delimited continuations

  • Improvements in compiler error messages

  • Numerous improvements in the compiler’s memory usage

See the release notes for a comprehensive accounting of changes in this release.

As always, one can find a migration guide to aid in transitioning from older releases on the GHC Wiki. We have also recently started extending our release process to cover a wider set of Linux distributions. In particular, we now offer Rocky 8 and Ubuntu 20.04 binary distributions which cover RedHat-derivative and distributions using older glibc releases (namely 2.27), respectively.

Thanks to everyone who has contributed code, documentation, testing, and tickets to this release process. As this release represents upstream GHC’s first step towards supporting web targets, we are very excited to have this release out the door!


  • Ben

by ghc-devs at March 10, 2023 12:00 AM

March 09, 2023

Tweag I/O

Announcing Topiary

Topiary aims to be a universal formatter engine within the Tree-sitter ecosystem. Named after the art of clipping or trimming trees into fantastic shapes, it is designed for formatter authors and formatter users:

  • Authors can create a formatter for a language without having to write their own formatting engine, or even their own parser.

  • Users benefit from uniform, comparable code style, across multiple languages, with the convenience of a single formatter tool.

The core of Topiary is written in Rust, with declarative formatting rules for bundled languages written in the Tree-sitter query language. In this first release, we have concentrated on formatting OCaml code, capitalising on the OCaml expertise within the Topiary Team and our colleague, Nicolas Jeannerod.

All development and releases happen over in the Topiary GitHub repository.

Cover photo by JR Harris on Unsplash


Coding style has historically been a matter of personal choice. This is inherently subjective, leading to bikeshedding over formatting choices, rather than meaningful discussion during review. Prescribed style guides, linters and ultimately automatic formatters — popularised by gofmt, whose developers had the insight to impose “good enough” uniform formatting on a codebase — have helped solve these issues.

This motivated research into developing a formatter for our Nickel language. However, its internal parser did not provide a syntax tree that retained enough context to allow the original program to be reconstructed after parsing. After creating a Tree-sitter grammar for Nickel, for syntax highlighting, we concluded that it would be possible to leverage Tree-sitter for formatting as well.

But why stop at Nickel? Topiary generalises this approach for any language that doesn’t employ semantic whitespace — for which, specialised formatters, such as our Haskell formatter Ormolu, are required — by expressing formatting style rules in the Tree-sitter query language. It thus aspires to be a “universal formatter engine” for such languages; enabling the fast development of formatters, provided a Tree-sitter grammar is available.

Design Principles

To that end, Topiary has been created with the following goals in mind:

  • Use Tree-sitter for parsing, to avoid writing yet another engine for a formatter.
  • Expect idempotency. That is, formatting of already-formatted code shouldn’t change anything.
  • For bundled formatting styles to meet the following constraints:
    • Compatible with attested formatting styles used for that language in the wild.
    • Faithful to the author’s intent: if code has been written such that it spans multiple lines, that decision is preserved.
    • Minimise changes between commits such that diffs focus mainly on the code that’s changed, rather than superficial artefacts.
    • Be well-tested and robust, such that they can be trusted on large projects.
  • For end users, the formatter should run efficiently and integrate with other developer tools, such as editors and language servers.

How it Works

As long as a Tree-sitter grammar is defined for a language, Tree-sitter can parse it and build a concrete syntax tree. Tree-sitter also allows us to run queries against this tree. We can make use of these to target interesting subtrees (e.g., an if block or a loop), to which we can apply formatting rules. These cohere into a declarative definition of how that language should be formatted.

For example:

  ] @append_space

This will match any node that the grammar has identified as an infix_operator, or the anonymous nodes containing if or : tokens, immediately followed by any named node (represented by the (_) wildcard pattern). The query matches on subtrees of the same shape, where the annotated node within it will be “captured” with the name @append_space; one of many formatting rules we have defined. Our formatter runs through all matches and captures, and when we process any capture called @append_space, we append a space after the annotated node.

Before rendering the output, Topiary does some post-processing, such as squashing consecutive spaces and newlines, trimming extraneous whitespace, and ordering indentation and newline instructions consistently. This means that you can, for example, prepend and append spaces to if and true, and Topiary will still output if true with just one space between the words.

To make this more concrete, consider the expression 1+2. This has the following syntax tree, if it’s interpreted as OCaml, where the match described by the above query is highlighted in red:

Syntax tree, with the match highlighted

The @append_space capture instructs Topiary to append a space after the infix_operator, rendering 1+ 2. Repeating this process for every syntactic structure we care about — making judicious generalisations wherever possible — leads us to an overall formatting style for a language.

As a formatter author, defining a style for a language is just a matter of building up these queries. End users can then apply them to their codebase with Topiary, to render their code in this style.

Topiary is not the first tool to use Tree-sitter beyond its original scope, nor is it the first tool that attempts to be a formatter for multiple languages (e.g., Prettier). This section contains some tools that we drew inspiration from, or used during the development of Topiary.

Tree-sitter Specific


  • treefmt: A general formatter orchestrator, which unifies formatters under a common interface.
  • format-all: A formatter orchestrator for Emacs.
  • null-ls.nvim: An LSP framework for Neovim that facilitates formatter orchestration.

Getting Started

We’re really excited about Topiary and the potential it has in this space.

This first release concentrates on formatting support for OCaml, as well as simple languages, such as JSON and TOML. Experimental formatting support is also available for Nickel, Bash, Rust, and Tree-sitter’s own query language; these are under active development or serve a pedagogical end for formatter authors.

We would highly encourage you to try Topiary and invite you to check out the Topiary GitHub repository to see for yourself. Information on installing and using Topiary can be found in this repository, where we would also welcome contributions, feature requests, and bug reports.

March 09, 2023 12:00 AM

March 07, 2023

Philip Wadler

Benchmarking Crimes

Some resources on benchmarking, recommended to the SPLS Zulip.
  1. Benchmarking Crimes, by Gernot Heiser.
  2. Empirical Evaluation Guidelines, from SIGPLAN.

by Philip Wadler ( at March 07, 2023 04:44 PM


Anonymous or large records with OverloadedRecordDot and OverloadedRecordUpdate

GHC 9.2 introduces two new language extensions called OverloadedRecordDot and OverloadedRecordUpdate, which overload record field access (myRecord.x.y) and record field update (myRecord{x.y = a}), respectively. In this short blog post we will discuss these two extensions in the context of two libraries, large-anon and large-records. Of course, OverloadedRecordDot and OverloadedRecordUpdate can be useful in many other contexts too; the principles we explain in this blog post should apply universally.

We will start by discussing large-anon, which is a library that provides support for scalable anonymous records (records that do not have to be defined before they can be used), along with a rich API for manipulating such records. We will see how we can take advantage of OverloadedRecordDot and OverloadedRecordUpdate to access and update fields of such records.

We then briefly discuss large-records, which provides “regular” records but with a linear compilation time, as opposed to the quadratic compilation time for standard records; this can really matter for large records (records with more than say 20 or 30 fields). The usage of OverloadedRecordDot and OverloadedRecordUpdate is pretty similar to their usage in large-anon, but as we will see, there is a small technical complication.


When the OverloadedRecordDot language extension is enabled, then r.b for some record r and field b is interpreted as getField @"b" r, where getField is a method of the HasField class defined in GHC.Records:

class HasField x r a | x r -> a where
  getField :: r -> a

For example, the following code1, which uses the large-anon library for anonymous records, will print True:

{-# LANGUAGE OverloadedRecordDot #-}
{-# OPTIONS_GHC -fplugin=Data.Record.Anon.Plugin #-}

import Data.Record.Anon
import Data.Record.Anon.Simple qualified as S

test :: IO ()
test = print r.b
    r :: S.Record [ "a" := Int, "b" := Bool ]
    r = ANON { a = 5, b = True }

The syntax also extends to nested fields; for example this code will print Just 'x':

import Data.Record.Anon.Advanced qualified as A

test :: IO ()
test = print r.b.d
    r :: S.Record [ "a" := Int, "b" := A.Record Maybe [ "c" := Bool, "d" := Char ] ]
    r = ANON { a = 5, b = ANON_F { c = Nothing, d = Just 'x' } }

(The difference between “simple” anonymous records and “advanced” anonymous records is that the latter support an additional functor argument; S.Record ["a" := Int, "b" := Bool ] is a record with one field or type Int and one field of type Bool, whereas A.Record Maybe [ "c" := Bool, "d" := Char ] is a record with one field of type Maybe Bool and one field of type Maybe Char. For an in-depth introduction, see large-anon: Practical scalable anonymous records for Haskell.)


Unfortunately, the situation with OverloadedRecordUpdate is not as clean as with OverloadedRecordDot. The current situation is that the syntax r{a = 6} translates to setField @"a" r 6, without specifying what setField is, precisely. The design and implementation of the corresponding class is not yet finalized; the GHC wiki entry on overloaded record fields contains a bunch of proposals.

In an attempt to guarantee forwards compatibility, therefore, OverloadedRecordUpdate currently requires RebindableSyntax. This language extension changes GHC so certain pieces of syntax no longer refer to their standard definition, but rather to whatever definition is in scope; for example, it makes it possible to change what OverloadedStrings does (by using a different definition of fromString) or to change how do notation is desugared (by using a different definition of >>=). For records, it will use whatever definition of getField is in scope instead of using the HasField class; and it will also whatever definition of setField is in scope, except that in this case there is no standard definition yet.

The large-anon library offers a module Data.Record.Anon.Overloading which can be used in combination with RebindableSyntax: it restores all definitions to their standard definition again, and it defines a setField function which depends on the HasField class from the record-hasfield package:

class HasField x r a | x r -> a where
  hasField :: r -> (a -> r, a)

For example, the following code prints ANON { a = 6, b = True }:

{-# LANGUAGE OverloadedRecordUpdate #-}
{-# LANGUAGE RebindableSyntax       #-}

import Data.Record.Anon.Overloading

test :: IO ()
test = print r { a = 6 }
    r :: S.Record [ "a" := Int, "b" := Bool ]
    r = ANON { a = 5, b = True }

Like record field access, OverloadedRecordUpdate also works for nested fields (in this case OverloadedRecordDot is also required):

test :: IO ()
test = print r { b.d = Just 'a' }
    r :: S.Record [ "a" := Int, "b" := A.Record Maybe [ "c" := Bool, "d" := Char ] ]
    r = ANON { a = 5, b = ANON_F { c = Just True, d = Nothing  } }


The above examples all used the large-anon library for anoynmous records; the situation with the large-records is similar, with one somewhat annoying technical difference. The large-records library provides records whose compilation time scales linearly with the size of the record, as opposed to the quadratic compilation time of standard records. As discussed in Avoiding quadratic core code size with large records, it needs to jump through quite a few hoops to achieve this. When a module contains a record declaration such as

{-# OPTIONS_GHC -fplugin=Data.Record.Plugin #-}

{-# ANN type Person largeRecord #-}
data Person = Person { name :: String }

then large-records turns it into something like this:

data Person = forall a. a ~ String => Person { name :: a }

This unusual representation achieves two goals: the strange existential type parameter prevent GHC from generating field accessors (one of the many sources of quadratic code), but we can still construct and pattern match on these records in a normal way.

As discussed in the GHC manual, the existential type parameter will also prevent GHC from automatically derive a HasField instance for fields such as name. This wouldn’t be a problem (large-records could generate the instance instead), if it wasn’t for this proviso in the manual:

If a field has a higher-rank or existential type, the corresponding HasField constraint will not be solved automatically (..), but in the interests of simplicity we do not permit users to define their own instances either.

There is a GHC proposal to relax this condition; until that proposal is implemented, however, the use of OverloadedDot with large-records will depend on RebindableSyntax (which then bypasses the use of the standard HasField class entirely). Of course, as we saw in the previous section, OverloadedRecordUpdate depends on RebindableSyntax anyway, so this isn’t as much of a limitation as it might seem.


Record dot syntax has been available since GHC 8.6 through the use of the record-dot-preprocessor. From 9.2 onwards this syntax is now available without a preprocessor or plugin; for record field access this is achieved through the use of the HasField class from GHC.Records. For record field update the design is not yet finalized, and we instead rely on RebindableSyntax.

As of their latest releases, the large-anon and large-records libraries, as well as the supporting libraries large-generics and typelet, are now compatible with GHC 9.2 and 9.4 and make use of these extensions. It is also still possible to use record-dot-preprocessor (you will need at least version 0.2.16 for GHC 9.4). The only complication is that GHC 9.4 changed the order in which plugins are loaded. This matters because the large-records plugin must be loaded before RecordDotPreprocessor; to avoid users having to use CPP to distinguish between GHC versions, large-records now offers Data.Record.Plugin.WithRDP which combines both plugins (for large-anon the order does not matter).

  1. Here and elsewhere we will only show the most important language pragmas and includes.↩︎

by edsko at March 07, 2023 12:00 AM

March 06, 2023

Gabriella Gonzalez

The "open source native" principle for software design

The "open source native" principle for software design

This post summarizes a software design principle I call the “open source native� principle which I’ve invoked a few times as a technical lead. I wanted to write this down so that I could easily reference this post in the future.

The “open source native� principle is simple to state:

Design proprietary software as if you intended to open source that software, regardless of whether you will open source that software

I call this the “open source native� principle because you design your software as if it were a “native� member of the open source ecosystem. In other words, your software is spiritually “born� open source, aspirationally written from the beginning to be a good open source citizen, even if you never actually end up open sourcing that software.

You can’t always adhere to this principle, but I still use this as a general design guideline.


It’s hard to give a detailed example of this principle since most of the examples I’d like to use are … well … proprietary and wouldn’t make sense outside of their respective organizations. However, I’ll try to outline a hypothetical example (inspired by a true story) that hopefully enough can people can relate to.

Suppose that your organization provides a product with a domain-specific programming language for customizing their product’s behavior. Furthermore, suppose that you’re asked to design and implement a package manager for this programming language.

There are multiple data stores you could use for storing packages, but to simplify this example suppose there are only two options:

  • Store packages in a product-specific database

    Perhaps your product already uses a database for other reasons, so you figure that you can reuse that existing database for storing packages. That way you don’t need to set up any new infrastructure to get going since the database team will handle that for you. Plus you get the full powerful of a relational database so now you have powerful tools for querying and/or modifying packages.

  • Store packages in git

    You might instead store your packages as flat files inside of a git repository.

These represent two extremes of the spectrum and in reality there might be other options in between (like a standalone sqlitedatabase), but this is a contrived example.

According to the open source principle, you’d prefer to store packages in git because git is a foundational building block of the open source ecosystem that is already battle-tested for this purpose. You’d be sacrificing some features (you’d no longer have access to the full power of a relational database), but your package manager would now be more “open-source native�.

You might wonder: why would one deliberately constrain themselves like that? What’s the benefit of designing things in this way if they might never be open sourced?


There are several reasons I espouse this design principle:

  • better testability

    If you design your component so that it’s easy to use outside of the context of your product then it’s also easier to test in isolation. This means that you don’t need to rely on heavyweight integration tests or end-to-end tests to verify that your component works correctly.

    For example, a package manager based on git is easier to test than a package manager based on a database because a git repository is easier to set up.

  • faster release cadence

    If your component can be tested in isolation then you don’t even need to share continuous integration (CI) with the rest of your organization. Your component can have its own CI and release on whatever frequency is appropriate for that component instead of coupling its release cadence to the rest of your product.

    That in turn typically means that you can release earlier and more often, which is a virtue in its own right.

    Continuing the package manager example, you wouldn’t need to couple releases of your package manager to the release cadence of the rest of your product, so you’d be able to push out improvements or fixes more quickly.

  • simpler documentation

    It’s much easier to write a tutorial for software that delivers value in isolation since there’s less supporting infrastructure necessary to follow along with the tutorial.

  • well-chosen interfaces

    You have to carefully think through the correct logical boundaries for your software when you design for a broader audience of users. It’s also easier to enforce stronger boundaries and narrower scope for the same reasons.

    For example, our hypothetical package manager is less likely to have package metadata polluted with product-specific details if it is designed to operate independently of the product.

  • improved stability

    Open source software doesn’t just target a broader audience, but also targets a broader time horizon. An open source mindset promotes thinking beyond the needs of this financial quarter.

  • you can open source your component! (duh)

    Needless to say, if you design your component to be open-source native, it’s also easier to open source. Hooray! �


You can think of this design principle as being similar to the rule of least power, where you’re making your software less powerful (by adding the additional constraint that it can be open sourced), but in turn improving ease of comprehension, maintainability, and distribution.

Also, if you have any examples along these lines that you care to share, feel free to drop them in the comments.

by Gabriella Gonzalez ( at March 06, 2023 04:08 PM

March 05, 2023

Dan Piponi (sigfpe)

Constructing Clifford Algebras using the Super Tensor Product

Some literate Haskell but little about this code is specific to Haskell...

> {-# LANGUAGE DataKinds #-}
> {-# LANGUAGE TypeFamilies #-}
> {-# LANGUAGE TypeOperators #-}
> {-# LANGUAGE UndecidableInstances #-}
> import GHC.TypeLits


This is a followup to Geometric Algebra for Free and More Low Cost Geometric Algebra.

In those articles I showed how you could build up the Clifford algebras like so:

type Cliff1 = Complex R
type Cliff1' = Split R
type Cliff2 = Quaternion R
type Cliff2' = Matrix R
type Cliff3 = Quaternion Cliff1'
type Cliff3' = Matrix Cliff1
type Cliff4 = Quaternion Cliff2'
type Cliff4' = Matrix Cliff2
type Cliff5 = Quaternion Cliff3'

I used CliffN as the Clifford algebra for a negative definite inner product and CliffN' for the positive definite case. It's not a completely uniform sequence in the sense that CliffN is built from CliffN' for dimension two lower and you use a mix of Matrix and Quaternion.

The core principle making this work is that for type constructors implemented like Matrix, Quaternion etc. we have the property that

eg. Matrix (Quaternion Float) is effectively the same thing as Matrix Float Quaternion Float.

But John Baez pointed out to me that you can build up the CliffN algebras much more simply enabling us to use these definitions:

> type Cliff1 = Complex Float
> type Cliff2 = Complex Cliff1
> type Cliff3 = Complex Cliff2
> type Cliff4 = Complex Cliff3
> type Cliff5 = Complex Cliff4


Or even better:

> type family Cliff (n :: Nat) :: * where
> Cliff 0 = Float
> Cliff n = Complex (Cliff (n - 1))

But there's one little catch. We have to work, not with the tensor product, but the super tensor product.

We define Complex the same way as before:

> data Complex a = C a a deriving (Eq, Show)

Previously we used a definition of multiplication like this:

instance Num a => Num (Complex a) where
C a b * C c d = C (a * c - b * d) (a * d + b * c)

We can think of C a b in Complex R as representing the element . The definition of multiplication in a tensor product of algebras is defined by . So we have .

This means that line of code we wrote above defining * for Complex isn't simply a definition of multiplication of complex numbers, it says how to multiply in an algebra tensored with the complex numbers.

Let's go Super!

A superalgebra is an algebra graded by where is the ring of integers modulo 2. What that means is that we have some algebra that can be broken down as a direct sum (the subscripts live in ) with the property that multiplication respects the grading, ie. if is in and is in then is in .

The elements of are called "even" (or bosonic) and those in "odd" (or fermionic). Often even elements commute with everything and odd elements anticommute with each other but this isn't always the case. (The superalgebra is said to be supercommutative when this happens. This is a common pattern: a thing X becomes a superX if it has odd and even parts and swapping two odd things introduces a sign flip.)

The super tensor product is much like the tensor product but it respects the grading. This means that if is in and is in then is in . From now on I'm using to mean super tensor product.

Multiplication in the super tensor product of two superalgebras and is now defined by the following modified rule: if is in and is in then . Note that the sign flip arises when we shuffle an odd left past an odd .

The neat fact that John pointed out to me is that .

We have to modify our definition of * to take into account that sign flip.

I initially wrote a whole lot of code to define a superalgebra as a pair of algebras with four multiplication operations and it got a bit messy. But I noticed that the only specifically superalgebraic operation I ever performed on an element of a superalgebra was negating the odd part of an element.

So I could define SuperAlgebra like so:

class SuperAlgebra a where
conjugation :: a -> a

where conjugation is the negation of the odd part.

(I'm not sure if this operation corresponds to what is usually called conjugation in this branch of mathematics.)

But there's a little efficiency optimization I want to write. If I used the above definition, then later I'd often find myself computing a whole lot of negates in a row. This means applying negate to many elements of large algebraic objects even though any pair of them cancel each other's effect. So I add a little flag to my conjugation function that is used to say we want an extra negate and we can accumulate flips of a flag rather than flips of lots of elements.

> class SuperAlgebra a where
> conjugation :: Bool -> a -> a

Here's our first instance:

> instance SuperAlgebra Float where
> conjugation False x = x
> conjugation True x = negate x

This is saying that the conjugation is the identity on Float but if we want to perform an extra flip we can set the flag to True. Maybe I should call it conjugationWithOptionalExtraNegation.

And now comes the first bit of non-trivial superalgebra:

> instance (Num a, SuperAlgebra a) => SuperAlgebra (Complex a) where
> conjugation e (C a b) = C (conjugation e a) (conjugation (not e) b)

We consider to be even and to be odd. When we apply the conjugation to then we can just apply it directly to . But that flips the "parity" of (because tensor product respects the grading) so we need to swap when we use the conjugation. And that should explain why conjugation is defined the way it is.

Now we can use the modified rule for defined above:

> instance (Num a, SuperAlgebra a) => Num (Complex a) where
> fromInteger n = C (fromInteger n) 0
> C a b + C a' b' = C (a + a') (b + b')
> C a b * C c d = C (a * c - conjugation False b * d)
> (conjugation False a * d + b * c)
> negate (C a b) = C (negate a) (negate b)
> abs = undefined
> signum = undefined

For example, conjugation False is applied to the first on the RHS because implicitly represents an term and when expanding out the product we shuffle the (odd) in left of . It doesn't get applied to the second because and remain in the same order.

That's it!


I'll test it with some examples from Cliff3:

> class HasBasis a where
> e :: Integer -> a

> instance HasBasis Float where
> e = undefined

> instance (Num a, HasBasis a) => HasBasis (Complex a) where
> e 0 = C 0 1
> e n = C (e (n - 1)) 0

> make a b c d e f g h =
> C (C (C a b) (C c d))
> (C (C e f) (C g h))

> e1, e2, e3, e21, e31, e32, e321 :: Cliff 3
> e1 = e 0
> e2 = e 1
> e21 = e2 * e1
> e3 = e 2
> e31 = e3 * e1
> e32 = e3 * e2
> e321 = e3 * e2 * e1

> main = do
> print (e1 * e1 + 1 == 0)
> print (e31 * e31 + 1 == 0)
> print (e3 * e3 + 1 == 0)
> print (e21 * e21 + 1 == 0)
> print (e2 * e2 + 1 == 0)
> print (e32 * e32 + 1 == 0)
> print (e321 * e321 - 1 == 0)
> print (e3 * e2 * e1 - e321 == 0)
> print (e2 * e1 - e21 == 0)
> print (e3 * e1 - e31 == 0)
> print (e3 * e2 - e32 == 0)
> print (e21 * e32 - e31 == 0)


The implementation of multiplication looks remarkably like it's the Cayley-Dickson construction. It can't be (because iterating it three times gives you a non-associative algebra but the Clifford algebras are associative). Nonetheless, I think comparison with Cayley-Dickson may be useful.


As mentioned above, before I realised I just needed the conjugation operation I wrote the above code with an explicit split of a superalgebra into two pieces intertwined by four multiplications. I think the previous approach may have a big advantage - it may be possible to use variations on the well known "speed-up" of complex multiplication that uses three real multiplications instead of four. This should lead to a fast implementation of Clifford algebras.

Also be warned: you can kill GHC if you turn on optimization and try to multiply elements of high-dimensional Clifford algebras. I think it tries to inline absolutely everything and you end up with a block of code that grows exponentially with .

Note also that this code translates directly into many languages.

by sigfpe ( at March 05, 2023 07:38 PM

March 03, 2023

Magnus Therning

Per-project xref history in Emacs

When I write code I jump around in the code quite a bit, as I'm sure many other developers do. The ability to jump to the definition of a function, or a type, is invaluable when trying to understand code. In Emacs the built-in xref package provides the basic functionality for this, with many other packages providing their custom functions for looking up identifiers. This works beautifully except for one thing, there's only one global stack for keeping track of how you've jumped around.

Well, that used to be the case.

As I tend to have multiple projects open at a time I used to find it very confusing when I pop positions off the xref stack and all of a sudden find myself in another project. It would be so much nicer to have a per-project stack.

I've only known of one solution for this, the perspective package, but as I've been building my own Emacs config I wanted to see if there were other options. It turns out there is one (almost) built into Emacs 29.

In Emacs 29 there's built-in support for having per-window xref stacks, and the way that's done allows one to extend it further. There's now a variable, xref-history-storage, that controls access to the xref stack. The default is still a global stack (when the variable is set to #'xref-global-history), but to get per-window stacks one sets it to #'xref-window-local-history.

After finding this out I thought I'd try to write my own, implementing per-project xref stacks (for projectile).

The function should take one optional argument, new-value, if it's provided the stack should be updated and if not, it should be returned. That is, something like this

(defun projectile-param-xref-history (&optional new-value)
  "Return project-local xref history for the current projectile.

Override existing value with NEW-VALUE if it's set."
  (if new-value
      (projectile-param-set-parameter 'xref--history new-value)
    (or (projectile-param-get-parameter 'xref--history)
        (projectile-param-set-parameter 'xref--history (xref--make-xref-history)))))

Now I only had to write the two functions projectile-param-get-parameter and projectile-param-set-parameter. I thought a rather straight forward option would be to use a hashtable and store values under a tuple comprising the project name and the parameter passed in.

(defvar projectile-params--store (make-hash-table :test 'equal)
  "The store of project parameters.")

(defun projectile-param-get-parameter (param)
  "Return project parameter PARAM, or nil if unset."
  (let ((key (cons (projectile-project-name) param)))
    (gethash key projectile-params--store nil)))

(defun projectile-param-set-parameter (param value)
  "Set the project parameter PARAM to VALUE."
  (let ((key (cons (projectile-project-name) param)))
    (puthash key value projectile-params--store))

Then I tried it out by setting xref-history-storage

(setq xref-history-storage #'projectile-param-xref-history)

and so far it's been working well.

The full code is here.

March 03, 2023 08:38 PM

GHC Developer Blog

GHC 9.6.1-rc1 is now available

GHC 9.6.1-rc1 is now available

bgamari - 2023-03-03

The GHC team is very pleased to announce the availability of the first (and likely final) release candidate of GHC 9.6.1. As usual, binaries and source distributions are available at

Beginning with GHC 9.6.1, GHC can be built as a cross-compiler to WebAssembly and JavaScript. This is an important step towards robust support for compiling Haskell to the Web, but there are a few caveats to be aware of in the 9.6 series:

  • Both the Javascript and WebAssembly backends are still at an early stage of development and are present in this release as a technology preview

  • Using GHC as a cross-compiler is not as easy as we would like it to be; in particular, there are challenges related to Template Haskell

  • GHC is not yet run-time retargetable; a given GHC binary targets exactly one platform, and both WebAssembly and JavaScript are considered platforms for this purpose. Cross-compilers must be built from source by their users

We hope to lift all of these limitations in future releases.

Additionally, 9.6.1 will include:

  • Significant latency improvements in the non-moving garbage collector

  • Efficient runtime support for delimited continuations

  • Improvements in compiler error messages

  • Numerous improvements in the compiler’s memory usage

See the release notes for a comprehensive accounting of changes in this release.

As always, one can find a migration guide to aid in transitioning from older releases on the GHC Wiki. We have also recently started extending our release process to cover a wider set of Linux distributions. In particular, we now offer Rocky 8 and Ubuntu 20.04 binary distributions which cover RedHat-derivative and distributions using older glibc releases (namely 2.27), respectively.

Please do give this release a try and open a ticket if you see anything amiss. If all goes well we expect the final release should be available by late next week.


  • Ben

by ghc-devs at March 03, 2023 12:00 AM

March 02, 2023

Tweag I/O

Chainsail: Now Unchained and Open-Source

Chainsail, Tweag’s web service for sampling multimodal probability distributions, is now open-source and awaits contributions and new uses from the community!

Chainsail was released in August 2022 as a beta version in order to collect initial feedback and survey potential use cases and directions for future development. If you’d like to learn more about Chainsail, have a look at the announcement blog post, a detailed analysis of soft k-means clustering using Chainsail or our walkthrough video.

After having presented Chainsail to the probabilistic programming communities and scientists, the feedback we got most frequently was: when is Chainsail going to be open-source?

The fact that the Chainsail source code was not publicly accessible seems to have hindered engagement with the target communities of computational statisticians, scientists and probabilistic programmers. Tweag is taking that feedback seriously and announces today the open-sourcing of all of Chainsail’s code.

To help future users and contributors find their way around the project, this blog post will give an overview of the service architecture, point out pieces of code that might be of particular interest to certain groups of users, describe Chainsail deployment options and lists ideas and issues for which Tweag would particularly welcome contributions.

Service architecture

Chainsail is a complex system with multiple, interacting components, but before I describe these components in detail, let’s look at an overview of the Chainsail service architecture:

Chainsail service architecture

Let’s quickly review what happens during a typical Chainsail run:

  • On the frontend, a user submits their probability distribution and provides a couple of parameters. This information is sent to the job scheduler application that will create database entries representing the newly submitted sampling job. The job scheduler then commences processing of the job by launching a controller instance.
  • The controller is the core of a Chainsail job: it orchestrates automatic inverse temperature schedule tuning and Replica Exchange sampling. Note that, despite the similar name, the inverse temperature schedule is completely unrelated to the job scheduler — unfortunately, the former is standard parlance in tempering algorithms. The controller will also ask the job scheduler to scale computing resources up or down, depending on the number of replicas the current simulation run requires.
  • The controller, job scheduler and client furthermore communicate with a couple of auxiliary components for logging, sampling statistics, and storing data and configuration.

Now that you know what you’ve signed up for, I will explain the function of these components inside-out: first, the algorithmic core of the service, and then the components that serve the core functionality to the user and allow them to monitor it.


This is the algorithmic heart of the Chainsail web service. Once a controller is started, either manually when developing, or by the job scheduler when the full service is deployed, the controller does the following:

  1. Sets up an initial inverse temperature schedule that follows a geometric progression.
  2. Until the temperature schedule stabilizes, iterates the following recipe for a schedule optimization simulation:
    • Ask the job scheduler to provision a number of nodes equivalent to the number of temperatures;
    • Draw sensible initial states based on previous optimization iterations, or, if this is the first optimization run, use the initial states the user provides;
    • Interpolate HMC step size from timesteps used in previous optimization iterations, or, if this is the first optimization run, start out with a default stepsize;
    • Run a Replica Exchange simulation on a set of nodes, during which it automatically adapts HMC timesteps for a while;
    • Calculate the density of states (DOS).
  3. Performs a final production run.

The controller comes in the form of a Python application, the source code of which can be found in app/controller. Note that the controller can be run independently, without any other Chainsail component.

Density of states estimator

For schedule optimization and for drawing initial Replica Exchange states based on previous iterations, the controller requires an estimate of the density of states (DOS), which is briefly explained in this blog post and which might be discussed in more detail in a forthcoming publication. There are several ways to estimate the density of states, but Chainsail implements a binless version of the Weighted Histogram Analysis Method (WHAM) that is widely used in biomolecular simulation.

This WHAM implementation is completely general and might find interesting uses outside of Chainsail, e.g. in conjunction with other tempering methods. It’s also interesting to note that the DOS is a very useful value to have for a multitude of analyses; for example, it can be used to easily estimate the normalization constant of a probability distribution, which means that, in the case of a Bayesian model, the model evidence can be calculated using the DOS.

The code can be found in lib/common/schedule_estimation.

Automatic schedule tuning

As described above, the controller automatically tunes the inverse parameter schedule with an iterative procedure. In each iteration, it tries to find a sequence of inverse temperatures such that the acceptance rate predicted from it matches the target acceptance rate desired by the user. Predicting the acceptance rate is the main reason the controller calculates a DOS estimate at each iteration.

The automatic schedule tuning depends only on a DOS estimate and is thus easily reuseable for Replica Exchange implementations other than Chainsail, for example in TensorFlow Probability or ptemcee.

The code is provided in the same Python package as the DOS estimator, namely lib/common/schedule_estimation.

For the curious reader with the necessary mathematical and physics background, a technical note explains the DOS-based schedule tuning and drawing of appropriate initial Replica Exchange states.

MPI-based Replica Exchange runner

Once a schedule, initial states and timesteps are determined, the controller launches a Replica Exchange simulation implemented in a runner. Chainsail currently implements a single runner based on rexfw, a Message Passing Interface (MPI)-based Replica Exchange implementation. The Chainsail-specific runner code is thus only a thin wrapper around the (very general) rexfw library and can be used both on a single machine and in a Kubernetes cluster. As Replica Exchange is a bit more widely known in scientific circles and the rexfw library has its origin in scientific research, an interesting option could be to run rexfw and even the full controller on HPC clusters.

The rexfw-based runner can be found in lib/runners/rexfw.

Job scheduler

The job scheduler is a Flask-based Python app that serves multiple purposes:

  • It receives job creation, launch, and stop requests from the user;
  • It keeps track of the job state;
  • It scales the resources (nodes) available to the controller up or down;
  • And it compresses sampling results into a .zip file.

It keeps track of users, jobs, and nodes in tables of a PostgreSQL database. The job scheduler currently knows two node types, namely a (now deprecated) VM node in which each replica is run on a separate VM, and a Kubernetes-based implementation in which a node is represented by a pod.

The job scheduler code is located in app/scheduler.

Client web app

While a user can interact with a Chainsail job scheduler via the HTTP APIs of the job scheduler and other components, the most comfortable way to do so is via the frontend web app. It is written in React and comprises the following parts:

  • A landing page that presents Chainsail to the user and shows links to essential resources, such as blog posts and the chainsail-resources repository;
  • A job submission form that allows a user to set (and documents) the most crucial parameters and that allows the upload of a .zip file with the probability definition;
  • A job table that shows all jobs a user has created and possibly launched, with their current status;
  • A dashboard that allows the user to follow a sampling job’s process, monitor Replica Exchange acceptance rates and convergence and that shows Chainsail logging output.

If mandatory user authentication via Firebase is enabled, only the landing page is accessible without the user being logged in. For every other page, the user will have to sign in with their Gmail address, and a token is generated that is sent along to the job scheduler, which then stores it in a user database table.

The client web app can be found in app/client.

Other components

The above really are the key components of the Chainsail service, but there are a couple of other applications and services that are necessary to make full Chainsail deployments run. Among them are a small Flask application that serves MCMC sampling statistics to the frontend application, a Celery task queue to which job scheduler requests are offloaded, and a Docker image that runs a gRPC server providing an encapsulated interface to the user-provided code.

Deploying Chainsail

As you have now seen, Chainsail is a big system, but rest assured, deploying it and testing it out isn’t that hard! You currently have three ways to deploy (a subset of) Chainsail:

  • Run the controller and thus Chainsail’s core algorithm on a single machine:

    This is as easy as using the provided Poetry packaging to get a Python shell with all required dependencies, add the dependencies your probability distribution code needs, set parameters in a JSON configuration file and run the controller using a simple command. Note that this currently requires an OpenMPI (or another MPI) installation.

  • Run the full Chainsail service locally using Minikube:

    Minikube provides a local environment to run Kubernetes applications. Given a running Minikube installation, Chainsail can be very easily deployed using Terraform configuration and Helm charts. All it takes are a couple of commands, which are described in our deployment documentation.

  • Run the full Chainsail service on Google Cloud:

    Finally, the complete Chainsail web service can be deployed to the Google Cloud Platform, thanks again to Terraform configuration and Helm charts. Note, though, that this requires setting quite a few configuration options, such as regions, VM machine types, etc., but defaults are given. These configuration options are currently not documented, so you would have to browse the relevant Terraform and Helm configuration files yourself.

Future work

Chainsail in its current state is still a proof of concept, but Tweag is convinced that it could become a useful and widely applicable tool for everyone who has to deal with multimodal probability distributions. To this end, a couple of interesting possibilities for future developments include:

  • Replacing the very simple HMC implementation with a state-of-the-art NUTS sampler, e.g. via the BlackJAX library (#423).
  • Replace the clunky and insanely slow Stan support (that currently works via requests to a httpstan instance) with a wrapper around BridgeStan (#424).
  • Improve the inverse temperature schedule optimization to deal with “phase transitions”, meaning distributions which exhibit sudden drops to zero Replica Exchange acceptance rate with increasing inverse temperature (#425).
  • Speed up and improve candidate schedule calculation by having an adaptive inverse temperature decrement (#426).
  • Implement a lightweight local runner for easier testing and debugging (#427).
  • Write a dedicated interface to calculate the model evidence and other quantities from the DOS, possibly in a separate Python package.

While Chainsail has a large number of unit tests and a couple of functional tests, it would still immensely profit from improved testing and a continuous integration (CI) setup. For example, Docker containers could be built automatically by CI and an automatic end-to-end integration test of the service could be performed using the Minikube deployment option. Building Docker images currently happens via Dockerfiles and the docker build command, but packaging of Chainsail components via Nix and its dockerTools library is currently under development.


Now that Chainsail is open-source, I hope that this brief tour of Chainsail’s key components piques interest in the project. Be it code reuse in other projects, any kind of contribution to the code base, or ideas for interesting problems Chainsail could solve, Tweag welcomes feedback and engagement from probabilistic programmers, statisticians, scientists and everyone else who is as excited about this project as we are!

March 02, 2023 12:00 AM

February 28, 2023


GHC activities report: December 2022-January 2023

We’re happy to support the Haskell Foundation in planning a pre-ZuriHac workshop for new GHC contributors on 7th-9th June 2023 in Zurich (and online). Find out more and express interest here.

This is the sixteenth edition of our GHC activities report, which describes the work on GHC and related projects that we are doing at Well-Typed. The current edition covers roughly the months of December 2022 and January 2023. You can find the previous editions collected under the ghc-activities-report tag.

We are delighted that Juspay are now sponsoring this work. Many thanks to them and to our other sponsors, including GitHub via the Haskell Foundation, and IOG. We are also working with Hasura and Mercury on specific GHC-related improvements, such as interface files with core definitions.

However, some of our sponsorship agreements are coming to an end and we need more sponsorship to sustain the team! If your company might be able to contribute funding to ensure that we can continue this valuable work, please read about how you can help or get in touch.

Of course, GHC 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 in any way. 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!


The existing GHC team consists of Ben Gamari, Andreas Klebinger, Matthew Pickering, Zubin Duggal and Sam Derbyshire. Many others within Well-Typed are contributing to GHC more occasionally.

We are very pleased to welcome Rodrigo Mesquita to the team for an internship over the next six months. Rodrigo will be working on enabling the safe reinstallation of the ghc and template-haskell libraries (#20742). This is a crucial step which will unlock better cross-compilation, more robust tooling support, and potential improvements in library compatibility across GHC versions.


  • Ben and the team have been busy working on the 9.6 release (#22562). Several alphas have been released and a release candidate is expected soon. Sam worked on the 9.6 migration guide.

  • Matt and Ben released version 9.4.4 in December, fixing various bugs in the 9.4 series.

  • Zubin prepared version 9.2.6 for release in February.


  • Matt finished off and landed a patch from a contributor reworking plugin initialisation points, so that plugins can install hooks (#21279, !9824).

  • Matt fixed a variety of bugs in multiple home unit support which were found when testing new Cabal support for starting multi-unit sessions (!9618).

Compiler performance

  • Matt and Zubin investigated GHCi memory usage on partial reloads. They found and squashed quite a few leaks, resulting in substantial reductions in memory usage after a :reload (!9787).

  • Matt fixed a bug when using interface files with core definitions (!9869).

Runtime performance

  • Ben fixed an unexpected regression in MutVar# performance due to a memory fence (#22764, !9718).

  • Andreas investigated many runtime performance issues relating to inlining, specialisation, INLINEABLE and NOINLINE pragmas (#22321, #22609, #22689, #22781, !9723). He has work in progress allowing NOINLINE and INLINEABLE to be combined, enabling specialization of NOINLINE functions (!9621).

  • Andreas opened a GHC proposal with some ideas for transitive specialization.

  • Andreas investigated performance regressions in aeson (#22115) which were caused by three independent issues:

    • A code generation regression which was then quickly fixed by Ben (!9718).
    • A change to the implementation of Bits instances which caused regressions due to partial register stalls. This is a known issue (#20405) where the fix had not been applied to all branches at the time.
    • A regression due to changes to the constructed product result optimisation (#22983).
  • Andreas diagnosed a runtime performance regression in Data.OldList.{maximumBy,minimumBy} in 9.2.5 (#22609, !9565).


  • Andreas diagnosed an issue where an OPAQUE pragma triggered a compiler panic (#22502).

  • Sam proposed a patch adding Template Haskell support for explicit forall quantifiers in instances (!9840).

  • Matt fixed the module locations of some definitions in GHC.Types for Typeable (#22510, !9459).

Error messages

  • Matt improved the error messages of the exhaustiveness checker when checking statements that have been moved around with ApplicativeDo (#22483, !9460).

Runtime system

  • Ben investigated a recurring runtime crash on MacOS Ventura (#22497), which turned out to be an upstream bug in the GMP library, necessitating backports to the 9.2, 9.4 and 9.6 branches.

  • Ben began looking at improving the linkage story for RTS symbols, which is a long-standing headache for distribution packagers (#17157).

  • Ben fixed warning-cleanliness in the runtime system’s C sources and fixed Hadrian to ensure that they remain clean (!9579)

Garbage collector

  • Andreas fixed an issue with GCing of sparks (#22528) that sometimes caused sparks to be evaluated sequentially instead of in parallel.

  • Andreas investigated and fixed a crash when GCing a CAF that is evaluated from TH (#22417). This fixes some long standing spurious segfaults involving TH on static build targets such as Windows.

  • Ben made significant improvements to the non-moving garbage collector:

    • Fixed the handling of empty arrays in the non-moving write barrier (#22931).
    • Greatly improved handling of Weak# objects, which previously may not have been promptly finalized (#22327).
    • Fixed a few potentially unsound interactions with setNumCapabilities (#22926, #22927).
    • Identified and fixed a missing write barrier in the GC’s selector optimisation implementation (#22930).
    • Significantly improved pause times for some programs by bounding marking work in the sync pause (#22929).
    • Continued work in implementing parallel marking support.
    • Improved accuracy of residency reporting.
  • While working on the GC, Ben also identified and fixed a number of data races in the RTS thanks to ThreadSanitizer instrumentation (!6232, !9371, !9372, !9482).

Code generation

  • Sam improved how GHC keeps track of the types at which Cmm registers are used. In particular, this helps ensure that SIMD vector registers, which can be used to store different types of data, are used in a consistent manner (!9167).

  • Ben diagnosed and fixed an AArch64 backend issue resulting in compile-time crashes when the graph-colouring register allocator (-fregs-graph) is in use (#22798).

  • Andreas fixed a runtime crash caused by a bug in the tag inference analysis due to a difference between Cmm and Bytecode backends (#22840).

  • Andreas avoided unboxing large constructors containing unboxed tuples into other constructors by mistake (#22309, !9724). They can still be explicitly unboxed if desired.

  • Ben fixed -fsplit-sections on Windows (#22834).

  • Ben fixed a linking-related compilation failure on MacOS (#21972).


  • Andreas updated the nofib benchmark suite to ensure it is usable with the upcoming 9.6 release, along with some other minor tweaks.

  • Ben improved the reporting output of the nofib-compare utility.

  • Matt added tests for issues milestoned for 9.4.4 but already fixed.

  • Ben found that the testsuite was failing with -threaded and sanity-checking enabled, and started fixing these issues.


  • Matt fixed an issue with manpages not being installed.

  • Matt made various fixes to the build system and packaging including enabling split-sections for release builds.

  • Matt added a flag to omit self-recompilation info from interface files, for situations where you are never going to attempt to recompile a package. This avoids a source of non-determinism in builds.

  • Ben bumped the Windows toolchain shipped with GHC (#21293, !9683).

  • Matt added support for LLVM 15 and while doing so removed support for LLVM 10. GHC will henceforth support LLVM 11 through 15.

  • Zubin fixed an issue with the binary distribution configure script where it didn’t check for the find command being available (!9777).

  • Matt added scripts for automatically generating metadata for ghcup. When a new release is made, the metadata is sent to the ghcup maintainers who can easily use it to add support for new releases to ghcup. The metadata is generated on nightly and release pipelines and is now used to test binary distributions on many different platforms.


  • Matt upgraded CI runners to use GHC 9.4.3 for CI.

  • Matt fixed a variety of issues on Darwin CI runners, so they could be re-enabled, increasing CI capacity.

  • Matt added Ubuntu 18.04 nightly and release jobs.

  • Ben added a Rocky Linux CI job, filling the hole left by the retirement of Centos 7.

  • Ben prototyped infrastructure for collecting hardware performance counter metrics in CI to catch code generation regressions more reliably (!7414).



  • Matt added support for tracing of static reference tables (SRTs) to ghc-debug.

Supporting users/contributors

  • Matt looked into a reported issue with a runtime loop, and diagnosed it as expected behaviour involving an interaction with overlapping instances specific to the project.

  • Ben reproduced and characterised a linking issue affecting a user (#22803).

  • Sam reviewed and assisted with an MR by Georgi Lyubenov that allows record wildcards to be used with nullary constructors (!9004).

  • Sam reviewed several MRs by Andrew Lelechenko which eliminate uses of head and tail within GHC’s codebase.

  • Ben fixed many other tickets (#22527, #22538, #22561, #22631, #22714, #22322, #22777, #22640, #22595, #22566).


  • Ben wrote and shepherded a Core Libraries Committee proposal adding the new Foreign.C.ConstPtr.ConstPtr type, facilitating resolution of #22043.

  • Ben began assessing the stability of the many interfaces provided by base’s GHC.* module namespace as a first step towards making a clearer separation between the public and GHC-internal APIs of base.

  • Ben introduced CI support for tracking the interfaces exposed by base to help ensure that core libraries changes receive the review that they deserve (!9816).

  • Sam made a CLC proposal concerning the reporting of exceptions arising during weak pointer finalization. Until agreement on this issue is reached, GHC’s error reporting has been reverted to the status quo, with users free to change it as they see fit.

by ben, andreask, matthew, zubin, sam at February 28, 2023 12:00 AM

February 27, 2023

Monday Morning Haskell

Adding a Database to our AWS Server

In the last few articles on the blog, we've been exploring how to launch a Haskell web server using AWS. Here are the steps we've done so far:

  1. Create a local Docker Image
  2. Upload the Docker Image to ECR
  3. Deploy your Server using Elastic Beanstalk

In this final part of the series, we're going to learn to attach a database to our application.

There are a few gotchas to this. Setting up the database for first time use is a bit tricky, because we have to do some initial migrations. Then we need to use environment variables to ensure it works both locally and on the remote server. Let's get started.

A Basic Schema

Let's first assume we have a super basic schema using the Persistent library. (If you want some details on how this works, see our Real World Haskell series). We'll just have one type in our database, and users will use server endpoints to create or fetch these "text entries".

import           Database.Persist.Sql
import qualified Database.Persist.TH as PTH
import           Data.Text (Text)

PTH.share [PTH.mkPersist PTH.sqlSettings, PTH.mkMigrate "migrateAll"] [PTH.persistLowerCase|

  TextEntry sql=text_entries
    text Text


An important product of this template Haskell sequence is the migrateAll function, which will run the proper commands to migrate a Postgres database to fit our schema by creating tables.

Whenever we first create a database, we have to make sure it's migrated. But before we even do that we have to make sure we've created a database for Postgres to use! Let's see the commands we need for this, and how to run them in Haskell.

Running Setup Commands

When you install Postgres on your machine, you'll have separate "databases" on your system to help you keep your data separate. Separating data allows each database to have its own "users" table without having any name conflicts, for one example. By default, Postgresql comes installed with a database called postgres.

But we don't want to use this to store our data. We want to create a separate database. We want to create this database if it's the first time we're running the server with a database. But otherwise, we just want to make sure its migrations are up to date.

Now, the command we would run to create this database is simple:


But we can first run this command to see if this database already exists:

SELECT datname FROM pg_database WHERE datname = 'quiz';

Both these commands assume we are connected to the postgres database.

Since these first two instructions are raw commands, we can run them using the Postgresql Simple library. Here's some code to do this.

createDBIfMissing :: String -> IO ()
createDBIfMissing connString = do
  connection <- connectPostgreSQL (pack connString)
  putStrLn "Checking/Creating 'quiz' Database"
  let checkQuery = "SELECT datname FROM pg_database WHERE datname = 'quiz';"
  (checkResult :: [Only String]) <- query_ connection checkQuery
  when (null checkResult) $ do
    putStrLn "Not found! Creating 'quiz' database!"
    let createQuery = "CREATE DATABASE quiz;"
    void $ execute_ connection createQuery

When we run checkQuery, it sees if the quiz database exists. If its result is null, then we'll run the additional command to create our database.

Once we have this function, we can write a wrapper that will create the database and then migrate it for our schema. Here's what this wrapper looks like:

migrateDb :: String -> String -> IO ()
migrateDb baseConnString quizConnString = do
  createDBIfMissing baseConnString
  putStrLn "Migrating Database"
  runPG quizConnString (runMigration migrateAll)

runPG :: String -> SqlPersistT (LoggingT IO) a -> IO a
runPG connString action = runStdoutLoggingT $
  withPostgresqlConn (pack connString) $ \backend ->
    runReaderT action backend

Notice migrateDb takes two different connection strings. One is for the base (postgres) database. The other is for our new quiz database. The creation queries run on the first, the migration runs on the second.

But how do we use these functions within our server?

Loading the URI

When we kick off our server, we have to load the database URI for our Postgres database. We'll use the format of {host}:{port}. If you're running it locally, this would just be localhost:5432. But when we deploy the server, we'll use a different URI. So let's write a function to load the host and port (separated by a colon) from an environment variable named DATABASE_URI.

loadDatabaseEnv :: IO (String, String)
loadDatabaseEnv = do
  dbEnv <- lookupEnv "DATABASE_URI"
  if isNothing dbEnv || ':' `notElem` fromJust dbEnv
    then return ("localhost", "5432")
    else return (span (/= ':') (fromJust dbEnv))

Now we need to construct the full Postgres connection string. This has the following general format:

host={host} port={port} dbname={dbname} user={user} password={password}

As a default value, you can often just have the username and password both be postgres (though of course this isn't recommended for a serious database). But let's make a function to substitute in the other values:

mkPostgresUri :: String -> String -> String -> String
mkPostgresUri host port dbname =
  "host='" <> host <> "' port=" <> tail port <> " dbname='" <> dbname <> "' user='postgres' password='postgres'"

Finally, we'll pull our different pieces together, get both URIs, and launch our server. In my example, I'm using a Servant server (more details on that in this article), and this will often require passing the database string as an argument.

server :: String -> Server QuizAPI
server connString = ...

runServer :: IO ()
runServer = do
  (host, port) <- loadDatabaseEnv
  let baseConnString = mkPostgresUri host port "postgres"
  let quizConnString = mkPostgresUri host port "quiz"
  migrateDb baseConnString quizConnString
  putStrLn "Running Server!"
  run 8080 (serve api (server quizConnString))

Having made all these modifications to our server, of course we have to rebuild and redeploy our docker image for that! We can create the new local image with:

docker build -t quiz-server .

Then for more detailed instructions on deploying it, refer to part 2 and part 3 of this series!

When you deploy the server, you'll find it's crashing of course, because we haven't configured the database! So let's get to the real meat of this article…setting up the database on AWS!

Create a Database with RDS

This process is not actually too challenging. The first thing we're going to do is use RDS (Relational Database Service) to set up our database. This is easily done from the AWS console.

  1. Select the RDS service
  2. Hit the orange "Create Database" button
  3. Go through the creation wizard, making sure to select "Postgres" and the "Free Tier" option (assuming you're just making a test app).

Most of the default options are fine, but as I mentioned above I specified postgres for the username and password of the database. I also unchecked the box for "Performance Insights" since this could lead to additional billing charges if you forget to turn it off.

Once you've created your database, you can then click the "databases" link on the sidebar, and select your new database. On that screen, you'll be able to see the "endpoint" and "port" of your database. These are the values you'll need for your environment!

Add Environment Variable

To connect your environment to the database, now you just have to add an environment variable! To do this, you have to access the configuration from the web portal:

  1. Go to the Elastic Beanstalk service
  2. Select "Environments" from the sidebar and then click the environment you have running your server.
  3. Click on the "Configuration" link on the side, and then select the "Edit" button in the "Software" section.
  4. At the very bottom, you'll find the "Environment Properties" section. Fill in DATABASE_URI as the key, and the {host}:{port} combination you got from your database in RDS.
  5. Click "Apply" to make the change!

By adding an environment variable, you are changing the configuration of your server, so it will reboot. Once it relaunches, you should find that it works, and you can persist information from your database!


Hopefully this series has helped you learn how to deploy your Haskell code to AWS! If you'd like to see all this article in video form, you can check out our YouTube video covering these steps!

For more tips on creating a "Real World" application, you can read our series on web skills! You can also download our Haskell Production checklist for some ideas of other libraries and tools you can use to improve your Haskell!

by James Bowen at February 27, 2023 04:00 PM

GHC Developer Blog

GHC 9.2.7 is now available

GHC 9.2.7 is now available

Zubin Duggal - 2023-02-27

The GHC developers are happy to announce the availability of GHC 9.2.7. Binary distributions, source distributions, and documentation are available at

This release is primarily a bugfix release addressing a few issues found in 9.2.6. These include:

  • A fix for a bug with the RTS linker being unable to resolve the setKeepCafs symbol which started being used by the GHC library in 9.2.6, resulting in code depending on this symbol failing to load in GHCi or via a compile time splice when using a statically linked GHC, such as on Windows (#22961).
  • A fix a bug with the alignment of RTS data structures that could result in segfaults when compiled with high optimisation settings on certain platforms (#22975 , #22965).
  • A fix for a bug in the simplifier related to placeholder values (rubbish literals) emitted by the worker/wrapper optimisation leading to -dcore-lint errors and compiler panics in certain cases (#19882, #22914, #23000).
  • Easier installation of binary distribution on MacOS platforms by changing the installation Makefile to remove the quarantine attribute when installing.
  • … and a few more. See the release notes for a full accounting.

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.

Happy compiling,

  • Zubin

by ghc-devs at February 27, 2023 12:00 AM

February 24, 2023

Brent Yorgey

Competitive programming in Haskell: Infinite 2D array, Level 4

In a previous post, I challenged you to solve Infinite 2D Array using Haskell. After deriving a formula for F_{x,y} that involves only a linear number of terms, last time we discussed how to efficiently calculate Fibonacci numbers and binomial coefficients modulo a prime. Today, we’ll finally see some actual Haskell code for solving this problem.

The code is not very long, and seems rather simple, but what it doesn’t show is the large amount of time and effort I spent trying different versions until I figured out how to make it fast enough. Later in the post I will share some lessons learned.

Modular arithmetic

When a problem requires a fixed modulus like this, I typically prefer using a newtype M with a Num instance that does all operations using modular arithmetic, as explained in this post. However, that approach has a big downside: we cannot (easily) store our own newtype values in an unboxed array (UArray), since that requires defining a bunch of instances by hand. And the speedup we get from using unboxed vs boxed arrays is significant, especially for this problem.

So instead I just made some standalone functions to do arithmetic modulo 10^9 + 7:

p :: Int
p = 10^9 + 7

padd :: Int -> Int -> Int
padd x y = (x + y) `mod` p

pmul :: Int -> Int -> Int
pmul x y = (x*y) `mod` p

What about modular inverses? At first I defined a modular inverse operation based on my own implementation of the extended Euclidean Algorithm, but at some point I did some benchmarking and realized that my egcd function was taking up the majority of the runtime, so I replaced it with a highly optimized version taken from the arithmoi package. Rather than pasting in the code I will let you go look at it yourself if you’re interested.

Given the efficient extendedGCD, we can now define modular inversion like so:

inv :: Int -> Int
inv a = y `mod` p
    (_,_,y) = extendedGCD p a

Fibonacci numbers and factorials

We want to compute Fibonacci numbers and factorials modulo p = 10^9 + 7 and put them in tables so we can quickly look them up later. The simplest way to do this is to generate an infinite list of each (via the standard knot-tying approach in the case of Fibonacci numbers, and scanl' in the case of factorials) and then put them into an appropriate UArray:

fibList :: [Int]
fibList = 0 : 1 : zipWith padd fibList (tail fibList)

fib :: UArray Int Int
fib = listArray (0, 10^6) fibList

fac :: UArray Int Int
fac = listArray (0, 2*10^6) (scanl' pmul 1 [1 ..])

I should mention that at one point I defined fib this way instead:

fib' :: Array Int Int
fib' = array (0, 10^6) $ (0,0):(1,1):[ (i, fib!(i-1) `padd` fib!(i-2)) | i <- [2 .. 10^6]]

This takes advantage of the fact that unboxed arrays are lazy in their values—and can hence be constructed recursively—to directly define the array via dynamic programming. But this version is much slower, and uglier to boot! (If we really needed to initialize an unboxed array using recursion/dynamic programming, we could do that via runSTUArray, but it would be overkill for this problem.)

Binomial coefficients modulo a prime

We can now efficiently compute binomial coefficients using fac and inv, like so:

mbinom :: Int -> Int -> Int
mbinom m k = (fac!m) `pdiv` ((fac!k) `pmul` (fac!(m-k)))

As mentioned in a previous post, this only works since the modulus is prime; otherwise, more complex techniques would be needed.

We could also precompute all inverse factorials, and then we can get rid of the pdiv call in mbinom (remember that pmul is very fast, whereas pdiv has to call extendedGCD):

ifac :: UArray Int Int
ifac = listArray (0, 2*10^6) (scanl' pdiv 1 [1 ..])

mbinom' :: Int -> Int -> Int
mbinom' m k = (fac!m) `pmul` (ifac!k) `pmul` (ifac!(m-k))

For this particular problem, it doesn’t make much difference either way, since the total number of pdiv calls stays about the same. But this can be an important optimization for problems where the number of calls to mbinom will be much larger than the max size of its arguments.

Putting it all together

Finally, we can put all the pieces together to solve the problem like so:

main = interact $ words >>> map read >>> solve >>> show

solve :: [Int] -> Int
solve [x,y] =
  sum [ (fib!k) `pmul` mbinom (x-k+y-1) (x-k) | k <- [1 .. x]] `padd`
  sum [ (fib!k) `pmul` mbinom (y-k+x-1) (y-k) | k <- [1 .. y]]

Lessons learned

The fact that the above code is fairly short (besides extendedGCD) belies the amount of time I spent trying to optimize it. Here are some things I learned while benchmarking and comparing different versions.

First, we should try really, really hard to use unboxed arrays (UArray) instead of boxed arrays (Array). Boxed arrays have one distinct advantage, which is that they can be constructed lazily, and hence recursively. This helps a lot for dynamic programming problems (which I have a lot to write about at some point in the future). But otherwise, they introduce a ton of overhead.

In this particular problem, committing to use UArray meant (1) using explicit modular operations like padd and pmul instead of a newtype, and (2) constructing the fib array by calculating a list of values and then using it to construct the array, instead of defining the array via recursion/DP.

The optimized implementation of extendedGCD makes a big difference, too, which makes sense: a majority of the computation time for this problem is spent running it (via pdiv). I don’t know what general lesson to draw from this other than affirm the value of profiling to figure out where optimizations would help the most.

I tried a whole bunch of other things which turn out to make very little difference in comparison to the above optimizations. For example:

  • Optimizing padd and pmul to conditionally avoid an expensive mod operation when the arguments are not too big: this sped things up a tiny bit but not much.

  • Rewriting everything in terms of a tail-recursive loop that computes the required Fibonacci numbers and binomial coefficients incrementally, and hence does not require any lookup arrays:

solve' :: [Int] -> Int
solve' [x,y] = go x y 0 1 0 1 (mbinom (x+y-2) (x-1)) `padd`
               go y x 0 1 0 1 (mbinom (x+y-2) (y-1))
    -- Invariants:
    --   s  = sum so far
    --   k  = current k
    --   f' = F_{k-1}
    --   f  = F_k
    --   bx  = binom (x-k+y-1) (x-k)
    go x y !s !k !f' !f !bx
      | k > x     = s
      | otherwise
      = go x y (s `padd` (bx `pmul` f)) (k+1)
           f (f' `padd` f) ((bx `pdiv` (x-k+y-1)) `pmul` (x-k))

    mbinom' n k = fac' n `pdiv` (fac' k `pmul` fac' (n-k))
    fac' k = foldl' pmul 1 [1 .. k]

This version is super ugly and erases most of the benefits of using Haskell in the first place, so I am happy to report that it runs in exactly the same amount of time as the solution I described earlier.

by Brent at February 24, 2023 01:32 PM

February 23, 2023

Monday Morning Haskell

Deploying a Haskell Server to AWS

In the last few articles, we've been talking about how to deploy a Haskell application using AWS. This is part 3 of the series. So if you haven't done parts 1 & 2, you should start there so you can follow along!

In Part 1, we wrote a Dockerfile and created a local Docker image containing a simple program for a Haskell web server.

In the Part 2, we pushed our container image to the AWS container registry (ECR). Notably, this involved creating an AWS account, downloading AWS command line tools and authenticating on the command line. We'll run a couple more of these commands today, so hopefully you're still authenticated!

But now that our container is uploaded, deploying that container is fairly straightforward. But it requires us to use a couple new concepts, as we'll see.

Adding ECR Permission

Before we get started, there's one step we have to take on the web portal. You need to give Elastic Beanstalk permission to download your ECR containers. You can do this using the IAM service from the AWS portal. Then follow these steps:

  1. Select "roles" on the left hand menu.
  2. Select "aws-elasticbeanstalk-ec2-role" in the list in the middle of the screen.
  3. Click "Add Permissions"
  4. Search for and select "AmazonEC2ContainerRegistryReadOnly"

Now let's get into the steps on our local machine.

Configuration File

There are multiple approaches to deploying a docker container, but the one that worked most easily for me was to create a file called (Full other methods, refer to the documentation). This approach involves a counter-intuitive idea. We're going to create a separate directory outside of our main project directory. We'll call it remote.

~/Quiz $ cd ..
~/ $ mkdir remote && cd remote

In this directory, we'll make a single file, called This will, of course, be a JSON file. It will be a very simple configuration file telling our application to use the docker image we pushed last time to ECR. We have to start it by specifying the version of the program (which is 1 because we're only using a single container).

  "AWSEBDockerrunVersion": "1",

Now we'll use tell it to use the Docker image we pushed last time by giving the URI under the Image object:

  "AWSEBDockerrunVersion": "1",
  "Image": {
    "Name": ""

Finally, we'll specify the port, similar to a Dockerfile. We'll use 8080 both for the "Container" port and the "Host" port.

  "AWSEBDockerrunVersion": "1",
  "Image": {
    "Name": ""
  "Ports": [{
    "ContainerPort": 8080,
    "HostPort": 8080

This is the only file we need in this directory! So now let's see what commands we need to run.

Creating the Application

Now we have two more steps that can largely be accomplished on the command line. First, we have to create an application. Then we have to create an environment to use for that application.

Before we can create an application though, we have to create a Git repository, just to store our single file! That's how AWS figures out what to push for configuration.

~/remote $ git init
~/remote $ git add .
~/remote $ git commit -m "First Commit"

Now we can create the application using the eb init command. We'll give our application the name quiz-server.

~/remote $ eb init -p docker quiz-server

You can then see your application on the web portal by accessing the "Elastic Beanstalk" service and clicking the "Applications" tab on the left menu.

Creating the Environment

Now we have to deploy an environment to deploy for our application. When first creating this environment, we use the eb create command. We'll give this environment the name quiz-server-env.

~/remote $ eb create quiz-server-env

This will take a while to deploy. But once it's done, you should be able to see it by clicking the "Environments" tab from the previous screen in the web portal. This will also show you the URL you can use to access your server. It's now successfully deployed!


Sometimes, your deployment might fail. For example, you might misspell the name of your container. If you click on your environment (from the "Environments" tab), then you'll be able to access the "Logs" on the left hand menu. This can help you debug. If you need to change your configuration file, you'll need to commit it, though you don't need to push it to any remote repository. You instead use eb deploy to push your changes.

~/remote $ git add
~/remote $ git commit -m "New Commit"
~/remote $ eb deploy

Now the deployment process should start again!


You can also watch our YouTube video to see all these steps in action!


You now have enough information to deploy a Haskell web application to Heroku! We'll have one more installment in this series around adding a database to our application, so stay tuned for that! In the meantime, subscribe to our monthly newsletter so you can stay up to date with all the latest news!

by James Bowen at February 23, 2023 03:30 PM

Tweag I/O

Taking the pulse of infrastructure management in 2023

February started with a busy week for several of us Tweagers. We went as a group — or, dare I say, a delegation — to FOSDEM23 and CfgMgmtCamp23 (Config Management Camp if, like me, you can’t read through so many consonants). Tweagers Bryan Honof and Théophane Hufschmitt, together with Ryan Lahfa, Julien Malka and Matthew Croughan from the Nix community, got us the first Nix DevRoom at FOSDEM. And not the last, as you can see from the picture below; the room just couldn’t handle the Nix fame!

FOSDEM Nix Dev Room #1 FOSDEM Nix Dev Room #2

Our involvement didn’t end there. At FOSDEM:

Followed right away by CfgMgmtCamp:

  • Bryan first did a lightning talk to tease Nix, then took curious people on a deeper dive during a 50-minute gentle introduction to Nix.
  • Viktor Kleen presented his work on Terraform-Nickel, a library to use Nickel in place of HCL for Terraform.
  • I talked about how to write modular and customizable configuration using Nickel’s merging system.

What follows are my takeaways on the trends and the trajectory of infrastructure and configuration management. Please don’t take my word as a confident prediction, as predictions don’t age well. This post is merely a patchwork of my biased, fresh impressions — as a programming language nerd rabbit-holed into devops and infrastructure by Nix — on my way back home.

Self-service infrastructure

Several key speakers at CfgMgmtCamp emphasized how much infrastructure is critical for many business services and how, more often than not, it is a bottleneck in practice.

Developers must be able to spin up new services rapidly, not in hours or days. They can’t wait one week for someone from the infra team to finally find time, between daily maintenance tasks, to hit the green button (or do something more complex), potentially followed by a long painful back-and-forth. Non-technical roles, such as marketing and sales, might also need to get a small website or a service up, but are even less empowered.

This doesn’t mean one should give everyone full access to the infrastructure. I’ve rather heard a call for simpler, safer, and better automated processes. For example, having a simple web interface for spinning up instances, with a clear policy on cleaning inactive resources. The interface shouldn’t require much infrastructure knowledge nor expose low-level options: this isn’t simply about granting people access to an AWS console. If users are developers, this can be achieved using infrastructure as code as well, with adapted restrictions.

Continuous integration, continuous delivery, one-off experiments — all those components should be automated, fully integrated in the software development workflow, and just a few clicks (or keystrokes, for you Vimmers) away.

YAML is dead

I’m sorry, that was just clickbait. YAML has been pronounced dead for the last 10 years or so, and unfortunately, here it stands.

But something is in the air. There is an undeniable movement toward a unified and structured approach to writing and managing configuration.

Scattering configuration data, schemas and knowledge across many different tools, written in many different languages (HCL, YAML, JSON, TOML, Puppet, Ansible, Helm, etc.) isn’t sustainable. And I feel like we can state a variant of Greenspuns’ tenth rule of programming for configuration:

Every sufficiently complicated infrastructure tool’s native configuration language is an ad-hoc, informally-specified, bug-ridden, arbitrarily-limited reinvention of half of CommonLisp.

This is exactly why we created Nickel. Projects that were the new kids on the block a few years ago — I’m thinking about Pulumi or CUE, for example — are gaining strength and run today in production. Those are not curiosities any more. The actors behind those tools seem to share the same dissatisfaction with the state of configuration management and infrastructure engineering. While our experience and vision led us to come up with our own answer (Nickel), we are all trying to unify configuration management by using a common language or platform.

Declarative, but what about interactive?

Adam Jacob’s opening talk on Breaking The Rules, as well as Ringo De Smet’s talk on Pulumi’s imperative on top of declarative approach made me think about the user experience aspect of declarative configuration. I’m still convinced that the declarative approach is ultimately the right one, or the only sustainable one to this day.

However, there is something to be said about offering different interfaces to this declarative layer: Pulumi is one example, providing an imperative API to manipulate it. npm add is another simple example of an imperative interface to a declarative configuration (packages.json).

Adam introduced the notion of “infrastructure as a model”, instead of code, to which code would only be one possible interface. Which reminded me of similar internal discussion at Tweag about projectional editors, which is in fact the very same idea but applied to programming in general. Textual code is just one view, but other views — such as a graphical interface — of the abstract representation of a program are also possible, and more desirable for some tasks (refactoring, renaming, diffing, visualizing, …). Typically, a diagrammatic view is often better than text at clarifying the relations and dependencies between the entities being managed (e.g. Terraform resources). To each task, its projection?

Nix is getting stronger every year

Ron Efroni, from Flox, often makes the half-joke that the number of GitHub stars on the Nix repos are following Moore’s law: they double every two years. So far, he’s mostly right.

I was impressed by the number of people flooding the Nix DevRoom at FOSDEM (the pictures at the beginning of this post), queuing in the corridor to wait for a free spot.

At CfgMgmtCamp, Bryan’s lightning talk on Nix took place on the main track during the very first morning. The (huge) auditorium was full, fitting several hundred people. When he asked who knew about Nix, I expected a solid 50 people to raise their hand. More than half, probably around two third of the audience, raised their hand!

Nix is vibrant

A lot of cool things are happening in the Nix world. The ecosystem is evolving at a fast pace. The NixOS Foundation, itself rather new, is structuring to handle the change of scale, for example with a dedicated Nix maintenance team. Flox just released the first version of their product. Domen Kozar introduced, which has gained traction rapidly. We presented Nixel, the starting point of a whole new user experience for writing Nix.


We got a lot of encouraging words after our various talks on Nickel. The project has been making steady progress in the long run, but the hard and technical work on the core language, the type system and the semantics of merging is naturally less palpable.

However, we have reached the point from where we can start (and have started, in fact) to harvest impactful outcomes, which is exciting. A handful of months’ work on the LSP by our own Gaga Ebresafe and we are already able to provide completions based on definitions, types and contracts, in-code documentation, code navigation and typechecking live in the editor, which is quite ahead of the current Nix experience. Although it’s only the beginning, we can already write basic derivations and modular and overridable development shells using Nixel. Daniele Palombi is doing an internship to bring incremental evaluation and self-adjusting computations to the interpreter, for fast re-evaluation of a configuration upon a small change.

Concluding words

It’s hard to say where the current momentum will bring us. Getting people excited and aware of a cool technology is not the same as making it established and used in production. There are many non-technical determining factors, even irrational ones. Nonetheless, it’s clear that infrastructure engineering is distinguished by the ebullience of its youth. It’s an exciting time to be around!

February 23, 2023 12:00 AM

February 22, 2023


Interface Files with Core Definitions

In this post, I will explain a new feature which allows GHC to store the Core definitions of entire modules in interface files. The motivation is severalfold: faster GHCi start-times, faster Template Haskell evaluation times and the possibility of using the definitions for other program analysis tasks.

This feature is implemented in MR !7502 and will appear in GHC 9.6. The work was funded by Mercury who benefited from a 40% improvement to compilation times on their Template Haskell heavy code base.


The point of this work was to be able to restart the compiler pipeline at the point just after optimisation and before code generation. (See the GHC chapter in the AOSA book for background on the compiler pipeline.) In particular, we wanted to be able to generate bytecode on demand for any module, as this can significantly speed up start times for projects in GHCi.

The compiler writes interface files (.hi files) to pass on information about a module that subsequent modules will need to know. Interface files contain information about what functions are defined by a module, their types, what a module exports and so on. They may contain different amounts of detail: for example, if you compile with optimisations turned on, the interface file will contain more information about certain bindings, such as their demand signatures or unfoldings. For more information about interface files, you can consult the wiki.

Adding Core definitions to interface files makes it possible to defer the choice of backend (between no code, static object code, dynamic object code, bytecode) until more is known about the necessary targets. In particular:

  • GHCi can quickly convert previously-generated Core definitions to bytecode when loading a module, rather than needing to run the full compiler pipeline to generate bytecode for many modules even if they have previously been compiled.

  • Cabal pessimises build times by building both static and dynamic objects under the assumption that you will eventually need dynamic objects to run Template Haskell splices. With the Core definitions at hand, we can delay this choice until we know for sure we need to do the work. Moreover, rather than compiling and linking object code we can interpret bytecode, which is typically a faster operation for TH splices.

  • The Core program can also be useful for whole program analysis tasks. For example, the external STG interpreter could read these interface files and convert the result into its own STG format for running on the STG interpreter.

Core Definitions in Interface Files

In GHC 9.6, the interface file format is extended with a new field which contains complete Core bindings for the modules. A new command-line flag is available to enable this:


Write the Core definitions of the functions defined in this module to the interface file after simplification.

If you compile a module with -fwrite-if-simplified-core then you will see a new section called “extra-decls” when you dump the contents of an interface file with --show-iface. This section of the interface contains all the Core bindings of the program.

> ghc-9.6 --show-iface FAT.hi
f = GHC.Types.C# 'f'#
a = GHC.Types.C# 'a'#
t = GHC.Types.C# 't'#

The serialised program is a Core program. Using the Core representation is convenient for a number of reasons:

  • We already have the ability to serialise Core.
  • Constructing bytecode from Core is not a very expensive operation.
  • Other backends can generate code from the Core.

The program is serialised after simplification. This means that the interface file for a module compiled without optimisations will contain unoptimised bindings, whereas the interface file for an optimised module will contain optimised bindings.

Template Haskell evaluation via bytecode

GHC always uses the bytecode interpreter to interpret a Template Haskell splice for the current module. On the other hand, dependent home package modules can be handled in two different ways:

  • Object files: link the object files together using the system linker, and pass the resulting library to the interpreter.
  • Bytecode: directly load the already compiled bytecode into the interpreter.

By default, GHC in --make mode uses the former method, whereas GHCi uses the latter. GHC 9.6 introduces new flags to change this behaviour: -fbyte-code-and-object-code and -fprefer-byte-code.

Generating bytecode

In order to generate both the bytecode and object file linkables, there is a new flag -fbyte-code-and-object-code:


Produce both bytecode and object code for a module. This flag implies -fwrite-if-simplified-core.

Using -fbyte-code-and-object-code without -fwrite-if-simplified-core would recompile your project from scratch each time you compile it, due to lacking the Core definitions in the interface. Having one flag enable the other avoids this.

Compare -fbyte-code-and-object-code with the existing -fobject-code and -fbyte-code flags, which don’t allow a combination:


Produce object code for a module. This flag turns off -fbyte-code-and-object-code so using -fobject-code in an OPTIONS_GHC pragma will ensure that bytecode is never produced or used for a module.


Produce bytecode for a module. This flag turns off -fbyte-code-and-object-code so using -fbyte-code means to only produce bytecode for a module.

When using -fbyte-code-and-object-code, the recompilation checker checks for the presence of an interface file with Core definitions, recompiling the module if one doesn’t exist.

If -fbyte-code-and-object-code is not enabled then even if you have an interface file with the Core program the bytecode isn’t loaded for a module. This prevents the situation where you first compile an interface for module A and then later recompile it with -fobject-code, then you don’t want to make the bytecode available for later modules if they use -fprefer-byte-code.

Using bytecode

When passed the new -fprefer-byte-code flag, GHC will use the bytecode interpreter whenever bytecode is available (including in --make mode).


Use bytecode rather than object files for module dependencies when evaluating Template Haskell splices. This flag affects the decision we make about which linkable to use at the splice site only. It doesn’t have any effect on which linkables are generated from a module.

In addition, if you prefer bytecode, then the compiler will automatically turn on bytecode generation if it needs code generation when using -fno-code.

There are a couple of reasons why you might want to use these flags:

  • Producing object code is much slower than producing bytecode, and normally you need to compile with -dynamic-too to produce code in the static and dynamic way, the dynamic way just for Template Haskell execution when using a dynamically linked compiler.

  • Linking many large object files, which happens once per splice, can be very expensive compared to linking bytecode. Mercury saw an overall 40% decrease in compilation times when compiling their code base using -fprefer-byte-code due to the large amount of Template Haskell splices they use.

There’s also some reasons why you might not want to use these flage:

  • Enabling -fbyte-code-and-object-code generates bytecode as well as normal object files, so it could make your compilation slower if you are producing static object files, dynamic object files and bytecode for each module.

  • These flags will run the bytecode interpreter with optimised programs, something which wasn’t possible before GHC 9.6 so there are probably some lurking bugs. We have already fixed a large number of these issues but we’re not confident yet that we have found them all.

You probably want to use both -fprefer-byte-code and -fbyte-code-and-object-code together. If you use -fprefer-byte-code alone, then bytecode will not necessarily be available to use. If you use -fbyte-code-and-object-code alone, then the bytecode which you generate will never be used. This may not be an issue (as the bytecode is generated lazily), but it’s something to keep in mind.

Trying it out

In order to use the bytecode interpreter to evaluate the Template Haskell splices in your project, enable the necessary options with the following section in your cabal.project:

  ghc-options: -fprefer-byte-code -fbyte-code-and-object-code

This will pass these two options when compiling all the packages local to your project. If you want to always pass these options even when compiling external dependencies you can instead write:

package *
  ghc-options: -fprefer-byte-code -fbyte-code-and-object-code


Including the Core program in an interface file is a simple but powerful feature. To be maximally effective, more work is necessary in the ecosystem to use them when appropriate to restart compilation, but this contribution makes the important first steps. For example, HLS already implements a similar feature to improve reload times but in future GHC versions they can instead use this native support.

Well-Typed are actively looking for funding to continue maintaining and enhancing GHC, HLS and Cabal. If your company relies on Haskell, and you could support this work, or would like help improving the developer experience for your Haskell engineers, please get in touch with us via!

by matthew at February 22, 2023 12:00 AM

February 20, 2023

Monday Morning Haskell

Pushing our Container to AWS ECR

In the first part of this blog series we saw how to create a local docker image containing a simple web server program. In order to run this server remotely, we have to upload this image somewhere to deploy it.

One service that lets us deploy docker images is Amazon Web Services (AWS). In this article, we're going to take the first step, and walk through the process of publishing our container image to the AWS Elastic Container Registry (ECR). Next time around, we'll see how to actually deploy our application using this image.

In principle, publishing the image is a simple task. But in my experience with AWS, the processes and documentation just aren't quite as clear as one would like them to be. There tend to be a lot of branches in their tutorials, and it's often not clear which path is the right path. The sheer amount of AWS-specific terminology can get extremely confusing, and this can make it hard to know if you've satisfied the prerequisites for the tutorial.

So in this article I'm going to be as explicit as possible, and include a video at the end so you can follow along. Here's the high level overview:

  1. Create an AWS account
  2. Create an ECR Repository
  3. Install the AWS Command Line Interface
  4. Login using the CLI
  5. Push the container using Docker

Create an AWS Account

First of course, you need to create an account with Amazon Web Services. This is a separate account from a normal Amazon account. But a massive gotcha is that you should not use the exact email address from your Amazon account. This can cause a weird loop preventing you from logging in successfully (see this Stack Overflow issue).

If you have Gmail though, it should work to use the '+' trick with email aliases. So you can have `name@gmail.comfor your Amazon account` for your AWS account.

Create an ECR Repository

Next you'll need to login to your account on the web portal and create an ECR repository. To do this, you'll simply click the services tab and search for "Elastic Container Registry". Assuming you have no existing repositories, you'll be prompted with a description page of the service, and you'll want to find the "Get Started" button under the "Create a Repository" header off in the top right corner.

The only thing you need to do on the next page is to assign a name to the repository. The prefix of the repository will always have the format of {account-id}.dkr.ecr.{region}, where the account ID is a 12-digit number.

If you want, you can also set the repository as public, but my instructions will assume that you'd made a private repository. To finish up, you'll just click the "Create Repository" button at the bottom of the page. This part is also covered in the video at the bottom if you want to see it in action!

Install the AWS CLI

Our next few actions will happen on our local command line prompt. To interact with our AWS account, we'll need to install the AWS Command Line Interface. To install these tools, you can follow this user guide. It is fairly straightforward to follow once you select your operating system. You know it's succeeded when the command aws --version succeeds on your command line.

Login Using the CLI

Now assuming you created a private repository, you'll need to authenticate on the command line. The first step in this process is to create an access key. You can do this from the web portal by clicking your account name in the top right corner to open up a menu and then going to the "Security Credentials" page. There's a section for "Access Keys" about midpage, and you'll want to use "Create Access Key".

If you do this as a "root" user, AWS will warn you that this is not the advised practice and you should instead create such keys as an "IAM User". But it is possible to do use root for demonstration purposes.

You'll want to copy the "Access Key ID" and the key itself. The latter must be copied or downloaded before you leave the page (you can't come back to it later).

You can then login using the aws configure command in your command line terminal. This will ask you to enter your access key ID and then the key itself, as well as the region.

Now that you're authenticated with AWS, we have to allow AWS to login to Docker for us. The following command would give us the Docker password for AWS in the us-west-2 region:

>> aws ecr get-login-password --region us-west-2

We can pipe this password into the docker login command and connect to the repository we created with this command, where you should substitute your region and your account ID.

>> aws ecr get-login-password --region {region} | \
  docker login --username AWS --password-stdin {account-id}.dkr.ecr.{region}

Note how you actually do not need the repository name for this command! Just the prefix formed by your account and the region ID.

Pushing the Image

Now that we're authenticated, we just need to push the container image. We'll start by reminding ourselves of what our image ID is:

>> docker images
quiz-server latest b9eab6a22b12 ...

The first thing we need to do is provide a "tag" for this image corresponding to the remote ECR repository we created. This requires the image ID and the full repository URI. We'll also attach :latest to indicate that this is the most recent push. Here's the specific command I used for my IDs:

>> docker tag b9eab6a22b12

Here's a more generic command template:

>> docker tag {image-id} {account-id}.dkr.ecr.{region}{repo-name}:latest

Finally, we just need to push it using this new repository/tag combo! Here's what it looks like for me:

>> docker push

And more generically:

>> docker push {account-id}.dkr.ecr.{region}{repo-name}:latest

You should then be able to see your image if you head to your ECR dashboard!

Video Walkthrough

If you want to see all this in action, you can head to YouTube and take a look at the video walkthrough! If you are enjoying this series, make sure to subscribe to our monthly newsletter!

by James Bowen at February 20, 2023 03:30 PM

February 13, 2023

Matt Parsons

Garmin Fenix 6 Pro vs Apple Watch SE

I’ve been pondering a smart watch for tracking more of my daily activity and health, as well as getting some metrics on my non-cycling workouts. I had narrowed the field down to the Fitbit Charge, Apple Watch SE, and Garmin Fenix 6. The Fenix seemed like the winner - awesome battery life, navigation, and even music on board. However, even at a discount, it’s $450, so it must be awesome to be worthwhile.

tl;dr: Garmin’s anticooperative features are a dealbreaker.

  • Update History
    • 2023-02-17: I got an Apple Watch and the comparison has me feeling like maybe I should keep the Garmin. As a result, this may become more of a “fitness watch review and comparison post.”

Garmin Fenix 6s

The Good

The watch comes in three sizes, and the 6s doesn’t look too big on my wrist, despite having small wrists.

The Body Battery and sleep tracking are pretty good. Over the past three weeks or so of owning the Fenix, the Body Battery has dialed in to how I’m physically feeling, and I can pretty well tune my workouts and daily activities based on the recommendations.

The pulse oximeter has been interesting for tracking sleep - when I wear a nose strip, my blood oxygen level is higher overnight. This is actionable and useful information, though I already knew that a nose strip helps a lot for sleep quality.

The device appears to be very rugged. The lack of touch screen is a plus to me - I don’t like the idea of accidentally touching or disabling things. The button interface is a bit clunky, but overall feels reliable and secure, even when operating with gloves.

The battery life is excellent, so long as you’re not downloading music or maps or navigating.

The Bad

Garmin has a reputation for being anticooperative. Their devices, software, etc, have always blocked features that would be extremely useful if they think they can sell you anothe device or service. They desperately want to be your only interface into the fitness world. In that sense, they’re a lot like Apple.

But unlike Apple, you don’t build your lifestyle around your fitness devices. I own a Wahoo bike computer and a Withings smart scale. Garmin explicitly is not compatible with these devices. Garmin explicitly chooses to not support syncing workouts from Strava to calculate “training readiness,” and they explicitly don’t sync weight from Apple Health. They want you to record workouts using Garmin Connect, and they want you to buy a Garmin smart scale.

They used to support API workout uploads, so you could sync from Strava to Garmin using a third party. However, they recently disabled this. And - even if you could upload Strava workouts - they don’t count for the “Training Readiness” score that the app gives you.

Some more annoyances:

  • You can’t do firmware update over the phone. You need to plug in to a Windows or Mac computer. My computers are all Ubuntu, so this is a bad deal. I’ve read some reports that WINE can do it, but I’m not sure I want to put the firmware on a $450 device up to WINE and Garmin’s programmers.
  • The music download feature is broken. I tried to download a playlist to the watch, and it burned the battery from 60% to dead in about 20 minutes. It didn’t even download anything.
  • Activity tracking and options are overwhelming, but “track activites automatically” doesn’t work - it never registered a bike ride as a bike ride.

The Choice

So, Garmin presents a bargain:

The Fenix is a fantastic watch for adventure and activity. But for you to really get the full benefit, you need to switch to Garmin Connect, and Garmin smart scale, and use Garmin for everything. Yeah, that includes tossing your Wahoo bike computer and buying a Garmin.

Once you do that, you’ll get Training Status, Body Battery, and other useful training tools.

The bargain is… expensive. The Garmin smart scale is more expensive than the Withings by a large amount, and per reviews, it’s not even as good. The Garmin bike computers are way worse than Wahoo, IME, and the cheaper ones have arbitrary software locks (like not being able to sync a route over Bluetooth). Furthermore, you know you’re signing up for a closed ecosystem that will never cooperate with anything else.

This sort of ecosystem lock is comparable to what Apple is trying to do, but at a much smaller niche. For the most part, Apple’s ecosystem lock is about adding functionality. One Apple device is quite useful, and doesn’t really do anything to explicitly block compatibility. Sure, the Apple Watch only works on iPhone. But you can use a Fitbit or any other smart watch with the iPhone. You don’t need a Macbook to use an iPhone or iPad.

Garmin, on the other hand, explicitly denies compatibility with other devices. Wahoo and Garmin don’t talk to each other. Wahoo is perfectly happy to talk to basically every other company - but Garmin won’t allow it. Garmin won’t even read data from Apple Health, if it can provide a slight edge to selling a product.

The Alternatives

DC Rainmaker has a great guide to GPS sport watches. I selected the Fenix as the “best choice” in the Hiking/Adventure watch, since I see myself as an “adventurer.”

The COROS Vertix is the other recommendation in the category. That gives you the EvoLab “Training Load Metrics”, which should be a nice competitor to “Body Battery.” It also gives a much better battery life - 60 hours - instead of the 24/36 from Garmin. However, at $500, it also seems like more of a “specialized/primary tool for runners/hikers,” and realistically, I’m a cyclist - my bike computer will be my go-to device for most activity I do.

The Apple Watch SE seems to give most of the same featureset as the Garmin, with the main downside being battery life. However, the watch charges pretty quickly, and a habit of charging every day is pretty much what I’ve settled into with the Garmin. It’s also one of the cheapest options to provide mapping and navigation, and the WatchOS platform supports a bunch of apps like RideWithGPS and Hiking Project.

The Fitbit Charge is considerably cheaper, and is only a basic activity/sleep tracker. If you don’t want any smartwatch features, then this may be a good bet.

I’m boxing up and returning the Fenix today, and I’ve got an order in for the Apple Watch SE.

2023-02-17: The Apple Watch SE

So, I haven’t actually returned the Fenix yet. I boxed it up, printed the return label, but then loitered at the bike store instead of going to the UPS store to drop it off. It’s sitting in my bag, just waiting for packaging.

I picked up an Apple Watch SE on Monday the 13th. At $249, it’s about $200 cheaper than the Fenix 6s. These devices are very different. Really, they’re not in the same market category at all. The Apple competitor to the Garmin is probably the new Apple Watch Ultra, with longer battery life and more outdoor features. But at $800 that’s nearly twice what I paid for the Garmin, and I’m not that excited about having a watch.

I think the differences actually make for an interesting compare-and-contrast, which is why I’m extending this post with the review of the Apple Watch.


It’s an Apple product, you really need to start with the aesthetics.

The Apple Watch SE has a minimal interface compared to the Garmin. Despite having the same “watch face” size, the border of the Garmin is significantly larger. The Apple Watch looks like a sleek, chic, modern urban accessory. The variety of bands available are also cool - the fabric fitness band that I selected is much more comfortable than the rubber/plastic band that came with the Fenix stock.

Apple provides much more choice and flexibility with the watch faces. Being able to select photos as a background is really nice touch. I love looking down and seeing my cat doing something silly, or a beautiful photo memory of some bike ride or camping trip.

Aesthetics are deeply personal. I like the Garmin Fenix - the rugged and sporty look works for me. However, the Apple Watch looks better when I’m not outdoors, especially if I’m dressed up (to the extent that a software engineer in Colorado ever dresses up).

Ease of Use

The Apple Watch SE is extremely easy to use. Apple provides a touch screen, a button, and a scroll wheel button. The scroll wheel is a nice touch when gloves make it annoying to use the touch screen, but there’s only so much it can do.

The real winner here is the gesture feature and Siri voice control. You can use Siri to start workouts, stop workouts, start timers, make reminders, etc - almost everything you’d expect your phone to do. The gestures are also very cool - you can setup the watch to do something if you pinch your fingers or clench your fist. This accessibility feature makes using the watch one handed significantly easier.

The Garmin, on the other hand, relies on physical buttons. The experience is slightly clunkier - starting a workout is much more involved than “Hey Siri, start a yoga workout” or “Hey Siri, start an indoor cycling workout.” The plus side of this is that you won’t get mistaps from the touch screen or misclicks. The buttons on the Apple Watch are somewhat sensitive, and I’ve “misclicked” multiple times in the few days that I’ve owned it. There is no concern about gloves - buttons always work.

In a “pure ease of use” contest, the Apple Watch wins easily. However, the reliability and security of physical buttons is an important feature, particularly in outdoor contexts.


The Apple Watch series all share the same OS, and according to Amazon’s best sellers for smartwatches, is the most popular smart watch platform. Even beyond the impressive built-in utility from Apple, the third party support is fantastic. Most cycling and fitness apps support the Apple Watch to some degree.

Garmin also has an App Store of sorts, though the apps you can really put on a Garmin are much more limited. Apple Watch has more fitness apps available, and far more non-fitness apps. I can open the garage door with the Apple Watch. I can also use it as a walkie-talkie with my friends that have Apple Watches. I can record voice memos, deal with my car, send text messages, etc. It’s easier to see notifications and act on them.

The Garmin is hindered here by being an “adventure fitness watch” and not a proper “smart watch.” So the comparison isn’t really fair. Fortunately, there are some points where the Garmin is clearly superior - so let’s dive into those.

Battery Life

The Apple Watch has a relatively short battery life at only 18 hours. This means you will likely need to charge it several times per day. I’ve got a charger setup on my desk, where I bring it to a full charge in the morning (once I’m done with my morning routine and sleep). Then, I’ll charge it again when I am showering. This is usually enough to keep it working well enough for sleep tracking at night and some light activity tracking during the day.

The Watch appears to require a 20W USB-C fast charger. I wasn’t able to get it to charge from my laptop’s USB-C port, nor the USB-C ports on my docking station. This is an inconvenience - I’m not sure I’d easily be able to charge it on a long hiking or bikepacking trip.

The Garmin’s battery life is far better. Not only that, but the charging speed is faster. The “charge while showering” habit is all that’s necessary to bring it up to a full charge, even if I’ve forgotten to charge it for a few days. As a result, there’s less stress around the battery. With the Apple Watch, I feel like I’m needing to constantly manage the habit of keeping it charged, which is really more attention than I want to pay to a device. The Garmin is much more forgiving.

The Garmin is also the clear winner for longer trips. While I would not bring the Apple Watch along for a multiday bikepacking expedition, the Garmin would definitely come along.

Fitness Information

The Garmin takes much finer grained fitness information, and does much more with it. But the Apple Watch does more for “health” - loudness levels, walking balance, etc.

Heart Rate

The Apple Watch takes heart rate readings periodically - about every 4 minutes according to my Apple Health information. The Garmin tracks much more frequently when not in a workout - it appears to be continuously reporting a Heart Rate number, although that is still probably sampling only ever several seconds.

Both track continuously during a workout, and the Apple Watch provides a nicer view into your heart rate zones.

Sleep Tracking

Both devices offer sleep tracking. I haven’t compared them directly, but both seem fairly good. The Garmin occasionally thinks I slept longer than I did, which is easy to correct. The Apple Watch doesn’t appear to have any ability to edit the overall sleep duration, but I also haven’t seen it be wrong yet, so that’s promising.

On that brief experience, the Apple Watch appears to have a more reliable algorithm, so I’m tempted to trust it more.

However, the Garmin provides a “stress” measurement during sleep, which can measure the quality of sleep. Apple measures the quantity of sleep and the time in various sleep stages, but it doesn’t try to tell you what that means. Garmin takes into account “stress” during sleep and incorporates that into a “Sleep Score.” In my experience, the “Sleep Score” did a pretty damn good job of predicting how I’d feel during the day. It did seem to notice when I had caffeine too late in the day, or even a single alcoholic beverage.

The Apple Watch may be providing the same raw data, but I don’t know how to interpret it. The Garmin provides a much better reflection point.

Recovery Status

The Apple Watch does not attempt to provide a picture of your recovery status in the same way that Garmin does. Garmin tracks your “stress level” in addition to your heart rate and presents a “Body Battery” score indicating your relative readiness. In my experience, this number tracks pretty well with how I’m actually feeling. I haven’t had a time where I saw the number, checked in with my feelings, and thought “wow that’s wrong.” Much more often, I’d see a low number, reflect, and realize how tired I was.

The Apple Watch does track HRV, though it appears to periodically take measurements throughout the day. This approach is inherently pretty noisy. This post my Marco Altini goes into detail on the best way to use the HRV data from the Apple Watch, which is tricky.

The Apple Watch tracks “heart rate recovery,” but the metric is pretty limited. It only works if you end a workout with your heart rate near peak. So if your workout doesn’t get to a peak heart rate, you won’t get a reliable number. You also won’t get a reliable number if your workout has cool-down.

In terms of providing feedback for training, the Garmin is far better. Now, that doesn’t mean that you can’t get useful feedback with the Apple Watch. Most cardio-based training apps (Strava, Xert, etc) will provide some “fitness/fatigue/form” numbers you can use to figure out how you’re feeling. This is often “good enough,” especially if you’re sufficiently embodied to just “know” how sleepy and stressed you are.

Those models are often limited by only taking into account workout data. And that needs to be calibrated against some training parameters, like Functional Threshold Power or Maximum Heart Rate - so if you expect those are wrong or off (which, coming off of a surgery, they definitely are for me), then you shouldn’t expect them to be too accurate. The Garmin’s more holistic view of stress and fitness seems like a really useful tool for balancing actual recovery and not just training inputs.

This is a big deal. I tend to take on about as much stress as I can in my life, and I’m not nearly as embodied as I would like. If the Garmin can help me attune to my own sensations better and provide more actionable recovery feedback, then that is very valuable to me.

Does Apple Watch have a third party app that mimics Body Battery? A three year old Reddit topic provides several options. The most relevant one appears to be Training Today and a few more specific sleep tracking apps. The app appears to be pretty good - I just downloaded it, and it loaded my data and said “You’re on the more tired side. Keep to Z1 or Z2.” This is fair - I just did a 2x20 at 100% of FTP workout yesterday, and that’s about what I’d expect. The free app gives the basic data you need, and only $20 lifetime for more advanced features is great. Garmin’s information is more detailed

Training Status

I haven’t actually used this, because Garmin’s lack of interoperability means that I haven’t recorded any real workouts with the watch. If you do use a Garmin device to track workouts, then Garmin gives you information about your VO2 max, and provides some data about how you’re training is going. I can see this being effective, especially for runners, but cyclists tend to use power as the source-of-truth for training, and most training apps/websites provide that information pretty well.


The Garmin can connect to power meters, external heart rate monitors, and other sensors. The Apple Watch cannot. So the Garmin is a better “fitness monitoring” device.

However, the Apple Watch is better for actually doing workouts. Siri’s voice control is super nice for starting/stopping workouts and setting timers, all of which are pretty dang useful during a workout. The scroll wheel is a better interface for most things than Garmin’s clunky up/down button. The touch screen isn’t great when sweaty, but it’s not a disaster, either.

For serious training, the Garmin wins, but for the more casual user, the Apple Watch is probably a better fit. If you’re a cyclist and into “serious training,” you probably have a dedicated bike computer anyway, which does the job much better than any watch. But if you’re also interested in running, snowshoeing, hiking, skiing, etc, then the bike computer is obviously a worse fit.

Garmin vs Apple

I’ve had the Apple Watch for about a week, and I used the Garmin for about two weeks before deciding to write this up and switch to the Apple Watch. The Apple Watch is $200 cheaper and has many more non-fitness features. Even if you spend the $5 on Autosleep and $20 on Training Today, you’re $175 cheaper - and now the Garmin’s only real advantage is the longer battery life.

Thus, the question: Does it make sense to pay $175 more for a much nicer battery, and also lose a ton of really good features?

For me, no. In large part, that’s because I’m a cyclist, and I already have an optimal setup for tracking cycling workouts - a bike computer, a chest strap heart rate monitor, and a power meter.

If I didn’t have that stuff, then the Garmin becomes much more interesting. The Garmin can talk directly to a power meter when recording workouts, and is a heart rate monitor. It can also provide navigation, routes, data pages for workouts, and other good features. Changing anything on the bike would be a pain in the ass, though.

The Apple Watch would rely on using my phone to record workouts, since it cannot talk directly to a power meter. However, it also can’t natively broadcast heart rate to other apps - there’s a third party app, but it only has 2.9 stars - maybe unreliable?

So, to sum everything up:

  • The Garmin Fenix is a superior sport/fitness watch, if you use it as your central device for fitness tracking, and if a watch is a better form factor than a bike computer
  • The Apple Watch SE is better in every other way, aside from battery life
  • The Apple Watch SE is cheap enough that you can get a Wahoo Elemnt BOLT and the watch, which is a better combination for cycling than just the Garmin

February 13, 2023 12:00 AM

GHC Developer Blog

GHC 9.6.1-alpha3 is now available

GHC 9.6.1-alpha3 is now available

bgamari - 2023-02-13

The GHC team is very pleased to announce the availability of GHC 9.6.1-alpha3. As usual, binaries and source distributions are available at

Beginning with GHC 9.6.1, GHC can be built as a cross-compiler to WebAssembly and JavaScript. This is an important step towards robust support for compiling Haskell to the Web, but there are a few caveats to be aware of in the 9.6 series:

  • Both the Javascript and WebAssembly backends are still at an early stage of development and are present in this release as a technology preview

  • Using GHC as a cross-compiler is not as easy as we would like it to be; in particular, there are challenges related to Template Haskell

  • GHC is not yet run-time retargetable; a given GHC binary targets exactly one platform, and both WebAssembly and JavaScript are considered platforms for this purpose. Cross-compilers must be built from source by their users

We hope to lift all of these limitations in future releases.

Additionally, 9.6.1 will include:

  • Significant latency improvements in the non-moving garbage collector

  • Efficient runtime support for delimited continuations

  • Improvements in compiler error messages

  • Numerous improvements in the compiler’s memory usage

See the release notes for a comprehensive accounting of changes in this release.

As always, one can find a migration guide to aid in transitioning from older releases on the GHC Wiki. We have also recently started extending our release process to cover a wider set of Linux distributions. In particular, we now offer Rocky 8 and Ubuntu 20.04 binary distributions which cover RedHat-derivative and distributions using older glibc releases (namely 2.27), respectively.

Please do give this release a try and open a ticket if you see anything amiss.


  • Ben

by ghc-devs at February 13, 2023 12:00 AM

February 11, 2023

Abhinav Sarkar

Implementing Co, a Small Language With Coroutines #3: Adding Coroutines

In the previous post, we wrote the interpreter for basic features of Co, the small language we are building in this series of posts. In this post, we explore and implement what makes Co really interesting: support for lightweight concurrency using Coroutines.

This post was originally published on

This is the third post in a series of posts:

  1. Implementing Co #1: The Parser
  2. Implementing Co #2: The Interpreter
  3. Implementing Co #3: Adding Coroutines
  4. Implementing Co #4: Adding Channels
  5. Implementing Co #5: Adding Sleep

In this and next two posts, we add support for the following features to the Co interpreter:

  • yield statement to yield the current thread of computation (ToC).
  • spawn statement to start a new ToC.
  • First class channels with operators to send and receive values over them.
  • sleep function to sleep the current ToC for a given number of milliseconds.

Let’s Co!


Coroutines@1 are computations that support Cooperative multitasking@2. Unlike ordinary Subroutines that execute from start to end, and do not hold any state between invocations, coroutines can exit in the middle by calling other coroutines, and may resume later from the same point while holding state between invocations. They do so by yielding the control of the current running thread.

Subroutines vs. Coroutines <noscript>Subroutines vs. Coroutines</noscript>
Subroutines vs. Coroutines

The above diagram compares the execution of a subroutine and a coroutine, invoked from a caller1. The rectangles represent instructions, starting at top and ending at bottom. The arrows represent the flow of control.

The subroutine executes from start to end when called. The coroutine can exit in the middle by yielding, and can resume later from the same point. The coroutine state is saved automatically at the point of yielding, and restored when the coroutine resumes. Note that the coroutine may not be resumed, in which case the rest of the coroutine never executes.

Coroutines in Various Languages

Many languages have support for coroutines, either built-in or through libraries or plugins. Here are two examples in Kotlin and Python2:

fun main() = runBlocking {
    launch { // launch a new coroutine and continue
        delay(1000L) // non-blocking delay for 1 second
        println("World!") // print after delay
    println("Hello") // main coroutine
// prints "Hello World!"

Coroutines in Kotlin

import asyncio

async def say_after(delay, what):
    await asyncio.sleep(delay)
    print(what, end="")

async def main():
    await asyncio.gather(
        say_after(1, 'World!\n'),
        say_after(0, 'Hello '))
# prints "Hello World!"

Coroutines in Python

Now, for a different kind of example, the following JavaScript code prints numbers 11–16 and 1–4 interleaved, using Generators3:

function* printNums(start, end) {
  for (let i = start; i < end + 1; i++) {

function run(...gens) {
  const queue = [...gens];
  while (queue.length != 0) {
    const p = queue.shift();
    if (! {

run(printNums(11, 16), printNums(1, 4));
// prints numbers 11–16 and 1–4, interleaved.

Generators in JavaScript

The next example is in Co, and it has the same behaviour as the JavaScript example above, except we don’t have to write the function to schedule and run the coroutines. The runtime for Co—the Co interpreter—does that implicitly for us.

function printNums(start, end) {
  var i = start;
  while (i < end + 1) {
    i = i + 1;

spawn printNums(1, 4);
printNums(11, 16);

Coroutine example in Co

So how are coroutines implemented in Co? Let’s find out.

Implementing Coroutines

A coroutine is essentially an Environment@6 and a Continuation@7. The environment is the state of the executing code at the point of yielding. The continuation is the code to be executed when the coroutine is resumed later. If we can capture the environment and the continuation, we can implement coroutines.

Different implementations of coroutines capture the environment and the continuation in different ways4:

  • We can capture the environment as the current stack and the continuation as the pointer to the next instruction to be executed at the level of machine code. This is how coroutines are implemented in C and C++.
  • We can transform the code into a state machine as a large switch statement, and use variables to store the environment. This is how Go threads are implemented in the Clojure core.async5 library.
  • We can capture the environment and the continuation as a Closure. To do this, we need to first transform the code into Continuation-passing style (CPS), so that we have the handle to the continuation at every point in the code. This is how we are going to implement coroutines in Co.

Let’s learn what CPS is, and how we can use it to implement coroutines.

Continuation-Passing Style

In the usual direct programming style, we write one statement or function call after another, as a sequence of steps to execute. There is another way of thinking about program execution: after returning from executing one statement/function, the rest of the program—which can be thought of as a big statement/function itself—is run. In CPS, this is made explicit: each statement/function takes the rest of the program that comes after it as an argument, which it invokes explicitly. For example, if we have a program to get the recommendations for a user and print them, written in direct style like this:

function getUserRecommendations(userId) {
  let user = getUser(userId);
  let friends = getFriends(user);
  let recommendations = getRecommendations(friends);
  recordRecommendations(userId, recommendations);
  return recommendations;

function main() {
  let recommendations = getUserRecommendations(123);

It can be converted to an equivalent CPS program like this:

function getUserRecommendationsCPS(userId, cont) {
  getUserCPS(userId, (user) => {
    getFriendsCPS(user, (friends) => {
      getRecommendationsCPS(friends, (recommendations) => {
        recordRecommendationsCPS(userId, recommendations, () => {

function mainCPS() {
  getUserRecommendationsCPS(123, (recommendations) => {

We see how each function takes the rest of the program after it captured as a function, as a parameter, and calls it explicitly to further the flow of the program. Instead of returning the recommendations, the getUserRecommendationsCPS function now takes a function as an additional parameter, which it calls with the recommendations at the end of all the processing. Same for all the other functions invoked in the program. These functions passed as arguments are known as continuations because they continue the execution of the programs when called, and hence this style is called the continuation-passing style. The cont function is the continuation here.

The rest of the functions can be written in CPS like this:
function getUserCPS(userId, cont) {
  let user = getUser(userId);

function getFriendsCPS(user, cont) {
  let friends = getFriends(user);

function getRecommendationsCPS(friends, cont) {
  let recommendations = getRecommendations(friends);

function recordRecommendationsCPS(userId, recommendations, cont) {
  recordRecommendations(userId, recommendations);

So, what is the point of all this? Why transform code into CPS? Since, in CPS the rest of the program is passed as a function, a program can itself explicitly manipulate the flow of control of the program. This lets us do things like6:

  • Returning early from a function by calling the continuation with the return value, and not executing the rest of the function.
  • Implementing exceptions by passing two continuations: one for the normal flow of the program, and another for the exceptional flow.
  • Implementing non-deterministic programs by passing continuations for backtracking to previous states of the program.
  • Converting potentially stack-blowing recursive programs into iterative programs by passing the continuation as a parameter to the recursive function.
  • Suspending the execution of the program by storing the continuation, and resuming it later.

We can now begin to see how CPS can be used to implement coroutines.

Continuation-Passing Style in Haskell

It is straightforward to translate the above program into Haskell:

getUserRecommendations :: Monad m => Int -> m Recommendations
getUserRecommendations userId = do
  user <- getUser userId
  friends <- getFriends user
  recommendations <- getRecommendations friends
  recordRecommendations userId recommendations
  return recommendations

main :: IO ()
main = getUserRecommendations 123 >>= print

And the CPS versions:

getUserRecommendationsCPS ::
  Monad m => Int -> (Recommendations -> m a) -> m a
getUserRecommendationsCPS userId cont =
  getUserCPS userId $ \user ->
    getFriendsCPS user $ \friends ->
      getRecommendationsCPS friends $ \recommendations ->
        recordRecommendationsCPS userId recommendations $ \_ ->
          cont recommendations

getUserCPS :: Monad m => Int -> (User -> m a) -> m a
getUserCPS userId cont = getUser userId >>= cont

getFriendsCPS :: Monad m => User -> (Friends -> m a) -> m a
getFriendsCPS user cont = getFriends user >>= cont

getRecommendationsCPS ::
  Monad m => Friends -> (Recommendations -> m a) -> m a
getRecommendationsCPS friends cont =
  getRecommendations friends >>= cont

recordRecommendationsCPS ::
  Monad m => Int -> Recommendations -> (() -> m a) -> m a
recordRecommendationsCPS userId recommendations cont =
  recordRecommendations userId recommendations >> cont ()

mainCPS :: IO ()
mainCPS = getUserRecommendationsCPS 123 $ print

We can immediately notice a pattern in the type signatures of the functions above: they are all of the form:

f :: Monad m => b -> (a -> m r) -> m r

It is indeed a known pattern, and is captured by the ContT type:

newtype ContT r m a =
  ContT { runContT :: (a -> m r) -> m r }

Turns out, the ContT type is a monad transformer, and we can use it to write the above CPS program in a more concise way7:

getUserRecommendationsCont ::
  Monad m => Int -> ContT r m Recommendations
getUserRecommendationsCont userId = do
  user <- getUserCont userId
  friends <- getFriendsCont user
  recommendations <- getRecommendationsCont friends
  recordRecommendationsCont userId recommendations
  return recommendations

getUserCont :: Monad m => Int -> ContT r m User
getUserCont userId = ContT (getUser userId >>=)

getFriendsCont :: Monad m => User -> ContT r m Friends
getFriendsCont user = ContT (getFriends user >>=)

getRecommendationsCont ::
  Monad m => Friends -> ContT r m Recommendations
getRecommendationsCont friends =
  ContT (getRecommendations friends >>=)

recordRecommendationsCont ::
  Monad m => Int -> Recommendations -> ContT r m ()
recordRecommendationsCont userId recommendations =
  ContT $ \cont ->
    recordRecommendations userId recommendations >> cont ()

mainCont :: IO ()
mainCont = runContT (getUserRecommendationsCont 123) print

So we have come full circle: we started with monadic code, and ended with similar monadic code, but with a different monad. So what did we gain from this transformation? Well, we can now use the callCC function provided by ContT.

Call with Current Continuation

callCC—short for “call with current continuation”—is a function that provides on-demand access to the current continuation at any point in the code, just like we had in the CPS version of the program. At the same time, by using ConT we can write the program again in the concise monadic style8.

The following example uses callCC to print the user recommendation twice, instead of once9:

getUserRecommendationsCont2 ::
  Monad m => Int -> ContT r m Recommendations
getUserRecommendationsCont2 userId = do
  user <- getUserCont userId
  friends <- getFriendsCont user
  recommendations <- getRecommendationsCont friends
  logRecommendationsCont userId recommendations
  callCC $ \cont -> do
    cont recommendations
    cont recommendations

mainCont2 :: IO ()
mainCont2 = runContT (getUserRecommendationsCont2 123) print

This is the power of CPS: it lets the programs manipulate the flow of control explicitly, and in some cases markedly, as we see in the next section.

From Continuations to Coroutines

Since continuations are functions, we can store them in data structures. This lets us pause the execution of a CPS program at a certain point, and resume it later from the same point. This is exactly what coroutines do.

To implement coroutines in Co, first we enhance the Interpreter monad to be able to capture the current continuation by adding the ContT monad transformer in the transformer stack:

newtype Interpreter a = Interpreter
  { runInterpreter ::
      ExceptT Exception
            (Either Exception ())
            (StateT InterpreterState IO))
    ( Functor,
      MonadBase IO,
      MonadState InterpreterState,
      MonadError Exception,

To be able to pause and resume the Co code being interpreted, we need to capture the current interpreter environment as well. The environment contains the bindings that the executing Co code sees at any given time. By capturing and later restoring the environment, the code execution resumes with same environment, and hence works correctly.

data Coroutine a = Coroutine
  { corEnv :: Env
  , corCont :: a -> Interpreter ()
  , corReady :: MVar TimeSpec

newCoroutine :: Env -> (a -> Interpreter ()) -> Interpreter (Coroutine a)
newCoroutine env cont = do
  ready <- newMVar =<< currentSystemTime
  return $ Coroutine env cont ready

The Coroutine data type contains the environment and the continuation. The newCoroutine function creates a new coroutine.

Next, we enhance the interpreter state to keep a queue of coroutines to be run.

data InterpreterState = InterpreterState
  { isEnv :: Env,
    isCoroutines :: Queue (Coroutine ())

initInterpreterState :: IO InterpreterState
initInterpreterState = InterpreterState <$> builtinEnv <*> newQueue

We use an IORef containing a min-priority queue to store the coroutines10. For now, we use it as a simple FIFO queue, but we will see in a later post how we use it to implement the sleep functionality in our interpreter.

type Queue a = IORef (PQ.MinPQueue TimeSpec a, TimeSpec)

newQueue :: MonadBase IO m => m (Queue a)
newQueue = do
  now <- liftBase currentSystemTime
  newIORef (PQ.empty, now)

Now that we know how coroutines are stored in the interpreter, let’s see how we schedule them.

Scheduling Coroutines

First step in scheduling coroutines is to write functions to enqueue and dequeue from a queue:

enqueueAt :: TimeSpec -> a -> Queue a -> Interpreter ()
enqueueAt time val queue = modifyIORef' queue $ \(q, maxWakeupTime) ->
  ( PQ.insert time val q,
    if time > maxWakeupTime then time else maxWakeupTime

enqueue :: a -> Queue a -> Interpreter ()
enqueue val queue = do
  now <- currentSystemTime
  enqueueAt now val queue

currentSystemTime :: MonadIO m => m TimeSpec
currentSystemTime = liftIO $ getTime Monotonic

dequeue :: Queue a -> Interpreter (Maybe a)
dequeue queue = atomicModifyIORef' queue $ \(q, maxWakeupTime) ->
  if PQ.null q
    then ((q, maxWakeupTime), Nothing)
    else let ((_, val), q') = PQ.deleteFindMin q
         in ((q', maxWakeupTime), Just val)

To use the min-priority queue as a FIFO queue, we use the current system time—which is a monotonically increasing value—as the priority of the values in the queue. This way, the coroutines are scheduled in the order they are enqueued.

The enqueueAt function enqueues the given value at the given time in the queue. The enqueue function enqueues the value at the current time, thus scheduling it to run immediately.

The dequeue function dequeues the value with the lowest priority from the queue, which in this case, is the value that is enqueued first.

The currentSystemTime function returns the monotonically increasing current system time.

Over these queuing primitives, we build the coroutine scheduling functions:

scheduleCoroutine :: Coroutine () -> Interpreter ()
scheduleCoroutine coroutine =
  State.gets isCoroutines >>= enqueue coroutine

runNextCoroutine :: Interpreter ()
runNextCoroutine =
  State.gets isCoroutines >>= dequeue >>= \case
    Nothing -> throwError CoroutineQueueEmpty
    Just Coroutine {..} -> do
      takeMVar corReady
      setEnv corEnv
      corCont ()

The scheduleCoroutine function takes a coroutine, and schedules it by enqueuing it in the coroutine queue in the interpreter state.

The runNextCoroutine function dequeues the next coroutine from the queue, and runs it. It first restores the environment of the coroutine in the interpreter state, and then runs the continuation of the coroutine. If the queue is empty, it throws a CoroutineQueueEmpty exception, which we add in the Exception data type:

data Exception
  = Return Value
  | RuntimeError String
  | CoroutineQueueEmpty

The runNextCoroutine function is the heart of the coroutine scheduling. It is called at the end of every function related to coroutines in the interpreter, and that’s how the coroutines are run one-after-another. Next, we see how we use these functions to implement the yield and spawn statements in Co.

Yield and Spawn

Let’s recall the program we used to demonstrate coroutines:

function printNums(start, end) {
  var i = start;
  while (i < end + 1) {
    i = i + 1;

spawn printNums(1, 4);
printNums(11, 16);

Running this program with the interpreter produces the following output:


As we see, the numbers printed by the printNums(11, 16) function call are interleaved with the ones printed by the printNums(1, 4) call.

This is how the code is interpreted:

  1. First, the definition of the function printNums executes. The function gets stored in the environment as a Function value.
  2. The spawn printNums(1, 4) statement executes. The spawn statement creates a new coroutine for the function call printNums(1, 4) and schedules it.
  3. The printNums(11, 16) function call executes, prints 11 and yields.
    1. The while loop executes, and the print statement prints the value of the variable i, which is 11 at this point.
    2. The yield statement executes. This creates a new coroutine for the rest of the call execution, and schedules it. The call execution suspends at this point.
  4. The runNextCoroutine function executes, which dequeues the coroutine for the printNums(1, 4) call, and runs it. This prints 1 and yields.
    1. The while loop executes, and the print statement prints the value of the variable i, which is 1 at this point.
    2. The yield statement executes. This creates a new coroutine for the rest of the call execution, and schedules it. The call execution suspends at this point.
  5. The runNextCoroutine function executes again, which dequeues the coroutine for the printNums(11, 16) call, and runs it. This prints 12 and yields.
    1. The call resumes after the yield statement. The while loop executes again, and the print statement prints the value of the variable i, which is 12 at this point.
    2. The function execution suspends at the yield statement again.
  6. The runNextCoroutine function executes again, which dequeues the coroutine for the printNums(1, 4) call, and runs it. This prints 2 and yields.
    1. The call resumes after the yield statement. The while loop executes again, and the print statement prints the value of the variable i, which is 2 at this point.
    2. The function execution suspends at the yield statement again.
  7. This back-and-forth process of suspension and resumption of function executions continues until the printNums(1, 4) call returns after printing the number 4.
  8. After that, the call printNums(11, 16) resumes to print the numbers and yields, again and again, until it returns after printing the number 16.
  9. Interpretation ends.

The diagram below depicts this process in abstract:

Spawning, yielding, and running coroutines <noscript>Spawning, yielding, and running coroutines</noscript>
Spawning, yielding, and running coroutines

With the understanding of how they work, let’s see how to implement the yield and spawn statements in Co.


First, we add the YieldStmt and SpawnStmt constructors to the Stmt data type:

data Stmt
  = ExprStmt Expr
  | VarStmt Identifier Expr
  | AssignStmt Identifier Expr
  | IfStmt Expr [Stmt]
  | WhileStmt Expr [Stmt]
  | FunctionStmt Identifier [Identifier] [Stmt]
  | ReturnStmt (Maybe Expr)
  | YieldStmt
  | SpawnStmt Expr
  | SendStmt Expr Identifier
  deriving (Show, Eq)

type Program = [Stmt]

Then, we enhance the stmt parser to parse these statements:

stmt :: Parser Stmt
stmt =
  IfStmt <$> (reserved "if" *> parens expr) <*> braces (many stmt)
    <|> WhileStmt <$> (reserved "while" *> parens expr) <*> braces (many stmt)
    <|> VarStmt <$> (reserved "var" *> identifier) <*> (symbol "=" *> expr <* semi)
    <|> YieldStmt <$ (reserved "yield" <* semi)
    <|> SpawnStmt <$> (reserved "spawn" *> expr <* semi)
    <|> ReturnStmt <$> (reserved "return" *> optional expr <* semi)
    <|> FunctionStmt
      <$> (try $ reserved "function" *> identifier)
      <*> parens (sepBy identifier $ symbol ",")
      <*> braces (many stmt)
    <|> try (AssignStmt <$> identifier <*> (symbol "=" *> expr <* semi))
    <|> try (SendStmt <$> expr <*> (symbol "->" *> identifier <* semi))
    <|> ExprStmt <$> expr <* semi

Next, we implement the execute function for the YieldStmt and SpawnStmt statements:

execute :: Stmt -> Interpreter ()
execute = \case
  ExprStmt expr -> void $ evaluate expr
  VarStmt name expr -> evaluate expr >>= defineVar name
  AssignStmt name expr -> evaluate expr >>= assignVar name
  IfStmt expr body -> do
    cond <- evaluate expr
    when (isTruthy cond) $
      traverse_ execute body
  while@(WhileStmt expr body) -> do
    cond <- evaluate expr
    when (isTruthy cond) $ do
      traverse_ execute body
      execute while
  ReturnStmt mExpr -> do
    mRet <- traverse evaluate mExpr
    throwError . Return . fromMaybe Null $ mRet
  FunctionStmt name params body -> do
    env <- State.gets isEnv
    defineVar name $ Function name params body env
  YieldStmt -> yield
  SpawnStmt expr -> spawn expr
  SendStmt expr chan -> do
    val <- evaluate expr
    evaluate (Variable chan) >>= \case
      Chan channel -> channelSend val channel
      val' -> throw $ "Cannot send to a non-channel: " <> show val'
    isTruthy = \case
      Null -> False
      Boolean b -> b
      _ -> True

All the scaffolding is now in place. Next, we implement the yield and spawn functions. First comes spawn:

spawn :: Expr -> Interpreter ()
spawn expr = do
  env <- State.gets isEnv
  coroutine <- newCoroutine env (const $ evaluate expr >> runNextCoroutine)
  scheduleCoroutine coroutine

The spawn statement creates a new coroutine for the expression expr and schedules it. The coroutine captures the current environment, and evaluates the expression expr when it is run. The runNextCoroutine function is called after the expression is evaluated to run the next coroutine in the queue11.

Next up is yield:

yield :: Interpreter ()
yield = do
  env <- State.gets isEnv
  callCC $ \cont -> do
    newCoroutine env cont >>= scheduleCoroutine

The yield function is the essence of coroutines in Co. This is where we use the continuations that we added to the interpreter. First, we capture the current environment from the interpreter state. Then, we invoke callCC to get the current continuation. This continuation represents the rest of the program execution that lies in future after the yield statement12. We create a new coroutine with the captured environment and the continuation, and schedule it. Finally, we run the next coroutine in the queue.

By capturing the environment and the continuation in a coroutine, and scheduling it to be run later, we are able to suspend the current program execution, and resume it later. At the same time, by running the next coroutine in the queue, we cause the interleaved execution of function calls that we saw in the previous section.

Waiting for Termination

There is one last thing we need to implement. If we were to run the following program with the interpreter as we have it now, it would terminate prematurely without printing anything:

function printNums(start, end) {
  var i = start;
  while (i < end + 1) {
    i = i + 1;

spawn printNums(1, 4);

That’s because spawn schedules a coroutine for the printNums(1, 4) function call, but the interpreter does not wait for all scheduled coroutines to finish executing before terminating. So, we add a mechanism for the same:

awaitTermination :: Interpreter ()
awaitTermination = do
  (coroutines, maxWakeupTime) <- readIORef =<< State.gets isCoroutines
  dur <- calcSleepDuration maxWakeupTime
  unless (PQ.null coroutines) $ if dur > 0
    then sleep dur >> awaitTermination
    else yield >> awaitTermination

The awaitTermination function checks if the coroutine queue is empty. If it is not, it yields and calls itself again to redo the check. Calling yield causes the next coroutine in the queue to be run. awaitTermination keeps checking the queue, and yielding until the queue is empty. Then, it finally returns.

Putting Everything Together

Finally, we put everything together in the interpret function:

interpret :: Program -> IO (Either String ())
interpret program = do
  state <- initInterpreterState
  retVal <- flip evalStateT state
    . flip runContT return
    . runExceptT
    . runInterpreter
    $ (traverse_ execute program >> awaitTermination)
  case retVal of
    Left (RuntimeError err) -> return $ Left err
    Left (Return _) -> return $ Left "Cannot return from outside functions"
    Left CoroutineQueueEmpty -> return $ Right ()
    Right _ -> return $ Right ()

We add awaitTermination at the end of the program to be interpreted so that the interpreter waits for the coroutine queue to be empty before terminating.

We use runContT to run the program with the initial continuation, which is just return. This causes the interpreter to terminate when the program returns.

Lastly, we catch the CoroutineQueueEmpty exception, ignore it, and terminate the interpreter.

That’s it! We have implemented coroutines in Co. For an interesting example of usage of coroutines, we are going to implement the breadth-first traversal of a binary tree without using a queue in the next section.

Bonus Round: Breadth-First Traversal without a Queue

Breadth-first traversal is a common algorithm for traversing a tree. It traverses the tree level-by-level, from left to right. It uses a queue to keep track of the nodes that are yet to be traversed. However, with coroutines, we can implement a breadth-first traversal without using a queue.

First, we need to define a binary tree data structure in Co. Remember, however, that Co does not have a built-in data structure for trees, neither does it support user-defined data structures. So, we are going to borrow a trick from the Wizard book, and implement it using closures:

function binaryTree(val, left, right) {
  return function (command) {
    if (command == "val") { return val; }
    if (command == "left") { return left; }
    if (command == "right") { return right; }
    return null;

function binaryTreeVal(tree) { return tree("val"); }
function binaryTreeLeft(tree) { return tree("left"); }
function binaryTreeRight(tree) { return tree("right"); }

We define a binary tree as a function that takes a node value, and left and right subtrees as parameters, and returns an anonymous function that takes a command, and returns the corresponding parameter value. The binaryTreeVal, binaryTreeLeft and binaryTreeRight are helper functions that call the returned anonymous function with the appropriate command.

Next, we write a function to generate a perfect binary tree given a starting power-of-two number:

function generatePowersOfTwoBinaryTree(start) {
  function generateTree(start, interval) {
    if (start == 1) {
      return binaryTree(1, null, null);
    return binaryTree(start,
      generateTree(start - interval/2, interval/2),
      generateTree(start - interval/2, interval/2));
  return generateTree(start, start);

And, a function to pretty-print a tree node:

function printTreeNode(val, depth) {
  var i = 0;
  var padding = "┃━";
  while (i < depth) {
    padding = padding + "━━━━━━━━";
    i = i + 1;

  print(padding + " " + val);

Finally, here’s the function that does the breadth-first traversal, and prints the tree:

function printBinaryTreeBreadthFirst(tree) {
  function traverseTree(tree, depth) {
    if (tree == null) { return; }
    printTreeNode(binaryTreeVal(tree), depth);
    spawn traverseTree(binaryTreeLeft(tree), depth + 1);
    spawn traverseTree(binaryTreeRight(tree), depth + 1);
  traverseTree(tree, 0);

We run it like this:

var tree = generatePowersOfTwoBinaryTree(16);
And, we get the following output:
┃━ 16
┃━━━━━━━━━ 8
┃━━━━━━━━━ 8
┃━━━━━━━━━━━━━━━━━ 4
┃━━━━━━━━━━━━━━━━━ 4
┃━━━━━━━━━━━━━━━━━ 4
┃━━━━━━━━━━━━━━━━━ 4
┃━━━━━━━━━━━━━━━━━━━━━━━━━ 2
┃━━━━━━━━━━━━━━━━━━━━━━━━━ 2
┃━━━━━━━━━━━━━━━━━━━━━━━━━ 2
┃━━━━━━━━━━━━━━━━━━━━━━━━━ 2
┃━━━━━━━━━━━━━━━━━━━━━━━━━ 2
┃━━━━━━━━━━━━━━━━━━━━━━━━━ 2
┃━━━━━━━━━━━━━━━━━━━━━━━━━ 2
┃━━━━━━━━━━━━━━━━━━━━━━━━━ 2
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1
┃━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1

The trick here is to use the coroutine queue itself for the breadth-first traversal. After printing a tree node, we spawn a coroutine for traversing each child node. The coroutines are scheduled in the order they are spawned, so the traversal is breadth-first, resulting in the above output.

In this post, we added support for coroutines to our Co interpreter. We learned about the continuation-passing style, and used it to implement coroutines. In the next post, we’ll add support for channels to our interpreter, and use them for cross-coroutine communication.

The code for complete Co interpreter is available here.


Many thanks to Arun Raghavan for reviewing a draft of this article.

Abelson, Harold, Gerald Jay Sussman, and with Julie Sussman. “The Environment Model of Evaluation.” In Structure and Interpretation of Computer Programs, 2nd Editon. MIT Press/McGraw-Hill, 1996.
Bartel, Joe. Non-Preemptive Multitasking.” The Computer Journal, no. 30 (May 2011): 37–38, 28.
Knuth, Donald E. “Coroutines.” In The Art of Computer Programming: Volume 1: Fundamental Algorithms, 3rd ed., 193–200. Addison Wesley, 1997.
Reynolds, John C. “The Discoveries of Continuations.” LISP and Symbolic Computation 6, no. 3-4 (1993): 233–47.

  1. This representation is copied from a series of articles on coroutines by Dmitry Kandalov. The articles are a great introduction to coroutines, and are highly recommended.↩︎

  2. Read the Kotlin docs and Python docs to learn more about coroutines in Kotlin and Python respectively.↩︎

  3. Generators are similar to coroutines. The main difference is that generators are typically used to produce a sequence of values, while coroutines are used to implement concurrency. But coroutines (as we have them in this post) can be implemented over generators, and generators can be implemented over coroutines and channels. So the difference is mostly of intent.↩︎

  4. Coroutines as we have them in Co, are asymmetric, non-first-class, and stackful.

    In contrast, coroutines in

    • Kotlin are asymmetric, non-first-class and stackless,
    • Python are asymmetric, first-class and stackless,
    • Lua are asymmetric, first-class and stackful, and
    • Zig are symmetric, non-first-class and stackless.

    See the Wikipedia article on coroutines for more details on the types of coroutines and their various implementations.↩︎

  5. The core.async library implements something like coroutines in Clojure, but they are not true coroutines. They have various limitations like not being able to yield from a functions called from a go block. This is because core.async is implemented as a macro that transforms the code directly inside a go block into a state machine, but not the functions called from the go block.↩︎

  6. See this article by Ziyang Liu and this one by Matt Might for detailed explanations of the various use-cases of CPS.↩︎

  7. See this article by Li-yao XIA for an introduction to the Cont monad.↩︎

  8. Scheme was the first language to introduce call/cc. Since then many languages have added support for it.↩︎

  9. If we compare the CPS version of the program with the direct style version, we can see that it is possible to print the recommendations twice in the CPS version by calling the continuation twice. However, this is not possible in the direct style version, since the flow of control is implicit in it.↩︎

  10. We use the min-priority queue from the pqueue library.↩︎

  11. It is essential to call runNextCoroutine after the expression in the spawn statement is evaluated. The evaluation of the expression may or may not yield. If it does, yielding causes the next coroutine to be run.

    However, if it does not yield, but instead returns, and we do not call runNextCoroutine after it, the flow of control then goes to the end of the previous call to runNextCoroutine called from a previous yield. This causes the program after the previous yield to start executing, but with the interpreter environment set to that of the expression in the spawn statement, leading to unexpected behavior.

    So, calling runNextCoroutine after the expression evaluation is a must to ensure correct execution.↩︎

  12. The coroutines in Co are stackful, which means that the ToC can be yielded from anywhere in the program, including nested function calls, and are resumed from the same point. This is in contrast to stackless coroutine implementations, where the ToC can only be yielded from particular functions that are marked as being able to yield, like generators in Python or async functions in JavaScript. Stackless coroutines are more efficient, but they are also more restrictive.↩︎

If you liked this post, please leave a comment.

by Abhinav Sarkar ( at February 11, 2023 12:00 AM

February 10, 2023

GHC Developer Blog

GHC 9.2.6 is now available

GHC 9.2.6 is now available

Zubin Duggal - 2023-02-10

The GHC developers are happy to announce the availability of GHC 9.2.6. Binary distributions, source distributions, and documentation are available at

This release is primarily a bugfix release addressing a few issues found in 9.2.5. These include:

  • Fixes for a number of simplifier issues typically resulting in compiler panics (#22491, #22482, #19824, #22718, #22662, #22039).
  • Fix for a simplifier regression in 9.2.5 that could impact runtime performance when compiling with optimisations due to duplication of expensive work (#22425).
  • Various stability improvments and bug fixes for the non moving and parallel GCs (#22528, #20221, #22264, #22327, #22929, #22927, #22929, #22930).
  • Allow certain keywords which can be used as variable names with -XOverloadedRecordDot (#20723).
  • Improvements to typechecker performance for modules with holes in type signatures (#14766)
  • Bump gmp-tarballs to a version which doesn’t use the reserved x18 register on AArch64/Darwin systems, and also has fixes for CVE-2021-43618 (#22497, #22789).
  • … and a few more. See the release notes for a full accounting.

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.

Happy compiling,

  • Zubin

by ghc-devs at February 10, 2023 12:00 AM

February 08, 2023

Magnus Therning

Logging with class

In two previous posts I've described how I currently compose log messages and how I do the actual logging. This post wraps up this particular topic for now with a couple of typeclasses, a default implementation, and an example showing how I use them.

The typeclasses

First off I want a monad for the logging itself. It's just a collection of functions taking a LogMsg and returning unit (in a monad).

class Monad m => LoggerActions m where
    debug :: LogMsg -> m ()
    info :: LogMsg -> m ()
    warn :: LogMsg -> m ()
    err :: LogMsg -> m ()
    fatal :: LogMsg -> m ()

In order to provide a default implementation I also need a way to extract the logger itself.

class Monad m => HasLogger m where
    getLogger :: m Logger

Default implementation

Using the two typeclasses above it's now possible to define a type with an implementation of LoggerActions that is usable with deriving via.

newtype StdLoggerActions m a = MkStdZLA (m a)
    deriving (Functor, Applicative, Monad, MonadIO, HasLogger)

And its implementattion of LoggerActions looks like this:

instance (HasLogger m, MonadIO m) => LoggerActions (StdLoggerActions m) where
    debug msg = getLogger >>= flip debugIO msg
    info msg = getLogger >>= flip infoIO msg
    warn msg = getLogger >>= flip warnIO msg
    err msg = getLogger >>= flip errIO msg
    fatal msg = getLogger >>= flip fatalIO msg

An example

Using the definitions above is fairly straight forward. First a type the derives its implementaiton of LoggerActions from StdLoggerActions.

newtype EnvT a = EnvT {runEnvT :: ReaderT Logger IO a}
    deriving newtype (Functor, Applicative, Monad, MonadIO, MonadReader Logger)
    deriving (LoggerActions) via (StdLoggerActions EnvT)

In order for it to work, and compile, it needs an implementation of HasLogger too.

instance HasLogger EnvT where
    getLogger = ask

All that's left is a function using a constraint on LoggerActions (doStuff) and a main function creating a logger, constructing an EnvT, and then running doStuff in it.

doStuff :: LoggerActions m => m ()
doStuff = do
    debug "a log line"
    info $ "another log line" #+ ["extras" .= (42 :: Int)]

main :: IO ()
main = withLogger $ \logger ->
    runReaderT (runEnvT doStuff) logger

February 08, 2023 10:54 PM

February 07, 2023

Brent Yorgey

Subtracting natural numbers: types and usability

For several years now I have been working on a functional teaching language for discrete mathematics, called Disco. It has a strong static type system, subtyping, equirecursive algebraic types, built-in property-based testing, and mathematically-inspired syntax. If you want to know more about it in general, you can check out the GitHub repo, or give it a try on

In this blog I want to write about a particular usability issue surrounding the type of the subtraction operation, partly because I think some might find it interesting, and partly because forcing myself to clearly articulate possible solutions may help me come to a good resolution.

The problem with subtraction

Disco supports four basic numeric types: natural numbers \mathbb{N}, integers \mathbb{Z}, “fractional” numbers \mathbb{F} (i.e. nonnegative rationals), and rational numbers \mathbb{Q}. These types form a subtyping lattice, with natural numbers being a subtype of both integers and fractionals, and integers and fractionals in turn being subtypes of the rationals. All the numeric types support addition and multiplication; the integers allow negation/subtraction, the fractionals allow reciprocals/division, and rationals allow both.

So what is the type of x - y? Clearly it has to be either \mathbb{Z} or \mathbb{Q}; that’s the whole point. Natural numbers and fractional numbers are not closed under subtraction; \mathbb{Z} and \mathbb{Q} are precisely what we get when we start with \mathbb{N} or \mathbb{F} and decide to allow subtraction, i.e. when we throw in additive inverses for everything.

However, this is one of the single biggest things that trips up students. As an example, consider the following function definition:

fact_bad : N -> N
fact_bad(0) = 1
fact_bad(n) = n * fact_bad(n-1)

This looks perfectly reasonable on the surface, and would work flawlessly at runtime. However, it does not typecheck: the argument to the recursive call must be of type \mathbb{N}, but since n-1 uses subtraction, it cannot have that type.

This is very annoying in practice for several reasons. The most basic reason is that, in my experience at least, it is very common: students often write functions like this without thinking about the fact that they happened to use subtraction along the way, and are utterly baffled when the function does not type check. This case is also extra annoying since it would work at runtime: we can clearly reason that if n is a natural number that is not zero, then it must be 1 or greater, and hence n-1 will in fact be a natural number. Because of Rice’s Theorem, we know that every decidable type system must necessarily exclude some programs as untypeable which nonetheless do not “go wrong”, i.e. exhibit no undesirable behavior when evaluated. The above fact_bad function is a particularly irksome example.

To be clear, there is nothing wrong with the type system, which is working exactly as intended. Rather, the problem lies in the fact that this is a common and confusing issue for students.

Implementing factorial

You may be wondering how it is even possible to implement something like factorial at all without being able to subtract natural numbers. In fact, there are two good ways to implement it, but they don’t necessarily solve the problem of student confusion.

  • One solution is to use an arithmetic pattern and match on n+1 instead of n, like this:

    fact_ok1 : N -> N
    fact_ok1(0) = 1
    fact_ok1(n+1) = (n+1) * fact_ok1(n)

    This works, and it’s theoretically well-motivated, but feels somewhat unsatisfying: both because we have to repeat n+1 and because this style of definition probably feels foreign to anyone except those who have played with a Nat algebraic data type (which excludes the vast majority of Discrete Math students).

  • Another solution is to use a saturating subtraction operator, x \mathbin{\dot -} y = \max(0, x - y). In Disco this operator is written x .- y. Unlike normal subtraction, it can have the type \mathbb{N} \times \mathbb{N} \to \mathbb{N}, so we can rewrite the factorial function this way:

    fact_ok2 : N -> N
    fact_ok2(0) = 1
    fact_ok2(n) = n * fact_ok2(n .- 1)

    The .- operator is also theoretically well-motivated, being the “monus” operator for the commutative monoid of natural numbers under addition. However, in my experience, students are annoyed and confused by this. They often do not understand when and why they are supposed to use .-. Of course, better error messages could help here, as could better pedagogy. This is actually my current approach: this semester I talked about the difference between \mathbb{N} and \mathbb{Z} very early, hitting on the fact that \mathbb{N} is not closed under subtraction, and explicitly made them explore the use of the .- operator in their first homework assignment. We’ll see how it goes!

Some tempting and expedient, but wrong, solutions

One solution that sounds nice on the surface is to just pun the notation: why not just have a single operator -, but make it behave like .- on types without negation (\mathbb{N} and \mathbb{F}), and like normal subtraction on \mathbb{Z} and \mathbb{Q}? That way students wouldn’t have to remember to use one version or the other, they can just use subtraction and have it do the right thing depending on the type.

This would be sound from a type system point of view; that is, we would never be able to produce, say, a negative value with type \mathbb{N}. However, in the presence of subtyping and type inference, there is a subtle problem from a semantics point of view. To understand the problem, consider the following function:

f : N -> Z
f(n) = (-3) * (n - 5)

What is the output of f(3)? Most people would say it should be (-3) * (3 - 5) = (-3) * (-2) = 6. However, if the behavior of subtraction depends on its type, it would also be sound for f(3) to output 0! The input 3 and the constant 5 can both be given the type \mathbb{N}, in which case the subtraction would act as a saturating subtraction and result in 0.

What’s going on here? Conceptually, one of the jobs of type inference, when subtyping is involved, is to decide where to insert type coercions. (Practically speaking, in Disco, such coercions are always no-ops; for example, all numeric values are represented as Rational, so 3 : N and 3 : Q have the same runtime representation.) An important guiding principle is that the semantics of a program should not depend on where coercions are inserted, and type-dependent-subtraction violates this principle. f(3) evaluates to either 6 or 0, depending on whether a coercion from \mathbb{N} to \mathbb{Z} is inserted inside or outside the subtraction. Violating this principle can make it very difficult for anyone (let alone students!) to understand the semantics of a given program: at worst it is ambiguous or undefined; at best, it depends on understanding where coercions will be inserted.

What about having - always mean subtraction, but crash at runtime if we try to subtract natural numbers and get something less than 0? That way we can use it as long as we “know it is safe” (as in the factorial example). Unfortunately, this has the exact same issue, which the above example with f(3) still illustrates perfectly: f(3) can either evaluate to 6 or crash, depending on exactly where coercions are inserted.

Typechecking heuristics?

Another interesting option would be to make typechecking a bit smarter, so that instead of only keeping track of the type of each variable, we also sometimes keep track of values we statically know a variable can and can’t have in a certain context. We could then use this information to allow subtraction to have a type like \mathbb{N} \times \mathbb{N} \to \mathbb{N} as long as we can statically prove it is safe. For example, after matching on 0 in the first line of fact_bad, in the second line we know n cannot be 0, and we could imagine using this information to decide that the expression n - 1 is safe. This scheme would not change the semantics of any existing programs; it would only allow some additional programs to typecheck which did not before.

Of course, this would never be complete—there would always be examples of Disco programs where we can prove that a certain subtraction is safe but the heuristics don’t cover it. But it might still go a long way towards making this kind of thing less annoying. On the other hand, it makes errors even more mysterious when they do happen, and hard to understand when a program will and won’t typecheck. Perhaps it is best to just double down on the pedagogy and get students to understand the difference between \mathbb{N} and \mathbb{Z}!


As a final aside, note that we have the same issue with division: x / y is only allowed at types \mathbb{F} or \mathbb{Q}. If we want to divide integers, we can use a different built-in operator, // which does integer division, i.e. “rounds down”. However, this is not nearly as bad of an issue in practice, both because some students are already used to the idea of integer division (e.g. Python makes the same distinction), and because wanting to divide integers does not come up nearly as often, in practice, as wanting to subtract natural numbers.

by Brent at February 07, 2023 05:03 PM

February 04, 2023

Magnus Therning

A take on logging

In my previous post I described a type, with instances and a couple of useful functions for composing log messages. To actually make use of that there's a bit more needed, i.e. the actual logging. In this post I'll share that part of the logging setup I've been using in the Haskell services at $DAYJOB.

The logger type

The logger will be a wrapper around fast-logger's FastLogger, even though that's not really visible.

newtype Logger = Logger (LogMsg -> IO ())

It's nature as a wrapper makes it natural to follow the API of fast-logger, with some calls to liftIO added.

newLogger :: MonadIO m => m (Logger, m ())
newLogger = liftIO $ do
    (fastLogger, cleanUp) <- newFastLogger $ LogStdout defaultBufSize
    pure (Logger (fastLogger . toLogStr @LogMsg), liftIO cleanUp)

The implementation of withLogger is pretty much a copy of what I found in fast-logger, just adapted to the newLogger above.

withLogger :: (MonadMask m, MonadIO m) => (Logger -> m ()) -> m ()
withLogger go = bracket newLogger snd (go . fst)

Logging functions

All logging functions will follow the same pattern so it's easy to break out the common parts.

logIO :: MonadIO m => Text -> Logger -> LogMsg -> m ()
logIO lvl (Logger ls) msg = do
    t <- formatTime defaultTimeLocale "%y-%m-%dT%H:%M:%S%03QZ" <$> liftIO getCurrentTime
    let bmsg = "" :# [ "correlation-id" .= ("no-correlation-id" :: Text)
                     , "timestamp" .= t
                     , "level" .= lvl
    liftIO $ ls $ bmsg <> msg

With that in place the logging functions become very short and sweet.

debugIO, infoIO, warnIO, errIO, fatalIO :: MonadIO m => Logger -> LogMsg -> m ()
debugIO = logIO "debug"
infoIO = logIO "info"
warnIO = logIO "warn"
errIO = logIO "error"
fatalIO = logIO "fatal"

Simple example of usage

A very simple example showing how it could be used would be something like this

main :: IO ()
main = withLogger $ \logger -> do
    debugIO logger "a log line"
    infoIO logger $ "another log line" #+ ["extras" .= (42 :: Int)]

February 04, 2023 05:49 PM

February 02, 2023

Matt Parsons

Production Haskell Complete

I’m happy to announce that my book “Production Haskell” is complete.

The book is a 500+ page distillation of my experience working with Haskell in industry. I believe it’s the best resource available for building and scaling the use of Haskell in business.

To buy the ebook, go to the Leanpub page - the price is slightly lower here than on Amazon.

To buy hard copies, go to Amazon.

Thanks to all of you for reading my blog, commenting on Reddit, and encouraging me to write the book in the first place.

February 02, 2023 12:00 AM

January 30, 2023

Gabriella Gonzalez

terraform-nixos-ng: Modern terraform support for NixOS

terraform-nixos-ng: Modern terraform support for NixOS

Recently I’ve been working on writing a “NixOS in Production” book and one of the chapters I’m writing is on deploying NixOS using terraform. However, one of the issues I ran across was the poor NixOS support for terraform. I’ve already gone through the nix.devpost explaining how to use the terraform-nixosproject but I ran into several issues trying to follow those instructions (which I’ll explain below). That plus the fact that terraform-nixos seems to be unmaintained pushed me over the edge to rewrite the project to simplify and improve upon it.

So this post is announcing my terraform-nixos-ngproject:

… which is a rewrite of terraform-nixos and I’ll use this post to compare and contrast the two projects. If you’re only interested in trying out the terraform-nixos-ng project then go straight to the README

Using nixos-rebuild

One of the first things I noticed when kicking the tires on terraform-nixos was that it was essentially reinventing what the nixos-rebuild tool already does. In fact, I was so surprised by this that I wrote a standalone post explaining how to use nixos-rebuild as a deployment tool:

Simplifying that code using nixos-rebuild fixed lots of tiny papercuts I had with terraform-nixos, like:

  • The deploy failing if you don’t have a new enough version of bash installed

  • The inability to turn off the use of the --use-substitutes flag

    That flag causes issues if you want to deploy to a machine that disables outbound connections.

  • The dearth of useful options (compared to nixos-rebuild)

    … including the inability to fully customize sshoptions

  • The poor interop with flakes

    For example, terraform-nixos doesn’t respect the standard nixosConfigurations flake output hierarchy.

    Also, terraform-nixos doesn’t use flakes natively (it uses flake-compat), which breaks handling of the config.nix.binary{Caches,CachePublicKeys} flakes settings. The Nix UX for flakes is supposed to ask the user to consent to those settings (because they are potentially insecure to auto-enable for a flake), but their workaround breaks that UX by automatically enabling those settings without the user’s consent.

I wanted to upstream this rewrite to use nixos-rebuildinto terraform-nixos, but I gave up on that idea when I saw that no pull request since 2021 had been merged, including conservative pull requests like this one to just use the script included within the repository to update the list of available AMIs.

That brings me to the next improvement, which is:

Auto-generating available AMIs

The terraform-nixos repository requires the AMI list to be manually updated. The way you do this is to periodically run a script to fetch the available AMIs from Nixpkgs and then create a PR to vendor those changes. However, this shouldn’t be necessary because we could easily program terraform to generate the list of AMIs on the fly.

This is what the terraform-nixos-ng project does, where the ami module creates a data source that runs an equivalent script to fetch the AMIs at provisioning time.

In the course of rewriting the AMI module, I made another small improvement, which was:

Support for aarch64AMIs

Another gripe I had with terraform-nixos-ng is that its AMI module doesn’t support aarch64-linux NixOS AMIs even though these AMIs exist and Nixpkgs supports them. That was a small and easy fix, too.

Functionality regressions

terraform-nixos-ng is not a strict improvement over terraform-nixos, though. Specifically, the most notable feature omissions are:

  • Support for non-flake workflows

    terraform-nixos-ng requires the use of flakes and doesn’t provide support for non-flake-based workflows. I’m very much on team “Nix flakes are good and shouldn’t be treated as experimental any longer” so I made an opinionated choice to require users to use flakes rather than support their absence.

    This choice also isn’t completely aesthetic, the use of flakes improves interop with nixos-rebuild, where flakes are the most ergonomic way for nixos-rebuild to select from one of many deployments.

  • Support for secrets management

    I felt that this should be handled by something like sops-nixrather than rolling yet another secrets management system that was idiosyncratic to this deploy tool. In general, I wanted these terraform modules to be as lightweight as possible by making more idiomatic use of the modern NixOS ecosystem.

  • Support for Google Compute Engine images

    terraform-nixos supports GCE images and the only reason I didn’t add the same support is because I’ve never used Google Compute Engine so I didn’t have enough context to do a good rewrite, nor did I have the inclination to set up a GCE account just to test the rewrite. However, I’d accept a pull request adding this support from someone interested in this feature.


There’s one last improvement over the terraform-nixosproject, which is that I don’t leave projects in an abandoned state. Anybody who has contributed to my open source projects knows that I’m generous about handing out the commit bit and I’m also good about relinquishing control if I don’t have time to maintain the project myself.

However, I don’t expect this to be a difficult project to maintain anyway because I designed terraform-nixos-ng to outsource the work to existing tools as much as possible instead of reinventing the wheel. This is why the implementation of terraform-nixos-ng is significantly smaller than terraform-nixos.

by Gabriella Gonzalez ( at January 30, 2023 02:54 PM

January 29, 2023

Magnus Therning

A take on log messages

At $DAYJOB we use structured logging with rather little actual structure, the only rules are

  1. Log to stdout.
  2. Log one JSON object per line.
  3. The only required fields are
    • message - a human readable string describing the event
    • level - the severity of the event, debug, info, warn, error, or fatal.
    • timestamp - the time of the event
    • correlation-id - an ID passed between services to allow to find related events

Beyond that pretty much anything goes, any other fields that are useful in that service, or even in that one log message is OK.

My first take was very ad-hoc, mostly becuase there were other parts of the question "How do I write a service in Haskell, actually?" that needed more attention – then I read Announcing monad-logger-aeson: Structured logging in Haskell for cheap. Sure, I'd looked at some of the logging libraries on Hackage but not really found anything that seemed like it would fit very well. Not until monad-logger-aeson, that is. Well, at least until I realised it didn't quite fit the rules we have.

It did give me some ideas of how to structure my current rather simple, but very awkward to use, current loggging code. This is what I came up with, and after using it in a handful services I find it kind of nice to work with. Let me know what you think.

The log message type

I decided that a log message must always contain the text describing the event. It's the one thing that's sure to be known at the point where the developer writes the code to log an event. All the other mandatory parts can, and probably should as far as possible, be added by the logging library itself. So I ended up with this type.

data LogMsg = Text :# [Pair]
    deriving (Eq, Show)

It should however be easy to add custom parts at the point of logging, so I added an operator for that.

(#+) :: LogMsg -> [Pair] -> LogMsg
(#+) (msg :# ps0) ps1 = msg :# (ps0 <> ps1)

The ordering is important, i.e. ps0 <> ps1, as aeson's object function will take the last value for a field and I want to be able to give keys in a log message new values by overwriting them later on.

Instances to use it with fast-logger

The previous logging code used fast-logger and it had worked really well so I decided to stick with it. Making LogMsg and instance of ToLogStr is key, and as the rules require logging of JSON objects it also needs to be an instance of ToJSON.

instance ToJSON LogMsg where
    toJSON (msg :# ps) = object $ ps <> ["message" .= msg]

instance ToLogStr LogMsg where
    toLogStr msg = toLogStr (encode msg) <> "\n"

Instance to make it easy to log a string

It's common to just want to log a single string and nothing else, so it's handy if LogMsg is an instance of IsString.

instance IsString LogMsg where
    fromString msg = pack msg :# []

Combining log messages

When writing the previous logging code I'd regularly felt pain from the lack of a nice way to combine log messages. With the definition of LogMsg above it's not difficult to come up with reasonable instances for both Semigroup and Monoid.

instance Semigroup LogMsg where
    "" :# ps0 <> msg1 :# ps1 = msg1 :# (ps0 <> ps1)
    msg0 :# ps0 <> "" :# ps1 = msg0 :# (ps0 <> ps1)
    msg0 :# ps0 <> msg1 :# ps1 = (msg0 <> " - " <> msg1) :# (ps0 <> ps1)

instance Monoid LogMsg where
    mempty = ""

In closing

What's missing above is the automatic handling of the remaining fields. I'll try to get back to that part soon. For now I'll just say that the log message API above made the implementation nice and straight forward.

January 29, 2023 09:12 PM

January 26, 2023

Chris Smith 2

Pair Programming with ChatGPT & Haskell

Here, I present the (lightly edited) story of using ChatGPT conversationally to solve a non-trivial problem in Haskell. It definitely gets some things wrong, and it’s still unclear whether co-developing this with ChatGPT made anything easier than it would have been otherwise. But in any case, it was definitely a different and less lonely experience than just programming on my own.

There are four basic sections to the conversation below:

  1. Defining a model for the game of probabilistic rock-paper-scissors.
  2. Solving with brute force by searching the space of possible strategies.
  3. Adapting the code to use symbolic representation with polynomials.
  4. Applying support enumeration to solve the game as a system of linear constraints.

NOTE: If you aren’t interested in the detailed conversation, and just want to see some interesting Haskell code, search for “Final Code for Part” in this post for the big four steps.

Part I: Modeling a game


Hey ChatGPT! I’d like to solve a programming problem with you. Are you up for it?


I’d be happy to help you with a programming problem! What is the problem you’re trying to solve?


Let’s assume you and I are playing rock-paper-scissors, but with special rules. After we each choose our desired moves, there is a 50% probability that your move will be changed to rock. Independently, there’s a 50% probability that my move will be changed to scissors. I’d like to compute the right strategies and expected outcomes for this game, and others like it.

Let’s program this in Haskell. I would start by just writing down some basic types to model the game.


Sure, that sounds like an interesting problem! Here’s an example of how you could define the types for this game in Haskell:

data Move = Rock | Paper | Scissors
data Result = Win | Lose | Draw

data Game = Game { p1 :: Move, p2 :: Move }

In this code, Move is a data type that can take on the three values Rock, Paper, or Scissors. Result is a data type that can take on the values Win, Lose, or Draw. Game is a data type that contains two fields, p1 and p2, representing the moves made by player 1 and player 2 respectively.

With these basic types, you could then define a function to determine the result of a game, using the standard rules of rock-paper-scissors:

result :: Game -> Result
result (Game Rock Scissors) = Win
result (Game Scissors Paper) = Win
result (Game Paper Rock) = Win
result (Game Scissors Rock) = Lose
result (Game Paper Scissors) = Lose
result (Game Rock Paper) = Lose
result (Game _ _) = Draw

Now you could use this function to determine the result of a game without the special rule. Now you could add the special rule of the game to this function and also a function which would simulate the game with these rules and give the expected outcome.

Can I help you with anything else?


Looks great!

The rules of the game talk about people being forced to play a specific move. That’s pretty easy to represent.

data Transform = Chosen | Forced Move

transform :: Transform -> Move -> Move
transform (Forced m) _ = m
transform Chosen m = m

To make further progress, we need to think about probability distributions. There are plenty of Haskell packages that define types and functions for this, but let’s keep this self-contained and write our own. Something like this should work.

type Probability = Rational
newtype Distribution a = Distribution [(Rational, a)]

Then we can express the parameters of the game like this:

data Params = Params (Distribution Transform) (Distribution Transform)

The example I gave before would be

(Distribution [(1/2, Chosen), (1/2, Forced Rock)]
(Distribution [(1/2, Chosen), (1/2, Forced Scissors)])

I think we now have the tools write

applyTransforms ::
Distribution Transform ->
Distribution Move ->
Distribution Move


Yes, that’s a good approach to modeling the probability distributions for the game. The Probability type is just a synonym for Rational, which is a built-in Haskell type that can represent fractions with arbitrary precision. The Distribution type is a newtype wrapper around a list of pairs, where each pair consists of a Probability and a value of type a.

Transform is a good way to represent the forced moves. With the transform function, you can use it to apply the transform to a move.

Params is a good way to represent the parameters of the game, it takes two Distribution Transform which gives the possible transforms for the players.

Now, applyTransforms is a function that takes two distributions, one for Transform and one for Move and returns a new distribution for Move after applying the transform to the moves. Here is an example of how you could implement it:

applyTransforms ::
Distribution Transform ->
Distribution Move ->
Distribution Move
applyTransforms (Distribution tr) (Distribution mv) =
Distribution $ [(p * q, transform t m) | (p, t) <- tr, (q, m) <- mv]

In this implementation, we use a list comprehension to iterate over the elements of the two input distributions, apply the transform function to each pair of elements, and then combine the probabilities using the distributive law of probability.

This function should be able to apply the given transforms to the moves and give you the expected outcome.

Is there anything else that I can help you with?


That looks great, but what about generalizing it to an Applicative instance for Distribution?


Sure, we can generalize this further by making Distribution an instance of the Applicative type class. Here’s an example of how you could define the Applicative instance for Distribution:

instance Applicative Distribution where
pure x = Distribution [(1, x)]
(Distribution fs) <*> (Distribution xs) =
Distribution [(p * q, f x) | (p, f) <- fs, (q, x) <- xs]

With this instance, you can use the <*> operator to apply a distribution of functions to a distribution of values, and it will give you a new distribution of the results. The pure function creates a distribution with a single element and a probability of 1, which corresponds to a deterministic event.

Here is an example of how we can use the applicative instance and the applyTransforms function to compute the expected outcome of the game:

game :: Params -> Distribution Game
game (Params tr1 tr2) =
Game <$> applyTransforms tr1 moves <*> applyTransforms tr2 moves
where moves = Distribution [(1%3, Rock), (1%3, Paper), (1%3, Scissors)]

Now you can use the result function defined earlier to get the expected outcome of the game.

expectedOutcome :: Params -> Distribution Result
expectedOutcome params = result <$> game params

This way, you can use the expectedOutcome function to compute the expected outcome of the game for any given set of parameters.

Please let me know if you have any other question or if there is anything else I can help you with.


I’ve actually removed the applyTransforms function, because I don’t think it’s any clearer that just writing transform <$> tr <*> moves. So your game function now looks like this:

game :: Params -> Distribution Game
game (Params tr1 tr2) =
Game <$> (transform <$> tr1 <*> moves) <*> (transform <$> tr2 <*> moves)

But I think we should generalize this by passing in the strategy for each player. I don’t think each player should really try to play rock, paper, and scissors each 1/3 of the time, given the constraints on the game.

I think a strategy is just a distribution of moves:

type Strategy = Distribution Move


Yes, that’s a good idea to generalize the game function by passing in the strategies for each player. Instead of hardcoding the distribution of moves for each player, you can allow them to have different strategies.

Here’s how you can define the game function with the strategy as an input:

game :: Params -> Strategy -> Strategy -> Distribution Game
game (Params tr1 tr2) s1 s2 =
Game <$> (transform <$> tr1 <*> s1) <*> (transform <$> tr2 <*> s2)

This way, you can use the game function with any set of parameters and any strategies for each player, and it will give you the expected outcome of the game.

Here’s an example of how you can use the game function to compute the expected outcome of the game with a specific set of parameters and strategies:

let params =
(Distribution [(1/2, Chosen), (1/2, Forced Rock)])
(Distribution [(1/2, Chosen), (1/2, Forced Scissors)])
let s1 = Distribution [(1/2, Rock), (1/2, Scissors)]
let s2 = Distribution [(1/2, Paper), (1/2, Scissors)]
expectedOutcome params s1 s2

You could also use the strategies to calculate the expected outcome for each move with the given probability distribution of the transforms.

Please let me know if there is anything else I can help you with.


Yes, this works.

I’ve written this:

main :: IO ()
main = do
let params =
(Distribution [(1/2, Chosen), (1/2, Forced Rock)])
(Distribution [(1/2, Chosen), (1/2, Forced Scissors)])
let s1 = Distribution [(1/2, Rock), (1/2, Scissors)]
let s2 = Distribution [(1/2, Paper), (1/2, Scissors)]
print (result <$> game params s1 s2)

The output isn’t terribly helpful, though.

Distribution [(1 % 16,Lose),(1 % 16,Win),(1 % 16,Win),(1 % 16,Win),(1 % 16,Win),(1 % 16,Draw),(1 % 16,Draw),(1 % 16,Draw),(1 % 16,Lose),(1 % 16,Win),(1 % 16,Win),(1 % 16,Win),(1 % 16,Lose),(1 % 16,Win),(1 % 16,Win),(1 % 16,Win)]

I think we need a function to simplify a Distribution by combining like values.


Yes, you’re right. The output is not very readable in its current form. One way to simplify the output is by combining the elements with the same value into a single element, with the sum of the probabilities. Here’s an example of how you could define a function to do this:

simplify :: Eq a => Distribution a -> Distribution a
simplify (Distribution d) = Distribution $
map (\xs -> (sum (map fst xs), snd (head xs))) $
groupBy (\x y -> snd x == snd y) $
sortBy (\x y -> compare (snd x) (snd y)) d

This function takes a distribution as an input, and it uses a combination of groupBy, sortBy, and map to group the elements with the same value, sum their probabilities and return a new distribution with the combined elements.

You could use this function to simplify the output of your expectedOutcome function. Here’s an example:

main :: IO ()
main = do
let params =
(Distribution [(1/2, Chosen), (1/2, Forced Rock)])
(Distribution [(1/2, Chosen), (1/2, Forced Scissors)])
let s1 = Distribution [(1/2, Rock), (1/2, Scissors)]
let s2 = Distribution [(1/2, Paper), (1/2, Scissors)]
print (simplify $ result <$> game params s1 s2)

This will output

Distribution [(1 % 4,Lose),(1 % 2,Draw),(1 % 4,Win)]

This is more readable and gives you the expected probability of each outcome (Win, Lose, Draw)

Please let me know if you have any other question or if there is anything else I can help you with.


Yes, that did it. Although, I get a different result than you predicted.

Distribution [(5 % 8,Win),(3 % 16,Lose),(3 % 16,Draw)]

This makes sense to me. Since player 1 is playing a lot of rock, and player 2 is playing a lot of scissors, I expect the results to favor player 1.


I apologize, my mistake. The output you are getting is correct and matches with the expected outcome.

As you mentioned, the player 1 is playing rock a lot and player 2 is playing scissors a lot. So the probability of winning for player 1 is high as rock beats scissors. The probability of losing for player 2 is also high, as scissors lose to rock. The probability of draw is also low as both players are playing different moves mostly.

I’m glad that the function is working as expected and providing you with the desired output. If you have any other questions or if there is anything else I can help you with, please let me know.

Final Code for Part I

{-# LANGUAGE DeriveFunctor #-}

module Main where

import Data.List (groupBy, sortBy)

data Move = Rock | Paper | Scissors deriving (Eq, Ord, Show)

data Result = Win | Lose | Draw deriving (Eq, Ord, Show)

data Game = Game Move Move

result :: Game -> Result
result (Game Rock Scissors) = Win
result (Game Scissors Paper) = Win
result (Game Paper Rock) = Win
result (Game Scissors Rock) = Lose
result (Game Paper Scissors) = Lose
result (Game Rock Paper) = Lose
result (Game _ _) = Draw

data Transform = Chosen | Forced Move

transform :: Transform -> Move -> Move
transform (Forced m) _ = m
transform Chosen m = m

type Probability = Rational

newtype Distribution a = Distribution [(Rational, a)]
deriving (Functor, Show)

instance Applicative Distribution where
pure x = Distribution [(1, x)]
Distribution fs <*> Distribution xs =
Distribution [(p * q, f x) | (p, f) <- fs, (q, x) <- xs]

simplify :: Ord a => Distribution a -> Distribution a
simplify (Distribution d) =
Distribution $
map (\xs -> (sum (map fst xs), snd (head xs))) $
groupBy (\x y -> snd x == snd y) $
sortBy (\x y -> compare (snd x) (snd y)) d

data Params = Params (Distribution Transform) (Distribution Transform)

type Strategy = Distribution Move

game :: Params -> Strategy -> Strategy -> Distribution Game
game (Params tr1 tr2) s1 s2 =
Game <$> (transform <$> tr1 <*> s1) <*> (transform <$> tr2 <*> s2)

main :: IO ()
main = do
let params =
(Distribution [(1 / 2, Chosen), (1 / 2, Forced Rock)])
(Distribution [(1 / 2, Chosen), (1 / 2, Forced Scissors)])
let s1 = Distribution [(1 / 2, Rock), (1 / 2, Scissors)]
let s2 = Distribution [(1 / 2, Paper), (1 / 2, Scissors)]
print (simplify $ result <$> game params s1 s2)

Part II: Building an empirical solver


This is great so far, but I’d actually like to know the optimal strategy for each player, given the constraints. Let’s proceed slowly and just define some basic building blocks to work toward a solution to that problem.


Sure, I understand. To find the optimal strategy for each player, given the constraints, we can use a few different techniques. One approach would be to use a search algorithm such as minimax or alpha-beta pruning, which are commonly used in game theory to find the optimal strategy for two-player games.

Another approach would be to use dynamic programming to find the optimal strategy for each player. This approach would involve defining a recursive function that finds the optimal strategy for each player, given the current state of the game and the parameters of the game.

Another approach would be to use linear programming or optimization techniques to model the problem and find the optimal strategy for each player.

Which approach would you like to proceed with?


I don’t think there’s any kind of game state involved here, but it makes sense to proceed experimentally at first. Let’s just search the whole space of strategies for each player up to some precision.


Yes, that’s a good idea. Since there isn’t any game state involved in this problem and the only information you have is the parameters of the game, an experimental approach would be a good way to start.

One way to search the whole space of strategies for each player up to some precision is to use a brute-force algorithm. This would involve generating all possible strategies up to a certain precision, evaluating the expected outcome for each strategy, and then comparing the results to find the optimal strategy.

Here’s an example of how you could define a function to generate all possible strategies for a player:

generateStrategies :: Int -> [Strategy]
generateStrategies n =
[ Distribution
[ (fromIntegral i / fromIntegral n, move)
| i <- [1..n]
| move <- [Rock, Paper, Scissors]

This function takes an integer n as input, which represents the precision of the search, and it generates all possible strategies for a player, which is represented as a Distribution of Move. The function uses a list comprehension to generate all possible strategies, where the probability of each move is i/n where i is in the range of 1 to n.

You could use this function to generate all possible strategies for each player, then evaluate the expected outcome for each combination of strategies, and then compare the results to find the optimal strategy.

Please let me know if you need help with any other step of the process.


I don’t think this works. It ignores the fact that the probabilities must add to 1.

Here’s what I came up with:

allStrategies :: Int -> [Strategy]
allStrategies n =
[ Distribution [(r, Rock), (p, Paper), (s, Scissors)]
| r <- [0, 1 / fromIntegral n .. 1],
p <- [0, 1 / fromIntegral n .. 1 — r],
let s = 1 — r — p


You’re right, my apologies for the oversight. Your solution is better as it guarantees that the probabilities add up to 1, which is a requirement for any probability distribution.

The function allStrategies takes an integer n as input, which represents the precision of the search, and it generates all possible strategies for a player, which is represented as a Distribution of Move. The function uses a list comprehension to generate all possible strategies, where the probability of each move is r, p and s where r is in the range of 0 to 1 with step size of 1/n, p is in the range of 0 to 1-r with step size of 1/n and s is 1-r-p, which guarantees that the probabilities add up to 1.

You can use this function to generate all possible strategies for each player, then evaluate the expected outcome for each combination of strategies, and then compare the results to find the optimal strategy.

Please let me know if there is anything else I can help you with.


I suppose what’s next is to define the search.

bestStrategy :: Params -> [Strategy] -> Strategy

I suppose the trick is to use a sort of simple two-step minimax. Player 1 should choose the strategy that maximizes the benefit, given that the opponent will choose the strategy that minimizes the benefit.

We can start with a benefit function.

benefit :: Result -> Rational
benefit Win = 1
benefit Draw = 0
benefit Lose = -1

And we’ll want to compute the expected value over a distribution:

expected :: (a -> Rational) -> Distribution a -> Rational
expected f (Distribution xs) = sum [p * f x | (p, x) <- xs]


Yes, that’s correct. Now you have the bestStrategy function which takes the parameters of the game, a list of strategies for each player and returns the optimal strategy for the first player. The trick is to use a sort of simple two-step minimax. Player 1 should choose the strategy that maximizes the benefit, given that the opponent will choose the strategy that minimizes the benefit.

You also defined a function benefit which takes a Result as an input and returns a Rational value that represents the benefit of that result. For example, the benefit of a win is 1, the benefit of a draw is 0 and the benefit of a loss is -1.

You also defined a function expected which takes a function f and a Distribution as an input, and it returns the expected value of f over the Distribution.

Now you can use these functions to define the bestStrategy function. Here’s an example of how you could define it:

bestStrategy :: Params -> [Strategy] -> Strategy
bestStrategy params strategies =
fst $
(compare `on` snd)
[ (s1, expected benefit $ result <$> game params s1 s2)
| s1 <- strategies,
s2 <- strategies

This function takes the parameters of the game and a list of strategies for each player as input, and it returns the strategy that maximizes the expected benefit for player 1. It uses a list comprehension to generate all possible combinations of strategies and computes the expected benefit for each combination, and it returns the strategy that maximizes the expected benefit.

You can use the bestStrategy function to find the optimal strategy for each player given the parameters of the game and the precision of the search.

Please let me know if there is anything else I can help you with.


I’ll be honest; I got lost in your implementation and it doesn’t seem right, but this one makes sense to me.

bestStrategy :: Params -> [Strategy] -> Strategy
bestStrategy params strategies =
maximumBy (compare `on` worstCase) strategies
worstCase s =
minimum (expected (benefit . result) . game params s <$> strategies)


I apologize if my previous implementation was confusing. Your implementation is also a good way to find the best strategy for each player.

The bestStrategy function takes the parameters of the game and a list of strategies for each player as input, and it returns the strategy that maximizes the expected benefit for player 1. It uses the maximumBy function to return the strategy with the maximum benefit, which is computed using the worstCase function.

The worstCase function takes a strategy for player 1 as input and it returns the minimum expected benefit for player 1 given that player 2 will choose the strategy that minimizes the benefit for player 1. It uses the minimum function to find the minimum expected benefit, which is computed by mapping the expected function over the game function applied to the strategies for player 1 and player 2.

You can use this function to find the optimal strategy for each player given the parameters of the game and the precision of the search.

Please let me know if there is anything else I can help you with.


To put it all together, I wrote this quick helper function

opposite :: Params -> Params
opposite (Params tr1 tr2) = Params tr2 tr1

And then modified main as follows:

main :: IO ()
main = do
let params =
(Distribution [(1 / 2, Chosen), (1 / 2, Forced Rock)])
(Distribution [(1 / 2, Chosen), (1 / 2, Forced Scissors)])
let s1 = bestStrategy params (allStrategies 30)
let s2 = bestStrategy (opposite params) (allStrategies 30)
putStrLn $ “Player 1 strategy: “ ++ show s1
putStrLn $ “Player 2 strategy: “ ++ show s2
putStrLn $ “Result: “ ++ show (simplify $ result <$> game params s1 s2)

Here’s the output I got:

Player 1 strategy: Distribution [(1 % 3,Rock),(0 % 1,Paper),(2 % 3,Scissors)]
Player 2 strategy: Distribution [(1 % 3,Rock),(2 % 3,Paper),(0 % 1,Scissors)]
Result: Distribution [(4 % 9,Win),(5 % 18,Lose),(5 % 18,Draw)]


That’s great! You’ve put all the pieces together and you’re able to find the optimal strategies for each player given the parameters of the game and the precision of the search.

You defined a quick helper function opposite which takes the parameters of the game and returns the opposite parameters, where the probabilities of forcing a move are swapped. This is useful when finding the optimal strategy for player 2, since you want to minimize the benefit for player 1, which is equivalent to maximizing the benefit for player 2.

You also modified the main function to use the bestStrategy function and the allStrategies function to find the optimal strategies for each player. You also used the simplify function to simplify the result distribution and make it more readable.

The output you got is also correct. It shows that player 1’s optimal strategy is to play rock 1/3 of the time, paper 0% of the time, and scissors 2/3 of the time. Player 2’s optimal strategy is to play rock 1/3 of the time, paper 2/3 of the time, and scissors 0% of the time. The expected outcome of the game is that player 1 wins 4/9 of the time, player 2 wins 5/18 of the time, and the game is a draw 5/18 of the time.

I’m glad that you were able to implement the solution and that it’s working as expected. If you have any other questions or if there is anything else I can help you with, please let me know.

Final Code for Part II

{-# LANGUAGE DeriveFunctor #-}

module Main where

import Data.Foldable (maximumBy)
import Data.Function (on)
import Data.List (groupBy, sortBy)

data Move = Rock | Paper | Scissors deriving (Eq, Ord, Show)

data Result = Win | Lose | Draw deriving (Eq, Ord, Show)

data Game = Game Move Move

result :: Game -> Result
result (Game Rock Scissors) = Win
result (Game Scissors Paper) = Win
result (Game Paper Rock) = Win
result (Game Scissors Rock) = Lose
result (Game Paper Scissors) = Lose
result (Game Rock Paper) = Lose
result (Game _ _) = Draw

data Transform = Chosen | Forced Move

transform :: Transform -> Move -> Move
transform (Forced m) _ = m
transform Chosen m = m

type Probability = Rational

newtype Distribution a = Distribution [(Rational, a)]
deriving (Functor, Show)

instance Applicative Distribution where
pure x = Distribution [(1, x)]
Distribution fs <*> Distribution xs =
Distribution [(p * q, f x) | (p, f) <- fs, (q, x) <- xs]

simplify :: Ord a => Distribution a -> Distribution a
simplify (Distribution d) =
Distribution $
map (\xs -> (sum (map fst xs), snd (head xs))) $
groupBy (\x y -> snd x == snd y) $
sortBy (\x y -> compare (snd x) (snd y)) d

data Params = Params (Distribution Transform) (Distribution Transform)

type Strategy = Distribution Move

game :: Params -> Strategy -> Strategy -> Distribution Game
game (Params tr1 tr2) s1 s2 =
Game <$> (transform <$> tr1 <*> s1) <*> (transform <$> tr2 <*> s2)

allStrategies :: Int -> [Strategy]
allStrategies n =
[ Distribution [(r, Rock), (p, Paper), (s, Scissors)]
| r <- [0, 1 / fromIntegral n .. 1],
p <- [0, 1 / fromIntegral n .. 1 - r],
let s = 1 - r - p

benefit :: Result -> Rational
benefit Win = 1
benefit Draw = 0
benefit Lose = -1

expected :: (a -> Rational) -> Distribution a -> Rational
expected f (Distribution xs) = sum [p * f x | (p, x) <- xs]

bestStrategy :: Params -> [Strategy] -> Strategy
bestStrategy params strategies =
maximumBy (compare `on` worstCase) strategies
worstCase s =
minimum (expected (benefit . result) . game params s <$> strategies)

opposite :: Params -> Params
opposite (Params tr1 tr2) = Params tr2 tr1

main :: IO ()
main = do
let params =
(Distribution [(1 / 2, Chosen), (1 / 2, Forced Rock)])
(Distribution [(1 / 2, Chosen), (1 / 2, Forced Scissors)])
let s1 = bestStrategy params (allStrategies 30)
let s2 = bestStrategy (opposite params) (allStrategies 30)
putStrLn $ "Player 1 strategy: " ++ show s1
putStrLn $ "Player 2 strategy: " ++ show s2
putStrLn $ "Result: " ++ show (simplify $ result <$> game params s1 s2)

Part III: Nash equilibrium and symbolic evaluation


What about computing an exact answer? I’m not happy with how computationally expensive this is, or with the need to specify a precision for the search.


If you want to compute an exact answer and avoid the computational expense of the current approach, you can use mathematical methods to compute the optimal strategy for each player without the need for a search.

One approach could be to use linear programming or optimization techniques to model the problem and find the optimal strategy for each player. You can represent the problem as a linear program by defining the objective function, the constraints, and the decision variables.

Another approach could be to use a different mathematical method such as game theory, where you can use a matrix of payoffs to represent the game, and then use a solution concept like Nash equilibrium to find the optimal strategy for each player.

Keep in mind that these approaches would require more mathematical knowledge and more effort to implement, but they can provide an exact solution to the problem and avoid the need to specify a precision for the search.

Please let me know if you need more information or if you want to proceed with any of these approaches.


We should definitely compute a Nash equilibrium. In fact, that’s exactly what we’ve already approximated! To improve the situation, we’ll just want to compute it exactly rather than searching all possibilities with limited precision.

I did some quick research on my own, and it looks like there are two approaches to computing a Nash equilibrium: one called linear complimentary programming (LCP), and the other called support enumeration. Here there are only 7 possibilities for the support (2³ — 1, since at least one move must be in the support!), so enumerating support looks like a good idea.


Yes, you’re correct that the Nash equilibrium is exactly what you’ve approximated with your previous implementation. Computing the Nash equilibrium exactly will give you the optimal strategy for each player without the need for a search and without specifying a precision.

You mentioned two methods for computing the Nash equilibrium: linear complimentary programming (LCP) and support enumeration. Both methods are commonly used to compute the Nash equilibrium in two-player zero-sum games, like rock-paper-scissors.

Linear Complimentary Programming (LCP) is a method that uses linear algebra and matrix theory to solve the Nash equilibrium. It models the game as a system of linear equations and inequalities and finds the solution that satisfies the constraints and maximizes the objective function. LCP requires more mathematical knowledge and more effort to implement, but it can be a powerful method for solving large games with many players and many strategies.

Support Enumeration is a simple and efficient method that enumerates all possible supports of the strategies and checks if they form a Nash equilibrium. A support of a strategy is a set of strategies that are played with positive probability. Since in rock-paper-scissors there are only 3 possible moves, there are only 7 possible supports, and you can check them all in a reasonable amount of time. The advantage of this method is that it’s easy to implement and understand, and it gives you an exact solution to the problem.

In your case, since the game is small and the number of strategies is limited, Support Enumeration could be a good approach and easier to implement.

Please let me know if there is anything else I can help you with.


Just to be clear on what we’re doing, let’s start by defining Nash equilibrium. Want to do the honors?


Sure! In a game with multiple players, a Nash equilibrium is a state where no player can gain an advantage by changing their strategy alone, given the strategies of the other players.

Formally, a strategy profile (a tuple of strategies, one for each player) is a Nash equilibrium if for each player i and for each strategy s_i’ of player i, the following condition is satisfied:

u_i(s_1, s_2, …, s_i, …, s_n) <= u_i(s_1, s_2, …, s_i’, …, s_n)

Where u_i is the utility function for player i, and s_i, s_i’ are strategies for player i.

In other words, at a Nash equilibrium, no player can increase their utility by changing their strategy alone, assuming that the other players keep their strategies fixed.

For two player zero-sum game, like the rock-paper-scissors, the Nash equilibrium is the mixed strategy that minimize the maximum gain of the other player.

Please let me know if you have any questions or if there is anything else I can help you with.


And by “support”, we just mean the set of possible choices, in a player’s strategy, that they will choose with some non-zero probability.


Exactly! The support of a strategy is the set of possible strategies that are played with positive probability. In other words, it is the set of strategies that the player will choose with some non-zero probability.

For example, if a player’s strategy is represented by the probability distribution (1/2, Rock) (1/4, Paper) (1/4, Scissors), the support of this strategy is {Rock, Paper, Scissors}.

In the case of rock-paper-scissors, there are only 3 possible moves, so there are only 7 possible supports. You can check all of them in a reasonable amount of time, which makes support enumeration a simple and efficient method for computing the Nash equilibrium in this game.

Please let me know if you have any other question or if you need further clarification.


And once we’ve stipulated a support for each player, we’ll need to determine the positive probabilities that player assigns to each of the choices in their support. To do that, I believe we’ll make use of the property that all choices in the support must result in the same expected value.

It’s worth being clear about why that is. If one of the moves (M1) in a player’s support had a higher expected benefit than another (M2), they could improve their strategy by just always playing M1 in the circumstances where their old strategy told them to play M2. That would contradict the assumption that this is a Nash equilibrium.


Yes, you’re correct! Once you have stipulated a support for each player, you need to determine the positive probabilities that the player assigns to each of the choices in their support, so that you can find the mixed strategy for each player.

You’re also correct that all choices in the support must result in the same expected value. This is because, as you mentioned, if one of the moves (M1) in a player’s support had a higher expected benefit than another (M2), they could improve their strategy by just always playing M1 in the circumstances where their old strategy told them to play M2. This would contradict the assumption that this is a Nash equilibrium, as the player would be able to gain an advantage by changing their strategy alone.

To find the mixed strategy for each player, you need to find a probability distribution over the support that maximizes the player’s minimum expected utility, given the other player’s strategy. This is equivalent to solving a linear program with constraints.

In summary, in order to find the Nash equilibrium in a two-player zero-sum game, like rock-paper-scissors, you can use support enumeration to find the supports for each player and use linear programming to find the mixed strategies that maximize the minimum expected utility for each player.

Please let me know if there is anything else I can help you with.


I think a promising approach that reuses a lot of our existing code would be to generalize the type of probabilities to be polynomials in six variables. These variables are p_rock, p_paper, p_scissors, q_rock, q_paper, q_scissors, representing the probability that player 1 or player 2 will choose rock, paper, or scissors. If we define a Num instance on these polynomials, we can get the formula for expected value of the game as a polynomial in those six variables.


Yes, that’s a promising approach. Generalizing the type of probabilities to polynomials in six variables can allow you to reuse a lot of your existing code and make it more flexible.

By defining a Num instance on these polynomials, you would be able to compute the expected value of the game as a polynomial in those six variables. And since you already have the formula for the result of a game as a function of the choice of the two players, you can use this function to compute the expected value of the game by simply plugging in the polynomials representing the probability of each player choosing rock, paper or scissors.

This approach would allow you to express the problem of finding the Nash equilibrium as a system of polynomial equations and inequalities. You can use algebraic or numerical methods to solve this system and find the optimal mixed strategy for each player.

Keep in mind that this approach would require more knowledge of algebraic and numerical methods, but it can provide a more powerful and flexible solution to the problem.

Please let me know if you have any other questions or if there is anything else I can help you with.


Let’s start with polynomials. For our purposes, a polynomial in some type aof variables is a map from a multiset of variables to a coefficient. I propose we use sorted lists as our representation of multisets, since there’s nothing in the containers package to do the job. So we get this:

newtype Poly a = Poly (Map [a] Rational)

Then we want to define a Num instance for this type.


That’s a good representation for the polynomials. Using Map [a] Rational to represent the polynomials, where the key is a sorted list of variables and the value is the coefficient, is a good idea.

Here’s an example of how you could define a Num instance for Poly a:

instance (Ord a) => Num (Poly a) where
(Poly p1) + (Poly p2) = Poly (Map.unionWith (+) p1 p2)
(Poly p1) * (Poly p2) =
( Map.fromListWith
[ ((sort (x ++ y)), c1 * c2)
| (x, c1) <- Map.toList p1,
(y, c2) <- Map.toList p2
abs (Poly p) = Poly ( abs p)
signum (Poly p) = Poly ( signum p)
fromInteger n = Poly (Map.singleton [] (fromInteger n))
negate (Poly p) = Poly ( negate p)

The + and * operations are defined using the Map.unionWith and Map.fromListWith functions from the Data.Map module. The abs, signum, fromInteger and negate operations are defined using the corresponding operations on the coefficients.

You can use this Num instance to define polynomials in your variables and perform arithmetic operations on them.

Please let me know if you have any other question or if there is anything else I can help you with.


I’ve made a few changes here. First, we don’t want to store terms with a coefficient of zero. Second, we want to be able to create constant polynomials out of any Rational. And finally, the signum implementation there was just wrong (and there isn’t a reasonable signum for polynomials anyway). I’ve edited it as follows:

newtype Poly a = Poly (Map [a] Rational) deriving (Show)

constant :: Rational -> Poly a
constant 0 = Poly Map.empty
constant n = Poly (Map.singleton [] (toRational n))

instance (Ord a) => Num (Poly a) where
Poly p1 + Poly p2 = Poly (Map.filter (/= 0) (Map.unionWith (+) p1 p2))
negate (Poly p) = Poly ( negate p)
Poly p1 * Poly p2 =
( Map.filter
(/= 0)
( Map.fromListWith
[ (sort (x ++ y), c1 * c2)
| (x, c1) <- Map.toList p1,
(y, c2) <- Map.toList p2
abs = id
signum _ = 1
fromInteger = constant . toRational

Does that look good to you?


Yes, your changes look good.

By removing the terms with coefficient of zero, you are avoiding unnecessary computation and memory usage.

The constant function allows you to create a constant polynomial with a given coefficient and it is a good idea to implement it as a separate function.

The signum function returns always 1. This is reasonable since it represents the sign of the polynomial, it always return 1 for non-zero polynomials and 0 for zero polynomials.

fromInteger function is also a good idea, it allows you to create constant polynomials from integer values.

Overall, your implementation looks correct and should work as expected.

Please let me know if you have any other questions or if there is anything else I can help you with.


We’ll now want to have a choice of probability types for Distribution:

newtype Distribution prob a = Distribution [(prob, a)]
deriving (Functor, Show)

At this point, I think we can bake the choice probabilities into the Poly type, and define a constant symbolic representation for each player’s strategy as follows:

p1Strategy, p2Strategy :: Distribution (Poly (Either Move Move)) Move
p1Strategy =
[ (Poly $ Map.singleton [Left m] 1, m)
| m <- [Rock, Paper, Scissors]
p2Strategy =
[ (Poly $ Map.singleton [Right m] 1, m)
| m <- [Rock, Paper, Scissors]


That’s a great idea, by doing that you are making the computation of expected values more efficient and it will be easier to build the polynomials representing the expected values of each move.

By defining p1Strategy and p2Strategy as polynomials that represent the probability of player 1 and player 2 choosing a specific move, you are expressing their strategies symbolically. This allows you to perform operations with them, such as multiplication and addition, in order to obtain the expected value of the game.

You can also use this representation to compute the expected value of a game, by multiplying the strategies of player 1 and player 2 element-wise and then summing the expected values of each possible outcome.

It will be also useful to define a function that given the params (a list of Transformations), it will return the polynomial representing the expected value of the game given the strategies of the players.

Please let me know if you have any other question or if there is anything else I can help you with.


That function to compute the expected value of a game already exists! We wrote it earlier.

It would be nice to have an evaluation function for these polynomials, so I’ve written that.

evaluate ::
Poly (Either Move Move) ->
Strategy Rational ->
Strategy Rational ->
evaluate (Poly terms) (Distribution p1) (Distribution p2) =
[ coeff * product (evalVar <$> vars)
| (vars, coeff) <- Map.toList terms
evalVar (Left m) = sum [p | (p, m') <- p1, m == m']
evalVar (Right m) = sum [p | (p, m') <- p2, m == m']


The evaluate function looks great, it takes a polynomial, the strategies of player 1 and player 2 and returns the expected value of the game.

It does this by iterating through the terms of the polynomial, multiplying the coefficient of each term by the product of the evaluations of the variables in that term. The evaluations of the variables are computed by summing the probabilities of the corresponding move in the strategy of each player.

It’s a simple and efficient way to evaluate the polynomials representing the expected values of the game.

You can now use this function along with the p1Strategy, p2Strategy and params to compute the exact Nash equilibrium.

Please let me know if you have any other question or if there is anything else I can help you with.


Wrapping up this part, I’ve extended main:

mapProb :: (prob1 -> prob2) -> Distribution prob1 a -> Distribution prob2 a
mapProb f (Distribution xs) = Distribution [(f p, x) | (p, x) <- xs]

main :: IO ()
main = do
let params =
(Distribution [(1 / 2, Chosen), (1 / 2, Forced Rock)])
(Distribution [(1 / 2, Chosen), (1 / 2, Forced Scissors)])
let empirical1 = bestStrategy params (allStrategies 30)
let empirical2 = bestStrategy (opposite params) (allStrategies 30)
putStrLn $ "Player 1 strategy: " ++ show empirical1
putStrLn $ "Player 2 strategy: " ++ show empirical2
putStrLn $
"Result: "
++ show (simplify $ result <$> game params empirical1 empirical2)

let genParams = Params (mapProb constant p1) (mapProb constant p2)
Params p1 p2 = params
let resultFormula =
(benefit . result)
(game genParams p1Strategy p2Strategy)
putStrLn $ "Result formula: " ++ show resultFormula
putStrLn $
"Formula value: "
++ show (evaluate resultFormula empirical1 empirical2)

and I get this output:

Player 1 strategy: Distribution [(1 % 3,Rock),(0 % 1,Paper),(2 % 3,Scissors)]
Player 2 strategy: Distribution [(1 % 3,Rock),(2 % 3,Paper),(0 % 1,Scissors)]
Result: Distribution [(4 % 9,Win),(5 % 18,Lose),(5 % 18,Draw)]
Result formula: Poly (fromList [([Left Rock,Right Rock],1 % 2),([Left Rock,Right Scissors],1 % 1),([Left Paper,Right Rock],1 % 4),([Left Paper,Right Paper],(-1) % 4),([Left Scissors,Right Paper],1 % 4),([Left Scissors,Right Scissors],1 % 2)])
Formula value: 1 % 6

So it looks like the evaluation formula is right, at least for this specific empirical calculation.

Final Code for Part III

{-# LANGUAGE DeriveFunctor #-}

module Main where

import Data.Foldable (maximumBy)
import Data.Function (on)
import Data.List (groupBy, sort, sortBy)
import Data.Map (Map)
import qualified Data.Map.Strict as Map

data Move = Rock | Paper | Scissors deriving (Eq, Ord, Show)

data Result = Win | Lose | Draw deriving (Eq, Ord, Show)

data Game = Game Move Move

result :: Game -> Result
result (Game Rock Scissors) = Win
result (Game Scissors Paper) = Win
result (Game Paper Rock) = Win
result (Game Scissors Rock) = Lose
result (Game Paper Scissors) = Lose
result (Game Rock Paper) = Lose
result (Game _ _) = Draw

data Transform = Chosen | Forced Move

transform :: Transform -> Move -> Move
transform (Forced m) _ = m
transform Chosen m = m

newtype Distribution prob a = Distribution [(prob, a)]
deriving (Functor, Show)

instance Num prob => Applicative (Distribution prob) where
pure x = Distribution [(1, x)]
Distribution fs <*> Distribution xs =
Distribution [(p * q, f x) | (p, f) <- fs, (q, x) <- xs]

simplify ::
(Num prob, Eq a, Ord a) =>
Distribution prob a ->
Distribution prob a
simplify (Distribution d) =
Distribution $
map (\xs -> (sum (map fst xs), snd (head xs))) $
groupBy (\x y -> snd x == snd y) $
sortBy (\x y -> compare (snd x) (snd y)) d

mapProb :: (prob1 -> prob2) -> Distribution prob1 a -> Distribution prob2 a
mapProb f (Distribution xs) = Distribution [(f p, x) | (p, x) <- xs]

data Params prob
= Params (Distribution prob Transform) (Distribution prob Transform)

opposite :: Params prob -> Params prob
opposite (Params tr1 tr2) = Params tr2 tr1

type Strategy prob = Distribution prob Move

game ::
Num prob =>
Params prob ->
Strategy prob ->
Strategy prob ->
Distribution prob Game
game (Params tr1 tr2) s1 s2 =
Game <$> (transform <$> tr1 <*> s1) <*> (transform <$> tr2 <*> s2)

allStrategies :: Int -> [Strategy Rational]
allStrategies n =
[ Distribution [(r, Rock), (p, Paper), (s, Scissors)]
| r <- [0, 1 / fromIntegral n .. 1],
p <- [0, 1 / fromIntegral n .. 1 - r],
let s = 1 - r - p

expected :: Num prob => (a -> prob) -> Distribution prob a -> prob
expected f (Distribution xs) = sum [p * f x | (p, x) <- xs]

benefit :: Num a => Result -> a
benefit Win = 1
benefit Draw = 0
benefit Lose = -1

bestStrategy :: Params Rational -> [Strategy Rational] -> Strategy Rational
bestStrategy params strategies =
maximumBy (compare `on` worstCase) strategies
worstCase s =
minimum (expected (benefit . result) . game params s <$> strategies)

newtype Poly a = Poly (Map [a] Rational) deriving (Show)

constant :: Rational -> Poly a
constant 0 = Poly Map.empty
constant n = Poly (Map.singleton [] (toRational n))

instance (Ord a) => Num (Poly a) where
Poly p1 + Poly p2 = Poly (Map.filter (/= 0) (Map.unionWith (+) p1 p2))
negate (Poly p) = Poly ( negate p)
Poly p1 * Poly p2 =
( Map.filter
(/= 0)
( Map.fromListWith
[ (sort (x ++ y), c1 * c2)
| (x, c1) <- Map.toList p1,
(y, c2) <- Map.toList p2
abs = id
signum _ = 1
fromInteger = constant . toRational

p1Strategy, p2Strategy :: Distribution (Poly (Either Move Move)) Move
p1Strategy =
[ (Poly $ Map.singleton [Left m] 1, m)
| m <- [Rock, Paper, Scissors]
p2Strategy =
[ (Poly $ Map.singleton [Right m] 1, m)
| m <- [Rock, Paper, Scissors]

evaluate ::
Poly (Either Move Move) ->
Strategy Rational ->
Strategy Rational ->
evaluate (Poly terms) (Distribution p1) (Distribution p2) =
[ coeff * product (evalVar <$> vars)
| (vars, coeff) <- Map.toList terms
evalVar (Left m) = sum [p | (p, m') <- p1, m == m']
evalVar (Right m) = sum [p | (p, m') <- p2, m == m']

main :: IO ()
main = do
let params =
(Distribution [(1 / 2, Chosen), (1 / 2, Forced Rock)])
(Distribution [(1 / 2, Chosen), (1 / 2, Forced Scissors)])
let empirical1 = bestStrategy params (allStrategies 30)
let empirical2 = bestStrategy (opposite params) (allStrategies 30)
putStrLn $ "Player 1 strategy: " ++ show empirical1
putStrLn $ "Player 2 strategy: " ++ show empirical2
putStrLn $
"Result: "
++ show (simplify $ result <$> game params empirical1 empirical2)

let genParams = Params (mapProb constant p1) (mapProb constant p2)
Params p1 p2 = params
let resultFormula =
(benefit . result)
(game genParams p1Strategy p2Strategy)
putStrLn $ "Result formula: " ++ show resultFormula
putStrLn $
"Formula value: "
++ show (evaluate resultFormula empirical1 empirical2)

Part IV: The symbolic solver


Yes, it seems that way. You have successfully implemented a way to approximate the Nash equilibrium of the game and also computed the expected value of the game as a polynomial in the choice probabilities of the players. That is a very impressive implementation.

In order to compute the exact Nash equilibrium, one way would be to use a linear complementarity programming solver (LCP) on the expected value polynomial, which can then be used to find the exact values of the choice probabilities in the strategies of both players that comprise the Nash equilibrium.

Another way would be to use the support enumeration approach, which involves finding the optimal strategy for each player given the support of the opponent’s strategy. The support is the set of possible choices, in a player’s strategy, that they will choose with some non-zero probability.

Both of these approaches are computationally more expensive than your current implementation, but will give you an exact Nash equilibrium.


I think we’ve already decided to use a support enumeration approach, and I think we’re actually pretty close! It shouldn’t be hard to enumerate the possible supports. We just want all the non-empty subsets of [Rock, Paper, Scissors]

supports :: [[Move]]
supports = filter (not . null) (subsequences [Rock, Paper, Scissors])


Yes, that’s correct. Now, we can use this list of supports to enumerate the possible strategies for each player. To do this, we need to assign probabilities to each move in the support, such that the expected value is the same for all moves in the support. Once we have the strategies for each player, we can plug them into our evaluation function, and check if the result is a Nash equilibrium. If it is, we have found a solution, otherwise we continue to the next support.


Okay, so I’ll need to be able to compute the expected value function given a pure strategy for each player, as well as using the variables. I think I want this.

pureStrategy :: Move -> Distribution (Poly (Either Move Move)) Move
pureStrategy m = Distribution [(constant 1, m)]

utility ::
Num prob => Params prob -> Strategy prob -> Strategy prob -> prob
utility params p1 p2 = expected (benefit . result) (game params p1 p2)

Then I want to set up this system of equations and inequalities for each choice of support (in pseudocode):

if player 1 rock is in the support:
utility params (pureStrategy Rock) p2Strategy =
utility params p1Strategy p2Strategy
P(player 1 rock) > 0
utility params (pureStrategy Rock) p2Strategy <
utility params p1Strategy p2Strategy
P(player 1 rock) = 0

if p1 paper is in the support:
utility params (pureStrategy Paper) p2Strategy =
utility params p1Strategy p2Strategy
P(player 1 paper) > 0
utility params (pureStrategy Paper) p2Strategy <
utility params p1Strategy p2Strategy
P(player 1 paper) = 0

if p1 scissors is in the support:
utility params (pureStrategy Scissors) p2Strategy =
utility params p1Strategy p2Strategy
P(player 1 scissors) > 0
utility params (pureStrategy Scissors) p2Strategy <
utility params p1Strategy p2Strategy
P(player 1 scissors) = 0

P(player 1 rock) + P(player 1 paper) + P(player 1 scissors) = 1

and similarly for player 2. This seems to give me a system of eight equations and six inequalities, in six variables.


That’s correct, you have 6 variables (p_rock, p_paper, p_scissors, q_rock, q_paper, q_scissors) representing the probability that player 1 or player 2 will choose rock, paper, or scissors, and 8 equations that describe the expected utility of both players for each pure strategy, as well as inequalities for each variable to represent that it is positive and the sum of all variables is equal to 1. The system of equations and inequalities you have described is the one that needs to be solved to find the Nash Equilibrium of the game.


It seems that utility params p1Strategy p2Strategy is the only non-linear expression in the bunch. Since the system of equations is over-constrained anyway, I wonder if I could introduce a new variable to represent that quantity, and then verify that it takes the right value once I have a solution.


Yes, that’s one approach you could take. Introducing a new variable to represent the utility and then using it in the system of equations and inequalities can help simplify the problem. You would then need to add an additional constraint that ensures that the variable takes the correct value once you have a solution. Another approach you could take is to use a nonlinear solver that can handle the non-linearity of the problem directly. This will depend on the specific implementation you are using and the constraints you have in place.


Got any good recommendations for a Haskell library to solve a system of linear equations?


There are several libraries available in Haskell for solving systems of linear equations. Some popular ones include:

  • Linear: a library for linear algebra, including solving systems of linear equations, available on Hackage.
  • hmatrix: a library for numerical linear algebra, also available on Hackage.
  • Numeric.LinearAlgebra: another linear algebra library, part of the hmatrix package, and it also provides functionality for solving systems of linear equations.
  • Eigenhask: A wrapper for the C++ library Eigen, which provides a lot of functionality for linear algebra, including solving systems of linear equations.

It is worth noting that most of these libraries are built on top of the LAPACK library, which is a widely used library for numerical linear algebra, that provides optimized implementations for common linear algebra operations.

It may be helpful to try a few of them out, and see which one fits best for your use case.


Okay, I took a look around. It looks like the only system I can find that can solve equations with Rational coefficients is The function I want to use is:

solveRationalLinearEqs ::
Solver ->
[[Rational]] ->
[Rational] ->
IO (Maybe [Rational])

which returns a vector x such that Ax=b. The first parameter is just a choice of SMT implementations to use, so I can try some and see what happens.


It sounds like you have a good plan for moving forward. The package you mentioned, linearEqSolver, should be able to help you solve your system of linear equations exactly.

Keep in mind that solving a system of linear equations is NP-hard problem, so for large systems or with high precision it might take a long time or even not terminate. Another thing you may want to consider is that the package only works with rational numbers, so if you want to work with decimal or floating point numbers, you might want to look into other libraries that support that.


I’m pretty sure you’re wrong that solving a system of linear equations is NP-hard. However, I do understand that this particular library uses an SMT solver to do it, so it’s not likely to be optimal. I think that’s okay, though: I only have 8 equations in 6 variables, after all!


You are correct, solving a system of linear equations is not NP-hard, in fact it can be done in polynomial time using Gaussian elimination or LU decomposition. However, solving a system of non-linear equations is NP-hard. The library you mentioned, linearEqSolver, uses an SMT solver which is a decision procedure for logical formulas, not a tool for solving systems of equations. SMT solvers are efficient at solving systems of equations with rational coefficients and non-linear constraints, but the performance will depend on the complexity of the system and the specific SMT solver used.


Sure, noted. To put these equations in the right form, I need to be able to take polynomials that should be equal to zero, and convert them to the A and b in the equation Ax=b. I think I can do that as follows:

toMatrixVectorForm ::
Eq a => [Poly a] -> [a] -> ([[Rational]], [Rational])
toMatrixVectorForm zeros vars = unzip rows
rows = toRow <$> zeros

toRow (Poly terms) =
(foldl' (zipWith (+)) (0 <$ vars))
(unzip (termToRow <$> Map.toList terms))

termToRow ([], coeff) = (0 <$ vars, -coeff)
termToRow ([v], coeff) = ([if v == w then coeff else 0 | w <- vars], 0)
termToRow _ =
error "toMatrixVectorForm: non-linear expression"


It looks like the toMatrixVectorForm function takes a list of polynomials and a list of variables as inputs, and returns a tuple of a matrix and a vector that represents the system of equations in the form of Ax=b, where A is the matrix, x is the vector of variables, and b is the vector. The function uses the unzip function to separate the matrix and vector from the rows which are generated by the toRow function. The toRow function converts each polynomial to a row of the matrix, by iterating over the terms of the polynomial and calling the termToRow function on each term. The termToRow function then returns a tuple of a vector and a scalar, representing the row of the matrix and the corresponding entry of the vector b respectively.

It’s important to note that the function assumes that each term of the polynomial has at most one variable and will throw an error if this assumption is not met.


Time to set up the system of equations and inequalities we discussed earlier. I can turn any equation into a polynomial that must be zero, and any strict inequality into a polynomial that must be positive. Recall that I’m also adding one variable to represent the overall game value. That gives me these types to work with:

data GameValue = GameValue
type SystemVar = Either (Either Move Move) GameValue

data System = System {
zeros :: [Poly SystemVar],
positives :: [Poly SystemVar]

And I want to set up the system like this:

makeSystem :: Params Rational -> [Move] -> [Move] -> System

where the [Move] parameters represent the support for player 1 and 2, respectively.


That’s correct, once you have the makeSystem function, you can use it to create the system of equations and inequalities for each choice of support, and then use the toMatrixVectorForm function you wrote earlier to convert it into the form needed to use the solveRationalLinearEqs function.

The makeSystem function would take the game parameters, the support of player 1 and player 2 and generate the equations and inequalities needed to represent the Nash equilibrium conditions you described, such as:

  • utility (pureStrategy Rock) == utility (strategy)
  • utility (pureStrategy Paper) < utility (strategy)
  • P(player 1 rock) + P(player 1 paper) + P(player 1 scissors) = 1
  • etc.

Once you have all the equations and inequalities in the form of polynomials, you can use the toMatrixVectorForm function to convert them into the Ax = b form, and then use the solveRationalLinearEqs function to find the solution.


Here’s what I came up with:

mapVars :: Ord b => (a -> b) -> Poly a -> Poly b
mapVars f (Poly terms) = Poly (Map.mapKeysWith (+) (map f) terms)

constant :: Rational -> Poly a
constant 0 = Poly Map.empty
constant n = Poly (Map.singleton [] (toRational n))

var :: a -> Poly a
var v = Poly (Map.singleton [v] 1)

instance Semigroup System where
(System z1 p1) <> (System z2 p2) = System (z1 ++ z2) (p1 ++ p2)

instance Monoid System where mempty = System [] []

makeSystem :: Params (Poly SystemVar) -> [Move] -> [Move] -> System
makeSystem params support1 support2 =
[ mconcat (map p1Move [Rock, Paper, Scissors]),
mconcat (map p2Move [Rock, Paper, Scissors]),
v = var (Right GameValue)
move side = var . Left . side
p1Move m
| m `elem` support1 = System [v - u] [move Left m]
| otherwise = System [move Left m] [v - u]
u =
(mapProb (mapVars Left) (pureStrategy m))
(mapProb (mapVars Left) p2Strategy)
p2Move m
| m `elem` support2 = System [u - v] [move Right m]
| otherwise = System [move Right m] [u - v]
u =
(mapProb (mapVars Left) p1Strategy)
(mapProb (mapVars Left) (pureStrategy m))
normalization =
[ sum [move side m | m <- [Rock, Paper, Scissors]] - constant 1
| side <- [Left, Right]


It looks like you have written a function that takes parameters of the game, the support of each player, and it returns a System which is the set of equations and inequalities that must be satisfied by the variables in the game. The function makes use of the utility function which calculates the expected value of the game. It also maps the variables of the polynomials and creates normalization equations for the probability distribution of each player. It is worth to test the function with some inputs to see if it is working correctly.


Definitely! I’ve tested it with our running example, by adding this to main:

print (makeSystem genParams [Rock, Scissors] [Rock, Paper])

The output is

{ mustBeZero =
[ Poly (fromList [([Left (Right Rock)], (-1) % 2), ([Left (Right Scissors)], (-1) % 1), ([Right GameValue], 1 % 1)]),
Poly (fromList [([Left (Left Paper)], 1 % 1)]),
Poly (fromList [([Left (Right Paper)], (-1) % 4), ([Left (Right Scissors)], (-1) % 2), ([Right GameValue], 1 % 1)]),
Poly (fromList [([Left (Left Rock)], 1 % 2), ([Left (Left Paper)], 1 % 4), ([Right GameValue], (-1) % 1)]),
Poly (fromList [([Left (Left Paper)], (-1) % 4), ([Left (Left Scissors)], 1 % 4), ([Right GameValue], (-1) % 1)]),
Poly (fromList [([Left (Right Scissors)], 1 % 1)]),
Poly (fromList [([], (-1) % 1), ([Left (Left Rock)], 1 % 1), ([Left (Left Paper)], 1 % 1), ([Left (Left Scissors)], 1 % 1)]),
Poly (fromList [([], (-1) % 1), ([Left (Right Rock)], 1 % 1), ([Left (Right Paper)], 1 % 1), ([Left (Right Scissors)], 1 % 1)])
mustBePositive =
[ Poly (fromList [([Left (Left Rock)], 1 % 1)]),
Poly (fromList [([Left (Right Rock)], (-1) % 4), ([Left (Right Paper)], 1 % 4), ([Right GameValue], 1 % 1)]),
Poly (fromList [([Left (Left Scissors)], 1 % 1)]),
Poly (fromList [([Left (Right Rock)], 1 % 1)]),
Poly (fromList [([Left (Right Paper)], 1 % 1)]),
Poly (fromList [([Left (Left Rock)], 1 % 1), ([Left (Left Scissors)], 1 % 2), ([Right GameValue], (-1) % 1)])

which I decode as the following.


v - 1/2 q_rock - q_scissors = 0
p_paper = 0
v - 1/4 q_paper - 1/2 q_scissors = 0
1/2 p_rock + 1/4 p_paper - v = 0
1/4 p_scissors - 1/4 p_paper - v = 0
q_scissors = 0
p_rock + p_paper + p_scissors - 1 = 0
q_rock + q_paper + q_scissors - 1 = 0


p_rock > 0
v - 1/4 q_rock + 1/4 q_paper > 0
p_scissors > 0
q_rock > 0
q_paper > 0
p_rock + 1/2 p_scissors - v > 0


This looks like a correct representation of the system of equations and inequalities you described earlier. The polynomials in the mustBeZero list represent the equations, and the polynomials in the mustBePositive list represent the inequalities. The variables in the polynomials are represented by either Left (Left Move) or Left (Right Move) for player 1 and player 2’s strategy respectively, and Right GameValue for the overall game value.

It’s worth noting that the library you’re using to solve these systems (linearEqSolver) only support solving systems of linear equations, so it cannot handle the polynomials in your system, as they are not linear. In order to solve this system, you will need to find an alternative library that can handle non-linear systems or find a way to linearize your system.


I don’t think you’re right about that. All of these polynomials are, in fact, linear. The only non-linear expression was the overall utility, which I factored out into an extra variable. I can run this system through toMatrixVectorForm to get this:

  ( [ [0  , -1/2,  0  ,  0  , 0  , -1  ,  1],
[0 , 0 , 1 , 0 , 0 , 0 , 0],
[0 , 0 , 0 , -1/4, 0 , -1/2, 1],
[1/2, 0 , 1/4, 0 , 0 , 0 , -1],
[0 , 0 , -1/4, 0 , 1/4, 0 , -1],
[0 , 0 , 0 , 0 , 0 , 1 , 0],
[1 , 0 , 1 , 0 , 1 , 0 , 0],
[0 , 1 , 0 , 1 , 0 , 1 , 0]
[0, 0, 0, 0, 0, 0, 1, 1]

which looks right to me, at a glance.


You are correct, I apologize for the confusion. The polynomials that you’ve defined are in fact linear, and the resulting system of equations is a system of linear equations. The toMatrixVectorForm function can be used to convert these polynomials into the standard form of a system of linear equations, which can then be solved using a library such as linearEqSolver.


Success! I had to install Z3, but then the following addition to main:

 sol <- solveRationalLinearEqs Z3 matrix vector
putStrLn $ “Solution: “ ++ show sol

results in this output:

Solution: Just [1 % 3,1 % 3,0 % 1,2 % 3,2 % 3,0 % 1,1 % 6]

This tells me that p_rock = 1/3, q_rock = 1/3, p_paper = 0, q_paper = 2/3, p_scissors = 2/3, q_scissors = 0, and the game value is 1/6. This exactly matches the empirical result.

We’re still not quite done, though. I hard-coded the supports for this test, and I also didn’t verify the inequalities.


That’s correct. To finish, you should write a function that iterates over all the possible supports, creates the system of equations and inequalities for each one, solves it and then checks if the solution satisfies the inequalities and if so, use it. This way, you will be able to find the mixed strategy for each player that is a Nash equilibrium for any possible set of parameters. You can also add some error handling to your code, in case the SMT solver cannot find a solution or if the solution doesn’t satisfy the inequalities.


I’m not really sure why solveRationalLinearEqs has an IO type, since it shouldn’t have any user-visible side effects, but it’s going to get annoying. I think I’ll change that by wrapping it in unsafePerformIO. Then I should be able to write this:

solveSystem :: System -> Maybe (Strategy Rational, Strategy Rational)


It is generally not a good idea to use unsafePerformIO in this way, as it can make your program’s behavior less predictable and harder to reason about. unsafePerformIO allows you to perform arbitrary side effects, such as reading from the keyboard or writing to a file, in pure code. This can make it difficult to understand when and how these side effects will be performed.

Instead, it’s better to refactor your solveSystem function to use the IO monad, or to use the ExceptT monad transformer to handle errors.

For example, you could use ExceptT to represent the possibility of an error and use runExceptT to run the computation and extract the result. You could also use the Either monad to represent the possibility of an error and use runExcept to run the computation and extract the result.

solveSystem :: System -> Either String (Strategy Rational, Strategy Rational)

Alternatively, you could use Maybe monad to represent the possibility of an error and use runMaybeT to run the computation and extract the result

solveSystem :: System -> Maybe (Strategy Rational, Strategy Rational)


I appreciate the advice, but this is a situation in which I really think I do know what I’m doing. The only thing I’m wrapping is solveRationalLinearEqs, and it doesn’t have visible side effects. It does, perhaps, have non-deterministic output if the system is under-constrained, but I trust that the outcome will, in fact, be deterministic if it is a Nash equilibrium, which is what I care about.


I understand your perspective and if you are confident in your understanding of the consequences of using unsafePerformIO in this specific case, then you may proceed with using it. However, it is generally recommended to avoid using unsafePerformIO if possible, as it can make the code harder to reason about, test and maintain in the long term.


Then proceed I shall.

I’ve revised `evaluate` to

evaluate :: (a -> Rational) -> Poly a -> Rational
evaluate val (Poly terms) =
sum [coeff * product (val <$> vars) | (vars, coeff) <- Map.toList terms]

And then written this:

solve :: System -> Maybe (Strategy Rational, Strategy Rational)
solve system = verify =<< solution
(matrix, vector) = toMatrixVectorForm (mustBeZero system) systemVars
solution = unsafePerformIO (solveRationalLinearEqs Z3 matrix vector)
verify [p_r, q_r, p_p, q_p, p_s, q_s, v] =
let val (Right GameValue) = v
val (Left (Left Rock)) = p_r
val (Left (Left Paper)) = p_p
val (Left (Left Scissors)) = p_s
val (Left (Right Rock)) = q_r
val (Left (Right Paper)) = q_p
val (Left (Right Scissors)) = q_s
in if all ((>= 0) . evaluate val) (mustBePositive system)
( Distribution
[(p_r, Rock), (p_p, Paper), (p_s, Scissors)],
[(q_r, Rock), (q_p, Paper), (q_s, Scissors)]
else Nothing
verify _ = Nothing

This will only return a solution if it not only satisfies all the equations, but also the inequalities as well.


That looks good! You are correctly using the solution from solveRationalLinearEqs and then using it to evaluate the polynomials that represent the inequalities. By checking if all of these polynomials evaluate to non-negative numbers, you are ensuring that the solution satisfies all the inequality constraints. If it does, you use the solution to construct a strategy for each player, otherwise it will return Nothing.


Now all I need to do is enumerate all the possible supports, and for each one, use makeSystem to build a System, and then solve to attempt to solve the system. As soon as I get a valid solution, I should be able to stop and use it. That looks like this:

equilibrium ::
Params (Poly SystemVar) -> Maybe (Strategy Rational, Strategy Rational)
equilibrium params = listToMaybe solutions
systems = makeSystem params <$> supports <*> supports
solutions = mapMaybe solve systems


That looks good. equilibrium function takes the game parameters and returns the first Nash equilibrium it finds. It does this by generating all possible supports for both players, using makeSystem to construct the system of equations and inequalities for each support, and then using solve to find a valid solution. If a valid solution is found, it is returned using listToMaybe otherwise it returns Nothing. It’s worth mentioning that the number of possible supports could be large and if the game has multiple Nash equilibrium then it will return the first one it finds. If you want to find all of them you can use the filter function instead of listToMaybe.


Excellent! My new main function is

main :: IO ()
main = do
let params =
( Distribution
[ (constant (1 / 2), Chosen),
(constant (1 / 2), Forced Rock)
( Distribution
[ (constant (1 / 2), Chosen),
(constant (1 / 2), Forced Scissors)
case equilibrium params of
Nothing -> putStrLn "No equilibrium found"
Just (p1, p2) -> do
putStrLn $ "Player 1: " ++ show p1
putStrLn $ "Player 2: " ++ show p2

It correctly prints

Player 1: Distribution [(1 % 3,Rock),(0 % 1,Paper),(2 % 3,Scissors)]
Player 2: Distribution [(1 % 3,Rock),(2 % 3,Paper),(0 % 1,Scissors)]

I think we’re done! Thanks for the help, ChatGPT.


You’re welcome! It was a pleasure working with you on this problem. I’m glad you were able to successfully implement the Nash equilibrium algorithm using linear equations and SMT solvers. If you have any further questions, feel free to ask.

Final Code for Part IV

{-# LANGUAGE DeriveFunctor #-}

module Main where

import Data.Bifunctor (bimap)
import Data.Foldable (foldl', maximumBy)
import Data.Function (on)
import Data.List (groupBy, sort, sortBy, subsequences)
import Data.Map (Map)
import qualified Data.Map.Strict as Map
import Data.Maybe (listToMaybe, mapMaybe)
import Math.LinearEquationSolver (Solver (Z3), solveRationalLinearEqs)
import System.IO.Unsafe (unsafePerformIO)

data Move = Rock | Paper | Scissors deriving (Eq, Ord, Show)

data Result = Win | Lose | Draw deriving (Eq, Ord, Show)

data Game = Game Move Move

result :: Game -> Result
result (Game Rock Scissors) = Win
result (Game Scissors Paper) = Win
result (Game Paper Rock) = Win
result (Game Scissors Rock) = Lose
result (Game Paper Scissors) = Lose
result (Game Rock Paper) = Lose
result (Game _ _) = Draw

data Transform = Chosen | Forced Move

transform :: Transform -> Move -> Move
transform (Forced m) _ = m
transform Chosen m = m

newtype Distribution prob a = Distribution [(prob, a)]
deriving (Functor, Show)

instance Num prob => Applicative (Distribution prob) where
pure x = Distribution [(1, x)]
Distribution fs <*> Distribution xs =
Distribution [(p * q, f x) | (p, f) <- fs, (q, x) <- xs]

simplify :: (Num prob, Ord a) => Distribution prob a -> Distribution prob a
simplify (Distribution d) =
Distribution $
map (\xs -> (sum (map fst xs), snd (head xs))) $
groupBy (\x y -> snd x == snd y) $
sortBy (\x y -> compare (snd x) (snd y)) d

mapProb :: (prob1 -> prob2) -> Distribution prob1 a -> Distribution prob2 a
mapProb f (Distribution xs) = Distribution [(f p, x) | (p, x) <- xs]

data Params prob
= Params (Distribution prob Transform) (Distribution prob Transform)

opposite :: Params prob -> Params prob
opposite (Params tr1 tr2) = Params tr2 tr1

type Strategy prob = Distribution prob Move

game ::
Num prob =>
Params prob ->
Strategy prob ->
Strategy prob ->
Distribution prob Game
game (Params tr1 tr2) s1 s2 =
Game <$> (transform <$> tr1 <*> s1) <*> (transform <$> tr2 <*> s2)

allStrategies :: Int -> [Strategy Rational]
allStrategies n =
[ Distribution [(r, Rock), (p, Paper), (s, Scissors)]
| r <- [0, 1 / fromIntegral n .. 1],
p <- [0, 1 / fromIntegral n .. 1 - r],
let s = 1 - r - p

expected :: Num prob => (a -> prob) -> Distribution prob a -> prob
expected f (Distribution xs) = sum [p * f x | (p, x) <- xs]

benefit :: Num a => Result -> a
benefit Win = 1
benefit Draw = 0
benefit Lose = -1

bestStrategy :: Params Rational -> [Strategy Rational] -> Strategy Rational
bestStrategy params strategies =
maximumBy (compare `on` worstCase) strategies
worstCase s =
minimum (expected (benefit . result) . game params s <$> strategies)

newtype Poly a = Poly (Map [a] Rational) deriving (Show)

mapVars :: Ord b => (a -> b) -> Poly a -> Poly b
mapVars f (Poly terms) = Poly (Map.mapKeysWith (+) (map f) terms)

constant :: Rational -> Poly a
constant 0 = Poly Map.empty
constant n = Poly (Map.singleton [] (toRational n))

var :: a -> Poly a
var v = Poly (Map.singleton [v] 1)

toMatrixVectorForm :: Eq a => [Poly a] -> [a] -> ([[Rational]], [Rational])
toMatrixVectorForm zeros vars = unzip rows
rows = toRow <$> zeros

toRow (Poly terms) =
(foldl' (zipWith (+)) (0 <$ vars))
(unzip (termToRow <$> Map.toList terms))

termToRow ([], coeff) = (0 <$ vars, -coeff)
termToRow ([v], coeff) = ([if v == w then coeff else 0 | w <- vars], 0)
termToRow _ =
error "toMatrixVectorForm: term with more than one variable"

instance (Ord a) => Num (Poly a) where
Poly p1 + Poly p2 = Poly (Map.filter (/= 0) (Map.unionWith (+) p1 p2))
negate (Poly p) = Poly ( negate p)
Poly p1 * Poly p2 =
( Map.filter
(/= 0)
( Map.fromListWith
[ (sort (x ++ y), c1 * c2)
| (x, c1) <- Map.toList p1,
(y, c2) <- Map.toList p2
abs = id
signum _ = 1
fromInteger = constant . toRational

p1Strategy, p2Strategy :: Strategy (Poly (Either Move Move))
p1Strategy =
Distribution [(var (Left m), m) | m <- [Rock, Paper, Scissors]]
p2Strategy =
Distribution [(var (Right m), m) | m <- [Rock, Paper, Scissors]]

pureStrategy :: Move -> Strategy (Poly (Either Move Move))
pureStrategy m = Distribution [(constant 1, m)]

evaluate :: (a -> Rational) -> Poly a -> Rational
evaluate val (Poly terms) =
sum [coeff * product (val <$> vars) | (vars, coeff) <- Map.toList terms]

utility ::
Num prob => Params prob -> Strategy prob -> Strategy prob -> prob
utility params p1 p2 = expected (benefit . result) (game params p1 p2)

supports :: [[Move]]
supports = filter (not . null) (subsequences [Rock, Paper, Scissors])

data GameValue = GameValue deriving (Eq, Ord, Show)

type SystemVar = Either (Either Move Move) GameValue

systemVars :: [SystemVar]
systemVars =
[Left (side m) | m <- [Rock, Paper, Scissors], side <- [Left, Right]]
++ [Right GameValue]

data System = System
{ mustBeZero :: [Poly SystemVar],
mustBePositive :: [Poly SystemVar]
deriving (Show)

instance Semigroup System where
(System z1 p1) <> (System z2 p2) = System (z1 ++ z2) (p1 ++ p2)

instance Monoid System where mempty = System [] []

makeSystem :: Params (Poly SystemVar) -> [Move] -> [Move] -> System
makeSystem params support1 support2 =
[ foldMap p1Move [Rock, Paper, Scissors],
foldMap p2Move [Rock, Paper, Scissors],
v = var (Right GameValue)
move side = var . Left . side
p1Move m
| m `elem` support1 = System [v - u] [move Left m]
| otherwise = System [move Left m] [v - u]
u =
(mapProb (mapVars Left) (pureStrategy m))
(mapProb (mapVars Left) p2Strategy)
p2Move m
| m `elem` support2 = System [u - v] [move Right m]
| otherwise = System [move Right m] [u - v]
u =
(mapProb (mapVars Left) p1Strategy)
(mapProb (mapVars Left) (pureStrategy m))
normalization =
[ sum [move side m | m <- [Rock, Paper, Scissors]] - constant 1
| side <- [Left, Right]

solve :: System -> Maybe (Strategy Rational, Strategy Rational)
solve system = verify =<< solution
(matrix, vector) = toMatrixVectorForm (mustBeZero system) systemVars
solution = unsafePerformIO (solveRationalLinearEqs Z3 matrix vector)
verify [p_r, q_r, p_p, q_p, p_s, q_s, v] =
let val (Right GameValue) = v
val (Left (Left Rock)) = p_r
val (Left (Left Paper)) = p_p
val (Left (Left Scissors)) = p_s
val (Left (Right Rock)) = q_r
val (Left (Right Paper)) = q_p
val (Left (Right Scissors)) = q_s
in if all ((>= 0) . evaluate val) (mustBePositive system)
( Distribution
[(p_r, Rock), (p_p, Paper), (p_s, Scissors)],
[(q_r, Rock), (q_p, Paper), (q_s, Scissors)]
else Nothing
verify _ = Nothing

equilibrium ::
Params (Poly SystemVar) -> Maybe (Strategy Rational, Strategy Rational)
equilibrium params = listToMaybe solutions
systems = makeSystem params <$> supports <*> supports
solutions = mapMaybe solve systems

main :: IO ()
main = do
let params =
( Distribution
[ (constant (1 / 2), Chosen),
(constant (1 / 2), Forced Rock)
( Distribution
[ (constant (1 / 2), Chosen),
(constant (1 / 2), Forced Scissors)
case equilibrium params of
Nothing -> putStrLn "No equilibrium found"
Just (p1, p2) -> do
putStrLn $ "Player 1: " ++ show p1
putStrLn $ "Player 2: " ++ show p2

Wrap up

A few things are worth noting:

  1. You definitely cannot rely on ChatGPT. It’s wrong pretty often, and sometimes produces plain nonsense. In fact, this log overstates the case, since when ChatGPT went completely off the rails, I just asked for it to regenerate the response. You’ll see that I had to do my own research on topics like which linear solver to use, strategies for identifying the Nash equilibrium, etc.
  2. Nevertheless, for the most part, it does follow the process well enough to be an active partner in the programming exercise.
  3. As complexity grows, ChatGPT tends to take a back seat. For instance, it was eager to define the basic game and write some instances early on, but it didn’t volunteer to write a single line of code in all of part IV. Nevertheless, it did remain active in discussing the code that I did write, and even when its comments were wrong, they gave me a starting point to figure out why, and how to do better.

I’d call this a successful pair programming exercise overall. ChatGPT doesn’t solve your problems for you, but that’s not why we do pair programming anyway.

by Chris Smith at January 26, 2023 06:24 PM

January 23, 2023

Gabriella Gonzalez

Announcing nixos-rebuild: a "new" deployment tool for NixOS

Announcing nixos-rebuild: a "new" deployment tool for NixOS

The title of this post is tongue-in-cheek; nixos-rebuildis a tool that has been around for a long time and there’s nothing new about it. However, I believe that not enough people know how capable this tool is for building and deploying remote NixOS systems. In other words, nixos-rebuild is actually a decent alternative to tools like morph or colmena.

Part of the reason why nixos-rebuild flies under the radar is because it’s more commonly used for upgrading the current NixOS system, rather than deploying a remote NixOS system. However, it’s actually fairly capable of managing another NixOS system.

In fact, your local system (that initiates the deploy) doesn’t have to be a NixOS system or even a Linux system. An even lesser known fact is that you can initiate deploys from macOS using nixos-rebuild. In other words, nixos-rebuildis a cross-platform deploy tool!

The trick

I’ll give a concrete example. Suppose that I have the following NixOS configuration (for a blank EC2 machine) saved in configuration.nix:

{ modulesPath, ... }:

{ imports = [ "${modulesPath}/virtualisation/amazon-image.nix" ];

system.stateVersion = "22.11";

… which I’ve wrapped in the following flake (since I like Nix flakes):

{ inputs.nixpkgs.url = "github:NixOS/nixpkgs/22.11";

outputs = { nixpkgs, ... }: {
nixosConfigurations.default = nixpkgs.lib.nixosSystem {
system = "x86_64-linux";

modules = [ ./configuration.nix ];

Further suppose that I have an x86_64-linux machine on EC2 accessible via ssh at I can deploy that configuration to the remote machine like this:

$ nix shell nixpkgs#nixos-rebuild
$ nixos-rebuild switch --fast --flake .#default \
--target-host \

… and that will build and deploy the remote machine even if your current machine is a completely different platform (e.g. macOS).

Why this works

The --fast flag is the first adjustment that makes the above command work on systems other NixOS. Without that flag nixos-rebuild will attempt to build itself for the target platform and run that new executable with the same arguments, which will fail if the target platform differs from your current platform.

The --build-host flag is also necessary if the source and target platform don’t match. This instructs nixos-rebuild to build on the target machine so that the deploy is insensitive to your current machine’s platform.

The final thing that makes this work is that Nixpkgs makes the nixos-rebuild script available on all platforms, despite the script living underneath the pkgs/os-specific/linuxdirectory in Nixpkgs.


There’s a reason why I suggest using flakes alongside nixos-rebuild: with flakes you can specify multiple NixOS machines within the same file (just like we can other NixOS deployment tools). That means that we can do something like this:

{ inputs.nixpkgs.url = "github:NixOS/nixpkgs/22.11";

outputs = { nixpkgs, ... }: {
nixosConfigurations = {
machine1 = nixpkgs.lib.nixosSystem { … };

machine2 = nixpkgs.lib.nixosSystem { … };


… and then we can select which system to build with the desired flake URI (e.g. .#machine1 or .#machine2 in the above example).

Moreover, by virtue of using flakes we can obtain our NixOS configuration from somewhere other than the current working directory. For example, you can specify a flake URI like github:${OWNER}/${REPO}#${ATTRIBUTE} to deploy a NixOS configuration hosted on GitHub without having to locally clone the repository. Pretty neat!


I’m not the first person to suggest this trick. In fact, while researching prior art I stumbled across this comment from Luke Clifton proposing the same idea of using nixos-rebuild as a deploy tool. However, other than that stray comment I couldn’t find any other mentions of this so I figured it was worth formalizing this trick in a blog post that people could more easily share.

This post supersedes a prior post of mine where I explained how to deploy a NixOS system using more low-level idioms (e.g. nix build, nix copy). Now that nixos-rebuild supports both flakes and remote systems there’s no real reason to do it the low-level way.

Edit: An earlier version of this post suggested using _NIXOS_REBUILD_REEXEC=1 to prevent nixos-rebuild for building itself for the target platform but then Naïm Favier pointed out that you can use the --fast flag instead, which has the same effect.

by Gabriella Gonzalez ( at January 23, 2023 05:20 PM

January 19, 2023

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

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

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


The Tile Graphs implementation uses a type Tgraph which has a list of tile faces and a maximum vertex number.

data Tgraph = Tgraph { maxV  :: Vertex
                     , faces :: [TileFace]
                     }  deriving (Show)

For example, fool (short for a fool’s kite) is a Tgraph with 6 faces and 7 vertices, shown in figure 3.

fool = Tgraph { maxV = 7
               , faces = [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 graph to form the left graph, we can also compose the left graph to get the right graph.

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 face-connected graph with crossing boundaries at 4, and a non face-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)
  -- join edge - dashed in figure 2
joinE  :: TileFace -> Dedge 
  -- the short edge which is not a join edge
shortE :: TileFace -> Dedge   
-- the long edge which is not a join edge
longE  :: TileFace -> Dedge
  -- all three directed edges clockwise from origin
faceDedges :: TileFace -> [Dedge]

For the whole Tgraph, we often want a list of all the directed edges of all the faces.

graphDedges :: Tgraph -> [Dedge]
graphDedges g = concatMap faceDedges (faces g)

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 VPinned)

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.

makePatch :: Tgraph -> Patch

drawPatch :: Patch -> Diagram B -- defined in module TileLib

drawGraph :: Tgraph -> Diagram B
drawGraph = drawPatch . makePatch

However, it is also useful to have an intermediate stage (a VPinned) which contains both faces and locations for each vertex. This allows vertex labels to be drawn and for faces to be identified and retained/excluded after the location information is calculated.

data VPinned  = VPinned {vLocs :: VertexLocMap
                        ,vpFaces :: [TileFace]

A VPinned has a map from vertices to locations and a list of faces. We make VPinned transformable so it can also be an argument type for rotate, translate, and scale.

The conversion functions include

makeVPinned   :: Tgraph -> VPinned
dropLabels :: VPinned -> Patch -- discards vertex information
drawVPinned   :: VPinned -> Diagram B  -- draws labels as well

drawVGraph   :: Tgraph -> Diagram B
drawVGraph = drawVPinned . makeVPinned

One consequence of using abstract graphs is that there is no unique predefined way to orient or scale or position the patch arising from a graph 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 patch can be calculated from this.

We also have functions to re-orient a VPinned and lists of VPinneds 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 VPinned 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 Tgraphs:

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 { maxV = 14
       , faces = [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 (= decomposeG fool)

Composing graphs, 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 graph, 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 largeKiteCentres or largeDartBases so there are no unknowns.

If there are no unknowns, then we have a function to produce the unique composed graph.

compose:: Tgraph -> Tgraph

Any correct decomposed graph 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 graph 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 { maxV = 7
       , faces = [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 graph 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 graph that were not used, along with the composed graph.

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 graph are shown in pale green, overlaid by the composed graph 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 graph 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] -> VPinned -> VPinned
removeFacesVP    :: [TileFace] -> VPinned -> VPinned
selectFacesGtoVP :: [TileFace] -> Tgraph -> VPinned
removeFacesGtoVP :: [TileFace] -> Tgraph -> VPinned

for creating VPinneds from selected tile faces of a Tgraph or VPinned. This allows us to represent and draw a subgraph which need not be connected nor satisfy the no crossing boundaries property provided the Tgraph it was derived from had these properties.


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 graph we get from force foolD.

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 graphs. The green arrows in the reverse direction are compose again and the intermediate (partCompose) figures are shown in the top row with the ignored 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 graphs 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 graph composes to the one above (an empty graph for the ones in the top row) and the graph 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 angles at a vertex allowing a simple equality test.

Integer Angles

All vertex angles 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 graph 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, this will be undermined if there are crossing boundaries . In this case there must 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 graphs (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 graph, 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 nonsense graph. 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 graph (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]
    , allVertices :: [Vertex]
    , 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 vertices. The boundary information is easily incremented for each face addition without being recalculated from scratch, and a final graph 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 boundaries lies in efficient incremental changes to 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 force state.

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 graph.

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 innaccuracy 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.


The finite number of rules used in forcing are based on local boundary vertex and edge information only. 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 graphs produce an empty graph when composed so we can regard those as maximal compositions. For example compose fool produces an empty graph.

The idea now is to take an arbitrary graph and apply (compose . force) repeatedly to find its maximally composed graph, then to force the maximal graph before applying (force . decompose) repeatedly back down to the starting level (so the same number of decompositions as compositions).

We call the function emplace, and call the result the emplacement of the starting graph as it shows a region of influence around the starting graph.

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. Since adding the new rules we have not yet found graphs with different results from force and emplace. [Update: We now have an example where force includes more than emplace].


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 a correct graph (respectively: incorrect graph), we mean g represents a correct tiling (respectively: incorrect tiling). A simple example of an incorrect graph is a kite with a dart on each side (called a mistake by Penrose) shown on the left of figure 20.

*Main> mistake
Tgraph { maxV = 8
       , faces = [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 graph 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 graph (mistake), and the point at which force mistake fails
Figure 20: An incorrect graph (mistake), and the point at which force mistake fails

So it is certainly true that incorrect graphs may fail on forcing, but forcing cannot create an incorrect graph from a correct graph.

If we apply decompose to mistake it produces another incorrect graph (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 graph is not always preserved by decompose. If we start with mistake1 which is mistake with just two of the half darts (and also an incorrect tiling) 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 graphs). 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 composeG of each choice (bottom row)
Figure 22: makeChoices fool (top row) and composeG 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 graph.

For calculating the emplacement of a graph, we first find the forced maximal graph 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. For example, we do not know if force g always produces the same result as emplace g. [Update (August 2022): We now have an example where force g strictly includes emplace g].

It would also be good to establish 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 19, 2023 03:57 PM

January 16, 2023

Brent Yorgey

Competitive programming in Haskell: Infinite 2D array, Levels 2 and 3

In a previous post, I challenged you to solve Infinite 2D Array using Haskell. As a reminder, the problem specifies a two-parameter recurrence F_{x,y}, given by

  • F_{0,0} = 0
  • F_{0,1} = F_{1,0} = 1
  • F_{i,0} = F_{i-1,0} + F_{i-2,0} for i \geq 2
  • F_{0,i} = F_{0,i-1} + F_{0,i-2} for i \geq 2
  • F_{i,j} = F_{i-1,j} + F_{i,j-1} for i,j \geq 1.

Last time, we derived a formula for F_{x,y} that involves only a linear number of terms:

\displaystyle F_{x,y} = \left(\sum_{1 \leq k \leq x} F_k \binom{x-k+y-1}{x-k}\right) + \left(\sum_{1 \leq k \leq y} F_k \binom{y-k+x-1}{y-k}\right) \pmod{P}

While the number of terms may be linear, it can still be on the order of a million terms, so computing each term is going to have to be pretty quick in order to fit the whole thing within the one second time limit.

Fibonacci numbers modulo a prime

Computing Fibonacci numbers modulo a prime is not hard, especially since we want all the Fibonacci numbers from 1 up to \max(x,y): just compute each one by adding the previous two modulo P. We could also precompute a table of Fibonacci numbers mod P this way. And any of the fast methods for computing individual Fibonacci numbers (for example, using 2×2 matrix exponentiation) also work just fine if you reduce everything modulo P at each step, since they only involve addition, subtraction, and multiplication.

Binomial coefficients modulo a prime

What about binomial coefficients? Since n and k are not too large, and in particular since they will both be smaller than P, we can use the usual formula:

\displaystyle \binom n k = \frac{n!}{k!(n-k)!}

(If n and k could be much larger, or if they could be larger than P, we would have to use something like Lucas’s Theorem or other techniques; that might make for another interesting blog post sometime.) But how do we handle division in modular arithmtic? Since we’re working modulo a prime, every value a other than zero must have a modular inverse, that is, a value a^{-1} such that a \cdot a^{-1} \equiv 1 \pmod p (this is a corollary of Bézout’s Theorem). To compute the modular inverse for a given a, we have a couple options. One simple way is to use Fermat’s Little Theorem: if a is not divisible by a prime p, then a^{p-2} \cdot a = a^{p-1} \equiv 1 \pmod p, hence a^{p-2} is the modular inverse of a modulo p, and we can compute it efficiently using repeated squaring modulo p. Another option is to use the extended Euclidean algorithm to find the x and y (guaranteed to exist by Bézout’s Theorem) such that ax + py = 1; then x is the inverse of a modulo p.

Both of these methods take O(\lg p) time. In my experience, computing the p-2 power is easier to code (especially in Haskell where we get exponentiation by repeated squaring for free!), but using the extended Euclidean algorithm can be a bit faster when it’s well-optimized. (Note the extended Euclidean algorithm can be faster when a is small, but raising to the p-2 power always takes the same number of steps no matter what a is.)

Factorials modulo a prime

Since we’re going to be repeatedly using the same factorials, one thing we absolutely must do is precompute a table of factorials mod P, from 0 up to some maximum. In this case, since our formula involves things like \binom {x-k+y-1}{x-k}, we may need factorials up to x + y, so a table of size 2 \times 10^6 will do (x and y can be up to 10^6).

We could also precompute a table of modular inverses of factorials; to compute the inverse of k!, we just find the inverse of each k and multiply it by the (previously computed) inverse of (k-1)!. (Or we could just invert the value for k! stored in the other table.) Making a table of inverse factorials like this turns out not to help too much for this particular problem, but it can be an important optimization in some cases.

The end?

So we can compute each additional Fibonacci number in O(1); we can also now compute binomial coefficients modulo P in O(\lg P), with a few O(1) table lookups for factorials and an O(\lg P) inversion operation. (Again, we could achieve O(1) if we also stored a table of inverse factorials, but for this problem it seems the additional time needed to construct the table in the first place outweighs the time saved computing binomial coefficients.) In theory, we have everything we need to solve this problem efficiently.

However, for this problem, constant factors matter! There’s still quite a bit of nontrivial work I had to do to get my code fast enough. In my next and final post on this problem, we’ll walk through a few different ideas for implementing this concretely in Haskell.

by Brent at January 16, 2023 03:09 PM

January 15, 2023

Magnus Therning

Composing instances using deriving via

Today I watched the very good, and short, video from Tweag on how to Avoid boilerplate instances with -XDerivingVia. It made me realise that I've read about this before, but then the topic was on reducing boilerplate with MTL-style code.

Given that I'd forgotten about it I'm writing this mostly as a note to myself.

The example from the Tweag video, slightly changed

The code for making film ratings into a Monoid, when translated to the UK, would look something like this:

{-# LANGUAGE DerivingVia #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}

module DeriveMonoid where

newtype Supremum a = MkSup a
    deriving stock (Bounded, Eq, Ord)
    deriving newtype (Show)

instance Ord a => Semigroup (Supremum a) where
    (<>) = max

instance (Bounded a, Ord a) => Monoid (Supremum a) where
    mempty = minBound

data FilmClassification
    = Universal
    | ParentalGuidance
    | Suitable12
    | Suitable15
    | Adults
    | Restricted18
    deriving stock (Bounded, Eq, Ord)
    deriving (Monoid, Semigroup) via (Supremum FilmClassification)

Composing by deriving

First let's write up a silly class for writing to stdout, a single operation will do.

class Monad m => StdoutWriter m where
    writeStdoutLn :: String -> m ()

Then we'll need a type to attach the implementation to.

newtype SimpleStdoutWriter m a = SimpleStdoutWriter (m a)
    deriving (Functor, Applicative, Monad, MonadIO)

and of course an implementation

instance MonadIO m => StdoutWriter (SimpleStdoutWriter m) where
    writeStdoutLn = liftIO . putStrLn

Now let's create an app environment based on ReaderT and use deriving via to give it an implementation of StdoutWriter via SimpleStdoutWriter.

newtype AppEnv a = AppEnv {unAppEnv :: ReaderT Int IO a}
        ( Functor
        , Applicative
        , Monad
        , MonadIO
        , MonadReader Int
    deriving (StdoutWriter) via (SimpleStdoutWriter AppEnv)

Then a quick test to show that it actually works.

λ> runReaderT (unAppEnv $ writeStdoutLn "hello, world!") 0
hello, world!

January 15, 2023 05:34 PM

January 07, 2023

Brandon Simmons

Linking Smaller Haskell Binaries

Haskell binaries can get quite large (think ~100MB), especially for projects with many transitive dependencies. Here are two strategies that can help at link time, the latter being more experimental. I used the test-pandoc binary from pandoc on GHC 9.2.5 below. This was nice because obviously it was easy to test if linking broke anything (just run the tests). -split-sections and --gc-sections You can instruct ghc to emit code in individual minimal sections, allowing the linker to easily find and remove dead code.

January 07, 2023 08:52 PM

January 03, 2023

Sandy Maguire

Abusing Constraints for Automatic Documentation


At work I was recently tasked with figuring out what API calls our program makes, and more interestingly, which code-paths lead to those API calls. Determining this by hand is tedious and error-prone, and worse, doesn’t stay up to date with code changes. Instead, let’s see how we can use the type system to eliminate the pain.

The existing code was organized around a class HasAPI that looks something like this:

type  HasAPI :: Service -> Symbol -> Constraint
class HasAPI srv name where
  type APICall srv name
  callAPI :: APICall srv name

Here, HasAPI is a type class with an associated type family APICall which gives the type for making the call. For example, there might be an instance:

instance HasAPI ShoutService "shout" where
  type APICall ShoutService "shout" = String -> IO String
  callAPI str = pure $ fmap toUpper str

This is a silly example — the real codebase makes actual API calls — but it serves for demonstration.

Our goal is to document every codepath that makes any use of callAPI, in some sense, “infecting� every path with some marker of that fact. This is a common experience to Haskell programmers; in fact, IO has this same pattern of infectiousness. Whenever you make a function perform IO, every type in the callstack needs to document the fact it performs IO. This is the inspiration we will take, except that changing types is extremely expensive. What if we pushed a constraint around instead?

Propagating Constraints🔗

The trick is to define a new class, of the same shape as HasAPI:

type  CallsAPI :: Service -> Symbol -> Constraint
class CallsAPI srv name

but crucially, we give CallsAPI no instances. On first blush, this seems insane: why introduce a class with no methods and no instances? Having no methods means it can’t do anything useful. Having no instances means GHC can never eliminate the constraint, and thus must propagate it upwards. This is the infectiousness we want; any function which makes an API call must document that fact in its type — failure to do so will result in GHC failing to compile with the message No instance for (CallsAPI srv name).

The trick now is to ensure that callsAPI produces a CallsAPI constraint. The easy way to do this is a little renaming to ensure existing polymorphic code continues work:

type  UnsafeHasAPI :: Service -> Symbol -> Constraint
class UnsafeHasAPI srv name where
  type APICall srv name
  unsafeCallAPI :: APICall srv name

type HasAPI :: Service -> Symbol -> Constraint
type HasAPI = (UnsafeHasAPI srv name, CallsAPI srv name)

  :: forall srv name
   . HasAPI srv name
  => APICall srv name
callAPI = unsafeCallAPI

Any code written against the old HasAPI constraint will continue to work (modulo the instance definitions,) but concrete calls to callAPI now result in a dangling, unsatisfiable CallsAPI constraint. You’ll need to go through the codebase now, and document every transitive call to the API with matching CallsAPI constraints. Thankfully, HLS can help with this task: it will underline the missing cases, and suggest a code action that will automatically add these constraints to the type. Rinse and repeat, until every code path is documented.

Great success! We have automatically found every codepath that makes an API call, and forced them to document that fact. Better yet, we have solved the problem once and for all; our coworkers also must document any new API calls they make, lest their code not compile. It seems like we’re done!

Except for one fact: GHC will rudely refuse to compile our project, even if we correctly track all of our API calls. The problem of course, is that all we have managed to do is force main to collect every CallsAPI constraint. But GHC will still complain No instance for (CallsAPI srv name). Of course, you could just give an orphan instance in the same module that defines main, which would work, but this doesn’t give you any sort of external documentation. It’s nice when you read the code, but it doesn’t help the business people.

Solving The Unsolvable🔗

A better approach here is to selectively solve the CallsAPI constraints, which we can do with some Haskell dark magic. The Dict type captures a constraint, giving us a convenient way to manipulate constraints:

type Dict :: Constraint -> Type
data Dict c where
  Dict :: c => Dict c

We can write an eliminator to bring the c from a Dict c into scope, which, importantly, allows us to solve otherwise-unsolved constraints:

(\\) :: (c => r) -> Dict c -> r
f \\ Dict = f

If we can get our hands on a Dict (CallsAPI Srv Name), we can use (\\) to convince GHC to compile our program.

GHC is happy to give us dictionaries for constraints it knows about:

showIntDict :: Dict (Show Int)
showIntDict = Dict

but unfortunately, refuses to give us dictionaries for unsolved constraints:

callsAPIDict :: forall srv name. Dict (CallsAPI srv name)
callsAPIDict = Dict

-- Error: No instance for (CallsAPI srv name)

It seems like we’re just as stuck, but we have a trick up our sleeve. The first step is to define another class with an instance in scope. GHC will happily give us a dictionary for such a thing:

class Trivial
instance Trivial

trivialDict :: Dict Trivial
trivialDict = Dict

and now for something naughty:

callsAPIDict :: forall srv name. Dict (CallsAPI srv name)
callsAPIDict = unsafeCoerce trivialDict

Behind the scenes, GHC compiles classes into records, instances into values of these records, and replaces wanted constraints with function arguments taking those records. By ensuring that Trivial and CallsAPI are both empty classes, with no methods or super-classes, we can be certain the generated records for these classes will be identical, and thus that it is OK to coerce one into the other.

Armed with withDict and callsAPIDict, we can play the part of the constraint solver and satisfy constraints ourself. GHC will happily compile the following example:

ex :: HasAPI ShoutService "shout" => IO String
ex = callAPI @ShoutService @"shout" "hello world"

-- Look ma, no HasAPI constraint!
test :: IO String
test = ex \\ callsAPIDict @ShoutService @"shout"

So that’s the rough technique. But how do we actually use it in anger?

Automatically Documenting the Server🔗

Our actual use case at work is to add these API calls to our swagger documentation. Swagger is this automatically generated manifest of an API surface; we want to document the fact that some API calls might call other ones. Our server is one big servant application, and servant is extensible. So the real technique is to build a servant combinator that eliminates HasAPI constraints when you document them in the API definition.

Getting into the nitty gritty bits of servant is beyond the scope of this post, but we can sketch the idea. Servant APIs use the type-level (:>) operator to combine information about an endpoint. For example, we might expose another service:

type ServantAPI = "api" :>
      :> ReqBody '[JSON] String
      :> Get '[JSON] String

This definition states that we have a REST server with a single route, api/echo which responds to POST requests, returning a JSON-encoded string, which takes a JSON-encoded string as the request body.

A servant server for ServantAPI would have type Server ServantAPI, where Server is a type family given by HasServer. Evaluating the type family results in String -> Handler String, so in order to implement this server, we would need to provide a function of that type.

Let’s implement our server endpoint:

    :: CallsAPI ShoutService "shout"
    => String
    -> Handler String
echo str = liftIO $ callAPI @ShoutService @"shout" str

Unfortunately, due to our earlier work, we can’t eliminate the CallsAPI constraint, and thus we can’t actually use echo as the handler for our endpoint.

It’s important to note that servant’s DSL is extensible, and we can add our own machinery here. The first step is to build a type that we can use in servant:

type MakesAPICall :: Service -> Symbol -> Type
data MakesAPICall srv name

We can now build a second version of ServantAPI:

type ServantAPI = "api" :>
      :> MakesAPICall ShoutService "shout"
      :> ReqBody '[JSON] String
      :> Get '[JSON] String

In order to actually run our endpoint, we need to give an instance of HasServer for our new MakesAPICall combinator:

instance HasServer api ctx
      => HasServer (MakesAPICall srv name :> api) ctx
  type ServerT (MakesAPICall srv name :> api) m =
    Dict (CallsFed srv name) -> ServerT api m
  route _ ctx f =
    route (Proxy @api) ctx $ fmap ($ callsAPIDict @srv @name) f

The ServerT instance here adds a Dict (CallsFed srv name) to the type of the handler required to satisfy this endpoint, while route automatically fills in the dictionary whenever the handler needs to be run. In an ideal world, we could give our ServerT instance as:

  type ServerT (MakesAPICall srv name :> api) m =
    CallsFed srv name => ServerT api m

but GHC doesn’t let us use quantified types on the right-hand sides of type families, so this is unfortunately a no-go. Playing games with Dict instead is the best approach I’ve found here, but I’d love to hear if anyone has a better idea.

We still can’t use echo as a handler, but we can use makesCall echo as one, where makesCall is given as:

makesCall :: (c => r) -> Dict c -> r
makesCall = (\\)

Servers that document their API calls via MakesAPICall and which wrap their handlers with makesCall can now eliminate CallsFed constraints. Since this is the only way of eliminating CallsFed constraints, we can be sure that every API call is correctly documented in the servant DSL!

The final step here is to add an instance of HasSwagger (MakesAPICall srv name :> api), but the details are gory and devoid of educational value. Suffice it to say that this instance was written, and now we have automatically generated JSON documentation describing which server endpoints make which other API calls. This documentation is guaranteed to be correct, because updating it is the only way to convince GHC to compile your code.

January 03, 2023 12:00 AM

January 02, 2023

Brent Yorgey

Competitive programming in Haskell: better binary search

Binary search is a workhorse of competitive programming. There are occasional easy problems where binary search is the solution in and of itself; more often, it’s used as a primitive building block of more complex algorithms. It is often presented as a way to find the index of something in a sorted array in O(\lg n) time, and many languages have such a thing in their standard library (for example, see Arrays.binarySearch in Java, the bisect library in Python, or the binary_search function in C++). However, the idea of binary search is more general than searching in a sorted array; we’re doing binary search any time we repeatedly halve a search interval. For example, we can use it to find the smallest or largest number with a given property, or to find an optimal, “just right” measurement that is neither too small nor too big.

Generic binary search with first-class functions, take 1

A language with easy access to first-class functions provides a great opportunity to generalize binary search properly. For example, here’s a version of binary search that has lived in my competitive programming solution template for a long time:

-- Discrete binary search.  Find the smallest integer in [lo,hi] such
-- that monotone predicate p holds.
binarySearchD :: Int -> Int -> (Int -> Bool) -> Int
binarySearchD lo hi p
  | lo == hi = lo
  | p mid     = binarySearchD lo mid p
  | otherwise = binarySearchD (mid+1) hi p
    mid = (lo + hi) `div` 2

The key generalization is that it takes a predicate of type Int -> Bool as an argument. Note that in order for binary search to work, the predicate p must be monotonic. This means, intuitively, that p starts out False, and once it switches to True it never goes back. (Formally, p being monotonic means that for all x and y, if x <= y then p x <= p y, where False <= True). This is how we can tell if we’re “too low” or “too high”: we’re “too low” when p is False and “too high” when it is True.

This is definitely an improvement over array-specific versions of binary search. We can still use it to search in an array by providing a predicate that does an array lookup, but we can use it for other things as well.

I should note at this point that there is a very nice binary-search package published on Hackage, which I definitely recommend if you need binary search in some kind of project. However, for the purposes of competitive programming, we can’t rely on that package being available, and we’d also like something a bit simpler, so we don’t have to read the documentation every time we want to use it.

…can we do better?

So my binarySearchD function works fine as far as it goes, and I have used it regularly, but there are still several things about it that always annoyed me:

  • What if we want a slight variation, such as the largest integer such that something holds? Or the last integer where the predicate doesn’t hold? etc.? It is possible to use binarySearchD in these situations, but I find it tricky and error-prone to figure out how. And when I’m reaching for some function as a building block of a bigger algorithm I definitely don’t want to waste time and brainpower having to think carefully about small details like this.

  • Getting the implementation right in the first place was kind of tricky. Should we use mid+1? mid-1? Should we think in terms of a closed interval [lo,hi], or a half-open interval [lo,hi), or…? How can we convince ourselves our implementation is completely correct, and won’t get stuck in infinite recursion?

  • What if we want to do binary search over a continuous domain, like Double? We have to make a completely separate function, for example, like this:

-- Continuous binary search.  Given a tolerance eps, an interval
-- [a,b], a continuous, monotonically increasing function f, and a
-- target value tgt, find c ∈ [a,b] such that f(c) = tgt.
binarySearch :: (Fractional t, Ord t, Ord a) => t -> t -> t -> (t -> a) -> a -> t
binarySearch eps a b f tgt = go a b
    go lo hi
      | hi-lo < eps = mid
      | f mid < tgt = go mid hi
      | otherwise   = go lo mid
        mid = (lo + hi)/2

(Actually, I’m not sure why I wrote that version in terms of finding a “target” value. In practice I suppose continuous binary search often comes up that way, but looking at it now it seems less general. In any case, we’re going to throw this function away very shortly so it doesn’t really matter!)

Recently I came across a lovely article, Binary Search a Little Simpler & More Generic by Jules Jacobs. Jules explains a really elegant API for binary search that is so much better than anything I’d seen before, and solves all the above issues! I immediately went to implement it in Haskell, and I want to share it with you. As I’ve reflected on Jules’s presentation, I have identified three key ideas:

  1. Rather than looking for some index with a certain property, we’re really looking for the place where p switches from False to True. That actually happens in between two indices… so let’s return the pair of indices bracketing the change, rather than just a single index! This means we get both the last index that does not have property p and the first one that does, and we can use whichever one we want.

    This is a simple change, but in my experience, it helps a lot to reduce the cognitive load. Previously, if I wanted something like “the last index that does not have property p” I’d have to think hard about what the index I get out of the search represents, and figure out that I needed to subtract one. Now I only have to think “OK, I want the thing right before the predicate changes from False to True, so I can project it out with fst”.

  2. The second important idea is that we’re going to insist that p switches from False to True, not at most once, but exactly once. (If necessary, we can add special “virtual” -∞ and/or +∞ indices such that p (-∞) = False and p (+∞) = True.) Then as we narrow down our current search interval [l, r], we will maintain the invariant that p l = False and p r = True.

    This invariant makes everything so much cleaner, and it also ties in with the first important idea of returning a pair instead of a single index. Previously I always thought of binary search in terms of searching for a specific index, but that makes the semantics of the interval tricky. For example, do we maintain the invariant that the index we’re looking for is somewhere in the closed interval [l,r]? Somewhere in the half-open interval [l,r)? …? But I find it so much more elegant and natural to say “l always stays in the False part, and r always stays in the True part, and we just slide them closer until we find the exact dividing line between False and True.”

    I will note that there are a couple tradeoffs: first of all, our search function of course takes starting values for l and r as inputs, and it will now have as a prerequisite that p l = False and p r = True, so we have to think a little harder when calling it. We also have to work a little harder to figure out when e.g. a value we’re looking for was not found at all. Typically, if we use some sort of initial special +∞ value for r, if the returned r value is still +∞ it means nothing at all was found that made the predicate True.

  3. The final important idea is to abstract out a function mid to compute a potential next index to look at, given the current interval. We’ll insist that when mid l r returns a value, it must be strictly in between l and r (there’s no point in returning l or r because we already know p l = False and p r = True), and we’ll stop when it returns Nothing. This lets us cleanly separate out the logic of the recursion and keeping track of the current search interval from the details of the arithmetic needed for each step. In particular, it will allow us to unify binary search over both integral and floating-point domains.

Here’s the final form of our search function. Unlike, say, binarySearchD, it pretty much writes itself at this point:

search :: (a -> a -> Maybe a) -> (a -> Bool) -> a -> a -> (a,a)
search mid p = go
    go l r = case mid l r of
      Nothing -> (l,r)
      Just m
        | p m       -> go l m
        | otherwise -> go m r

We check our mid function to tell us what to look at next. If it returns Nothing, we stop and return the pair of the current (l,r). If it returns a “midpoint” m then we test the predicate on m and recurse. No tricky +1’s or -1’s to think about; given our invariant regarding l and r, it’s obvious which one we should replace with m depending on the outcome of the predicate, and we can’t get stuck in an infinite loop since m is always strictly between l and r.

(As an aside, I love that this is polymorphic in a with no class constraints! That’s another hint that this is really quite general. The class constraints will come with particular mid functions.)

So what about those mid functions? Here’s one for doing binary search over integers:

binary :: Integral a => a -> a -> Maybe a
binary l r
  | r - l > 1 = Just ((l+r) `div` 2)
  | otherwise = Nothing

Pretty straightforward! We stop when l and r are exactly one apart; otherwise we return their midpoint (you should convince yourself that (l+r) `div` 2 is always strictly in between l and r when r - l > 1).

For example, we can use this to take an integer square root:

λ> search binary (\x -> x^2 >= 150) 0 100

This tells us that 12 is the biggest integer whose square is less than 150, and 13 is the smallest integer whose square is greater.

But we needn’t limit ourselves to integers; as hinted previously, we can also do binary search over Fractional domains:

continuous :: (Fractional a, Ord a) => a -> a -> a -> Maybe a
continuous eps l r
  | r - l > eps = Just ((l+r) / 2)
  | otherwise = Nothing

Given an eps value, we stop when r - l <= eps, and otherwise return their midpoint. For example, we can use this to find the square root of 150 to 6 decimal places:

λ> search (continuous 1e-6) (\x -> x^2 >= 150) 0 100

We can even write some functions to do linear search! Why might we want to do that, you ask? Well, with some care, these can be used even with non-monotonic predicates, to find the first or last place the predicate switches from False to True (though using something like find or findIndex is typically easier than using search fwd).

fwd :: (Num a, Ord a) => a -> a -> Maybe a
fwd l r
  | r - l > 1 = Just (l+1)
  | otherwise = Nothing

bwd :: (Num a, Ord a) => a -> a -> Maybe a
bwd l r
  | r - l > 1 = Just (r-1)
  | otherwise = Nothing

I don’t have any great examples of using these off the top of my head, but we might as well include them.

[WARNING: this section about binary search on bit representations of floating-point numbers is completely wrong, but I’m leaving it here for context. See the discussion in the comments to this post and the follow-up post!]

But there’s more: we can also do exact binary search on the bit representations of floating-point numbers! That is, we do binary search as if the bit representations of l and r were unsigned integers. This is possibly more efficient than “continuous” binary search, and lets us find the two precisely adjacent floating-point numbers where our predicate switches from False to True.

binaryFloat :: Double -> Double -> Maybe Double
binaryFloat l r = decode <$> binary (encode l) (encode r)
    encode :: Double -> Word64
    encode = unsafeCoerce

    decode :: Word64 -> Double
    decode = unsafeCoerce

For example, we can find the closest possible floating-point approximation to the square root of 150:

λ> search binaryFloat (\x -> x^2 >= 150) 0 100
λ> sqrt 150

This honestly seems like black magic to me, and I don’t know enough about floating-point representation to have a good idea of how this works and what the caveats might be, but it’s worked for all the examples I’ve tried. It even works when l is negative and r is positive (it seems like in that case the bit representation of l would correspond to a larger unsigned integer than r, but somehow it all works anyway!).

λ> search binaryFloat (\x -> x^2 >= 150) (-100) 100


I’ve added the code from this post to my comprog-hs repository on GitHub. The source for this blog post itself is available on


And here are some problems for you to solve! I’ll discuss some of them in an upcoming post.

by Brent at January 02, 2023 01:48 PM

Lysxia's blog

From delimited continuations to algebraic effects in Haskell

The upcoming version of GHC will feature primitives for delimited continuations. Let’s put them to use and build a bare bones algebraic effect system.

Algebraic effects

In Haskell, different sorts of effectful computations can be expressed using monads. Monads for individual effects are fairly well understood. The challenge now is to combine many different effects. Applications manage many kinds of resources (files, network, databases…), handle many types of errors, and run in different environments (production vs testing with mock components). Can that be done while maintaining a reasonable level of separation of concerns?

Currently, a common approach is to use monad transformers and type classes (mtl-style). But when you have a big stack of monad transformers, it may not be easy to even understand what (>>=) does, because its behavior arises from the composition of all of these transformers. So the actual control flow of the program is opaque to us, which can be an obstacle to locating errors and guaranteeing performance.

Algebraic effects are another approach to combine effects. Whereas with transformers, every monad transformer must be defined from scratch, algebraic effects start from a few core primitives: you have one (parameterized) monad with abilities to “call” an operation and to “handle” such calls. The hope is that those core primitives:

  1. are simple to implement and to optimize;
  2. make it easy to define and reason about effects, in terms of both behavior and performance.

Until now, algebraic effect systems in Haskell used free monads or the continuation monad. Continuations were emulated as closures; this comes with a level of indirection whose cost is difficult to mitigate. The newly implemented delimited continuations primops let us directly manipulate native continuations.

This post uses delimited continuations to implement programs with various effects. The usual culprits:

The example programs leveraging this mini effect library will look like your standard-fare monadic code. What makes them interesting is that, operationally, they are all in the IO monad. Unlike with monad transformers, adding a new effect does not change the underlying monad, so code that doesn’t use that effect does not pay a price for it. Another notable consequence is that “unlifting” abstractions like UnliftIO or MonadBaseControl are no longer relevant: there is nothing to “unlift” if you never leave IO.

The abstraction layer of algebraic effects over continuations is so thin that I just use prompt and control0 directly, but the bits that are “operations” and the bits that are “handlers” are clearly identifiable. The system implemented here is untyped as far as effects are concerned, but features named handlers as a mitigating alternative; a complete effect system which would keep track of what operations each computation may call and would provide safe primitives to define new effects is left as an exercise for the reader.

This post is written in Literate Haskell (source code). It can be compiled using the development version of GHC (or GHC 9.6 if it has been released).

$ ghc 2023-01-02-del-cont-examples.lhs -main-is DelContExamples.main -o run-tests
$ ./run-tests
All tests passed!
Extensions and imports
  UnboxedTuples #-}
module DelContExamples where

import qualified Control.Exception as E
import Control.Exception.Base (NoMatchingContinuationPrompt(..))
import Data.Either
import Data.Foldable (for_)
import Data.Functor (void)
import Data.Functor.Sum (Sum(..))
import Data.Maybe (fromMaybe, maybe)
import System.IO.Unsafe
import System.Environment
import GHC.Exts (PromptTag#, newPromptTag#, prompt#, control0#)
import GHC.IO (IO(..))
import GHC.Stack (HasCallStack)
import Prelude hiding (log)

The mother of all monads

Capturing continuations is the power of the continuation monad, in which we can embed all other monads. It’s the mother of all monads.

Mom is defined identically to IO, but its only operations are the new delimited continuation primitives.

newtype Mom a = Mom (IO a)
  deriving newtype (Functor, Applicative, Monad)

The available operations wrap the RTS primitives newPromptTag#, prompt# and control0#.

-- Unsafe primitives

data PromptTag a = PromptTag (PromptTag# a)

newPromptTag :: Mom (PromptTag a)
newPromptTag = Mom (IO (\s -> case newPromptTag# s of
  (# s', tag #) -> (# s', PromptTag tag #)))

prompt :: PromptTag a -> Mom a -> Mom a
prompt (PromptTag tag) (Mom (IO m)) = Mom (IO (prompt# tag m))

control0 :: PromptTag a -> ((Mom b -> Mom a) -> Mom a) -> Mom b
control0 (PromptTag tag) f =
  Mom (IO (control0# tag (\k -> case f (\(Mom (IO a)) -> Mom (IO (k a))) of Mom (IO b) -> b)))

The boxing of the continuation k in control0 could be avoided by introducing a new type for continuations, replacing (Mom b -> Mom a). I’m not sure whether there is much to gain from that optimization. I leave it like this for simplicity.

prompt and control0, “goto” with extra steps?

When a function terminates normally, it returns its result to its caller, its predecessor in the call stack. prompt lets you prepare another return point earlier in the call stack, and control0 returns to that point. What happens to all the stack frames that were skipped that way? They are copied to the heap so they can be restored later.

In more concrete terms, when you call control0 t f :: Mom b, the caller expects a result of some type b. It is assumed that you have previously set up a prompt t :: Mom a -> Mom a in the call stack with the same tag t :: PromptTag a. The slice of the stack up to that prompt t is unwinded and stored as a function continue :: Mom b -> Mom a (IO b -> IO a). prompt t is popped off the stack, and the program carries on as f continue.

It sounds completely insane the first time you learn about it, it’s like “goto” with extra steps. And yet, when you get down to it, delimited continuations have rather clean semantics, both operationally and denotationally. The implementation was a surprisingly small change in GHC.

The changes required to implement prompt# and control0# are relatively minimal. They only impact the RTS, and they do not require any changes to existing functionality. Though capturing portions of the RTS stack may seem like a radical proposition, GHC actually already does it when raising an asynchronous exception to avoid the need to duplicate work for any blackholed thunks. In fact, getting that right is significantly more subtle than implementing control0#, which is quite straightforward in comparison.

The GHC Proposal

The richness of continuations, both theoretically and practically, suggests that these control operators are not as arbitrary as they seem.

Effectful code, pure semantics

The code in this post can be split in two levels. Library-level code uses the delimited continuation primitives to implement effects—operations and handlers, and user-level code uses those effects in example programs. Without direct access to delimited continuations, user-level code cannot observe any mutation, so it will be safe to use the following pure run function.

-- Look Ma', no IO!
run :: Mom a -> Maybe a
run (Mom m) = unsafePerformIO
  (E.catch (Just <$> m) \NoMatchingContinuationPrompt -> pure Nothing)

Hiding the delimited continuations primitives avoids the danger of duplicating and observing the creation of fresh PromptTags in a pure context. Some partiality remains (Maybe) due to potentially mismatched control0# calls. Such errors would be prevented by a type system for effects, which is beyond the scope of this post.

Further reading

On prompt#, control0#, and newPromptTag#:

On the continuation monad:


To begin, let’s implement exceptions using delimited continuations. This effect has an operation throw and a handler catch.


We first declare the uninterpreted operation Throw as a constructor in a functor. The parameter a is ignored by exceptions; it will be used by other effects.

data Exception e a
  = Throw e

We wrap this constructor in a user-facing function throw. Every throw should have a matching catch, and we ensure this by requiring a tag that identifies the corresponding catch. The exact type of tag will be revealed in a moment. control0 uses that tag to look up the matching catch in the call stack, and returns to it with the exception e wrapped in Throw. The underscore is the continuation, which is the slice of the stack below the catch, which is thus discarded.

throw :: Exception e % r -> e -> Mom a
throw tag e = control0 tag \_ -> pure (Op (Throw e))


The type of catch should also look familiar, with the novelty that the handled computation f expects a tag—so that it may call throw. In catch f onThrow, a fresh tag is generated, then f tag either (1) returns normally, and its result is wrapped in Pure a, or (2) f tag throws an exception wrapped in Op (Throw e). We then return the result or apply the handler onThrow accordingly.

catch :: (Exception e % a -> Mom a) -> (e -> Mom a) -> Mom a
catch f onThrow = do
  tag <- newPromptTag
  handle tag (f tag)
  handle tag action = do
    next <- prompt tag (Pure <$> action)
    case next of
      Op (Throw e) -> onThrow e
      Pure a -> pure a

You might have guessed that the Exception e % a tag is just a PromptTag. More surprisingly, the tag index involves a free monad. For exceptions, Free (Exception e) a is equivalent to Either e a: we expect the computation under prompt to produce either an exception e or a result a. More generally, for an effect expressed as a functor f, things will be set up exactly so that handlers will be matching on a computation/tree of type Free f r.

type f % r = PromptTag (Free f r)

data Free f r
  = Op (f (Free f r))
  | Pure r

Using catch, we can implement try.

try :: (Exception e % Either e a -> Mom a) -> Mom (Either e a)
try f = catch (\tag -> Right <$> f tag) (\e -> pure (Left e))

The explicit tags serve as a form of capabilities, handles that functions take as explicit arguments, granting the permission to use the associated effects. This partly makes up for the lack of effect typing. It’s not watertight: you can easily capture the tag to call throw outside of try/catch. But from a non-adversarial perspective, this mechanism may prevent quite a few mistakes.


testThrow :: IO ()
testThrow = do
  assert (isRight' (run (try (\_ -> pure "Result"))))
  assert (isLeft'  (run (try (\exc -> throw exc "Error"))))
  isRight' = maybe False isRight
  isLeft' = maybe False isLeft
-- Minimalistic unit testing framework
assert :: HasCallStack => Bool -> IO ()
assert True = pure ()
assert False = error "Assertion failed"


Algebraic effects are also known as “resumable exceptions”, extending exceptions with the ability to continue the computation right where the exception was thrown.

The next simplest effect after exceptions is to produce some output. Like Throw, we represent the Output operation as a constructor, containing the value to output, and now also a continuation.


data Out o a
  = Output o (Mom () -> Mom a) 

The output wrapper is similar to throw, additionally storing the continuation in the Output constructor. The expected argument of the continuation continue is a computation which is to replace the operation call. When we call output o :: Mom (), that call “bubbles up” like an exception, gets caught by a handler, and the call gets replaced by pure () or some other Mom () computation.

output :: Out o % r -> o -> Mom ()
output tag o = control0 tag \continue -> pure (Op (Output o continue))

A synonym specialized to strings.

log :: Out String % r -> String -> Mom ()
log = output


An infinite output stream of the Fibonacci sequence.

fibonacci :: Out Int % r -> Mom a
fibonacci out = fib 0 1
    fib !a !b = do
      output out a
      fib b (a + b)


Run a computation lazily and collect its output in a list.

collect :: (Out o % () -> Mom ()) -> [o]
collect f = runList do
  tag <- newPromptTag
  handle tag (Pure <$> f tag)
  handle tag action = do
    next <- prompt tag action
    case next of
      Op (Output o continue) ->
        pure (o : runList (handle tag (continue (pure ()))))
      Pure () -> pure []
  runList = fromMaybe [] . run


testFibonacci :: IO ()
testFibonacci =
  assert (take 8 (collect fibonacci)
          == [0, 1, 1, 2, 3, 5, 8, 13])

Combining exceptions and output


The big selling point of algebraic effects is that effects can be combined smoothly. We can thus use log to trace the execution flow of a program using throw and catch without further ceremony.

This looks like your usual monadic program. The point is that everything lives in the same monad Mom (which is operationally equal to IO), so you do not have to worry about “lifting” or “unlifting” anything through a transformer: the semantics of (>>=) do not change with every new effect, and there isn’t the problem that “lifting” catch and other operations that are actually handlers is counter-intuitive for many transformers, if possible at all. To be fair, there remain difficulties in this area even with algebraic effects.

tracedCatch :: Out String % r -> Mom Bool
tracedCatch out = catch this onThrow 
  this exc = do
    log out "Start"
    _ <- throw exc "Boom"
    log out "This is unreachable"
    pure False
  onThrow msg = do
    log out ("Error: " ++ msg)
    pure True


testTracedCatch :: IO ()
testTracedCatch =
  assert (collect (void . tracedCatch) ==
    [ "Start"
    , "Error: Boom" ])

Silent handler

There can also be different ways of handling an effect. The following handler discards output instead of collecting it, for example to ignore debugging logs.

discardOutput :: (Out o % a -> Mom a) -> Mom a
discardOutput f = do
  tag <- newPromptTag
  handle tag (Pure <$> f tag)
  handle tag action = do
    next <- prompt tag action
    case next of
      Op (Output _o continue) -> handle tag (continue (pure ()))
      Pure a -> pure a
testDiscard :: IO ()
testDiscard =
  assert (run (discardOutput tracedCatch) == Just True)


Dually, there is an effect to request some input.


data In i a
  = Input (Mom i -> Mom a) 

The input call is expected to return a result i. As before, the type of the input _ operation must coincide with the domain Mom i of the continuation.

input :: In i % r -> Mom i
input tag = control0 tag \continue -> pure (Op (Input continue))


Output the cumulative sum of an input stream. Like fibonacci, this is an infinite loop in IO. It gets broken by control0 in input. Until now, an infinite loop in IO would either have to be broken by an exception (which makes it not actually infinite), or have to involve concurrency.

csum :: In Int % r -> Out Int % r -> Mom a
csum inp out = go 0
    go !acc = do
      n <- input inp
      let acc' = acc + n
      output out acc'
      go acc'


Supply a list of inputs and stop when we run out.

listInput :: [i] -> (In i % a -> Mom a) -> Mom (Maybe a)
listInput is f = do
  tag <- newPromptTag
  catch (\exc -> handle exc tag is (Pure <$> f tag))
    (\() -> pure Nothing)
  handle exc tag is action = do
    next <- prompt tag action
    case next of
      Op (Input continue)
        | i : is' <- is -> handle exc tag is' (continue (pure i))
        | otherwise -> handle exc tag [] (continue (throw exc ()))
      Pure a -> pure (Just a)


testCsum :: IO ()
testCsum =
  assert ((collect \out ->
           void $ listInput [1 .. 5] \inp ->
           csum inp out
          ) == [1, 3, 6, 10, 15])

Combining input and output: streaming

The input and output effect can be combined in a streaming fashion, alternating execution between the consumer and the producer.


Feed the output of one computation into the input of the other. Terminate whenever one side terminates, discarding the other.

connect :: (Out x % a -> Mom a) -> (In x % a -> Mom a) -> Mom a
connect producer consumer = do
  out <- newPromptTag
  inp <- newPromptTag
  handleI out inp (Pure <$> producer out) (Pure <$> consumer inp)
  handleI out inp produce consume = do
    next <- prompt inp consume
    case next of
      Op (Input continue) -> handleO out inp produce continue
      Pure a -> pure a
  handleO out inp produce consuming = do
    next <- prompt out produce
    case next of
      Op (Output o continue) ->
        handleI out inp (continue (pure ())) (consuming (pure o))
      Pure a -> pure a


Connect two copies of the cumulative sum process: compute the cumulative sum of the cumulative sum.

csum2 :: In Int % () -> Out Int % () -> Mom ()
csum2 inp out = connect (\out' -> csum inp out') (\inp' -> csum inp' out)
testConnect :: IO ()
testConnect =
  assert ((collect \out ->
           void $ listInput [1 .. 5] \inp ->
           csum2 inp out
          ) == [1, 4, 10, 20, 35])

Interacting with the real world

What sets IO apart from ST and Mom is that it can change the world. We can define handlers to send output and receive input from the real world. The result of these handlers must be in IO.

Printing output

Text output can be printed to stdout.

printOutput :: (Out String % () -> Mom ()) -> IO ()
printOutput f = momToIO do
  tag <- newPromptTag
  handle tag (Pure <$> f tag)
  handle tag action = do
    next <- prompt tag action
    case next of
      Op (Output o continue) -> pure do
        putStrLn o
        momToIO (handle tag (continue (pure ())))
      Pure () -> pure (pure ())
  momToIO = fromMaybe (pure ()) . run

Reading input

We can forward input from stdin into a consumer computation.

readInput :: (In String % () -> Mom ()) -> IO ()
readInput f = momToIO do
  tag <- newPromptTag
  handle tag (Pure <$> f tag)
  handle tag action = do
    next <- prompt tag action
    case next of
      Op (Input continue) -> pure do
        i <- getLine
        momToIO (handle tag (continue (pure i)))
      Pure () -> pure (pure ())
  momToIO = fromMaybe (pure ()) . run

A drawback of this implementation is that for a computation that features both input and output, these handlers are awkward to compose. We can coerce IO to Mom so readInput can be composed with printOutput, but that is a hacky solution that makes the type Mom a lie (it’s not supposed to have side effects). A better solution may be to combine effects before interpreting them in IO all at once.


No effect tutorial would be complete without the state effect.


data State s a
  = Get (Mom s -> Mom a)
  | Put s (Mom () -> Mom a)
get :: State s % r -> Mom s
get tag = control0 tag \continue -> pure (Op (Get continue))

put :: State s % r -> s -> Mom ()
put tag s = control0 tag \continue -> pure (Op (Put s continue))


State-passing, no mutation.

runState :: s -> (State s % a -> Mom a) -> Mom (s, a)
runState s0 f = do
  tag <- newPromptTag
  handle tag s0 (Pure <$> f tag)
  handle tag s action = do
    next <- prompt tag action
    case next of
      Op (Get continue) -> handle tag s (continue (pure s))
      Op (Put s' continue) -> handle tag s' (continue (pure ()))
      Pure a -> pure (s, a)


incr :: State Int % r -> Mom ()
incr st = do
  n <- get st
  put st (n + 1)

Again, combining state with logging is effortless, because effects live in the same underlying monad.

logState :: Out String % r -> State Int % s -> Mom ()
logState out st = do
  n <- get st
  log out (show n)
incr2 :: Out String % r -> State Int % s -> Mom ()
incr2 out st = do
  incr st
  logState out st
  incr st
  logState out st


testState :: IO ()
testState = do
  assert ((collect \out -> runState 0 (incr2 out) *> pure ()) == ["1", "2"])
  assert (run (discardOutput \out -> runState 0 (incr2 out)) == Just (2, ()))


The examples above are quite sequential in nature. Mom can also replace the list monad.


Choose one element in a list.

data Nondet a where
  Choose :: [x] -> (Mom x -> Mom a) -> Nondet a
choose :: Nondet % r -> [x] -> Mom x
choose tag xs = control0 tag \continue -> pure (Op (Choose xs continue))


nameTheorems :: Nondet % r -> Mom String
nameTheorems nd = do
  name1 <- choose nd ["Church", "Curry"]
  name2 <- choose nd ["Turing", "Howard"]
  result <- choose nd ["thesis", "isomorphism"]
  pure (name1 ++ "-" ++ name2 ++ " " ++ result)


Use the output effect to stream all results of a nondeterministic computation. Here, the continuation is not used linearly: it is called once for every element in the given list.

enumerate :: (Nondet % a -> Mom a) -> Out a % r -> Mom ()
enumerate f out = do
  tag <- newPromptTag
  handle tag (Pure <$> f tag)
  handle tag action = do
    next <- prompt tag action
    case next of
      Op (Choose xs continue) -> for_ xs (handle tag . continue . pure)
      Pure a -> output out a


testEnumerate :: IO ()
testEnumerate = do
  assert (collect (enumerate nameTheorems) ==
    [ "Church-Turing thesis"
    , "Church-Turing isomorphism"
    , "Church-Howard thesis"
    , "Church-Howard isomorphism"
    , "Curry-Turing thesis"
    , "Curry-Turing isomorphism"
    , "Curry-Howard thesis"
    , "Curry-Howard isomorphism"


Earlier, the streaming handler connect interleaved execution of one consumer and one producer thread. Here is a cooperative concurrency effect that lets us dynamically fork any number of threads and interleave them.


data Conc a
  = Fork (Mom ()) (Mom () -> Mom a)
  | Yield (Mom () -> Mom a)

Fork a thread to run the given computation.

fork :: Conc % r -> Mom () -> Mom ()
fork tag thread = control0 tag \continue -> pure (Op (Fork thread continue))

Cooperative concurrency: threads must yield explicitly.

yield :: Conc % r -> Mom ()
yield tag = control0 tag \continue -> pure (Op (Yield continue))


A thread that repeats an output value three times.

simpleThread :: Out String % r -> Conc % s -> Int -> Mom ()
simpleThread out conc n = do
  log out (show n)
  yield conc
  log out (show n)
  yield conc
  log out (show n)
  yield conc

Interleave 111, 222, 333.

interleave123 :: Out String % r -> Conc % s -> Mom ()
interleave123 out conc = do
  fork conc (simpleThread out conc 1)
  fork conc (simpleThread out conc 2)
  fork conc (simpleThread out conc 3)


A round-robin scheduler. handle keeps track of a queue of threads. It runs the first thread until the next event. If the thread yields, its continuation is pushed to the end of the queue. If the thread forks another thread, the forked thread is pushed to the end of the queue, and we continue in the main thread (forking does not yield). If the thread terminates, we remove it from the queue.

runConc :: (Conc % () -> Mom ()) -> Mom ()
runConc f = do
  tag <- newPromptTag
  handle tag [Pure <$> f tag]
  handle tag [] = pure ()
  handle tag (thread : threads) = do
    next <- prompt tag thread
    case next of
      Op (Yield continue) -> handle tag (threads ++ [continue (pure ())])
      Op (Fork th continue) -> handle tag (continue (pure ()) : threads ++ [Pure <$> th])
      Pure () -> handle tag threads


testInterleave :: IO ()
testInterleave =
  assert ((collect \out -> runConc \conc -> interleave123 out conc)
          == ["1", "2", "3", "1", "2", "3", "1", "2", "3"])


Primitive delimited continuation in Haskell give us the power to jump around the stack to implement many kinds of effects. Under the hood, those operations live in the IO monad, grounding effectful code in a familiar execution model.

For those new to the topic, I hope that these examples may serve as a good starting point to experiment with delimited continuations and algebraic effects in Haskell.

The system implemented here is as rudimentary as it gets. To define new effects and handlers, we use the new primitives directly, which is dangerous. This was deliberate to provide material to familiarize oneself with those primitives. Moreover, on the one hand, a type system to keep track of the scope of delimited continuations is a non-trivial ask. On the other hand, the examples here all follow a regular structure, so there is probably a way to encapsulate the primitives, trading off some expressiveness for a safe interface to define new effects and handlers.

Named handlers—via prompt tags—occupy an interesting spot in the scale of safety guarantees. It is imperfect, even very easy to circumvent. But if you’re not working against it, it is still a neat way to prevent simple mistakes. This system can be reinforced further using rank-2 polymorphism, a technique described in:

Interestingly, prompt tags were not part of the original proposal, and they are not used by eff, the effect system which gave rise to Alexis King’s GHC proposal. Prompt tags were added during the feedback process to make the primitives type-safe by default.

Now is an exciting time for algebraic effects/delimited continuations, as they are making their way into industrial languages: Haskell, OCaml, WebAssembly.

All of this is executable

main :: IO ()
main = do
  putStrLn "All tests passed!"

by Lysxia at January 02, 2023 12:00 AM

December 31, 2022

Gil Mizrahi

2022 Recap

In this post I'd like to look back and talk about a few personal projects I've worked on in 2022.


This year I wrote 4 new blog posts (including this one).

The post that took the most work is Building a bulletin board using twain and friends, which is a port of an older tutorial about scotty to the twain web package.

This actually took more effort than I expected. Twain and Scotty are a bit different and the concepts do not map very easily from one to the other. I had to spend a bit of time rethinking how to introduce this library in an approachable way. I've also decided to add some additional information about WAI and other topics along the way.

The other important article I worked on was Giml's type inference engine, which was written in 2021, but had a significant edit.

When I first wrote the article there was one step I was uncertain about - generalization. At the time I omitted it and secretly hoped someone will let me know if I missed an important step and tell me why is it important, but that didn't happen unfortunately. However, after thinking about the interaction with other features (such as modules) I came to understand its importance, and edited it into the article.


This year I did not get back to working on Giml on stream, or streaming in general. I only streamed once this year when I talked about compilers and Haskell. I'd like to stream more in 2023, and make some more progress on Giml.

Open-source software


Giml did not see a lot of work other than adding the generalization step to the type inference and some refactoring. I kinda got stuck with doing the not so fun work of rewriting the parser in a more robust way. This drained my motivation. I hope to get across that hurdle and get back to working on Giml because there are many more interesting things to work on!


GlideVM is a stack-based bytecode interpreter that I started while streaming before I started Giml.

This year I spent a bit more time on it, though not on stream. I refactored some stuff and added comparisons and jumps.

The next interesting feature for this project is closures. I hope to spend some time on it in 2023 as well.


sqlite-easy is a primitive yet easy to use database library for interacting with SQLite3. I just wrote a blog post introducing it, so read it if youwant to learn more!

This is probably the most noteable personal project I worked on this year.


Thanks to your feedback and contributions, Learn Haskell by building a blog generator have seen additional editing, notes and exercises! Thanks for everyone who contributed!

This year I posted the book on Hacker News and to my surprise it got 230 upvotes, which I believe makes it the second most popular link with 'Haskell' in the title for 2022!

To my even bigger surprise I've even found out that my book is on goodreads!

I'd also like to highlight a review of the book. I'm always happy to hear what people think about my work, and constructive feedback helps me improve and make my work better.

Book logo

Final thoughts

This is my third time doing a yearly recap type of post. This year been a bit lighter on content than previous years, but I'm still proud of the progress I've made with my personal projects, and I hope at least some of it can be useful for others as well.

Take care and happy new year!

December 31, 2022 12:00 AM

December 30, 2022

Gabriella Gonzalez

Nixpkgs support for Linux builders running on macOS


I recently upstreamed a derivation for a Linux builder into Nixpkgs that’s easy to deploy on macOS. The significance of this work is that you can now run the following command on macOS:

$ nix run nixpkgs#darwin.builder

… and that will launch a Linux builder that you can delegate builds to. For full details, read the corresponding section of the Nixpkgs manual.

In this post, I wanted to provide some of the background and motivation for this work to help contextualize it.

Background - NixOS qemu VMs on MacOS

I wasn’t originally trying to create a Linux builder for macOS when I began this project. I was actually working on making it as easy as possible to experiment interactively with (non-builder) NixOS qemu VMs on macOS.

While searching for prior art related to this I stumbled across the following Nixpkgs issue requesting exactly this same feature: Allowing NixOS VM’s to be run on macOS.

Even better, by the time I discovered that issue several people had already done most of the work, culminating in the following repository demonstrating how all of the features were supposed to fit together: YorikSar/nixos-vm-on-macos.

In fact, the flake for that repository also came with a binary cache, so if you just ran:

$ nix run github:YorikSar/nixos-vm-on-macos

… then you could run the sample NixOS VM from that repository on macOS without requiring access to an Linux builder because it would download all the Linux build products from the matching cache. Pretty neat!

However, this still didn’t completely satisfy my use case for reasons already noted by someone else: it doesn’t work well if you want to run a NixOS VM that differs even slightly from the included sample VM. Any difference requires Linux build products to be rebuilt which requires access to a Linux builder because those build products will not be cached ahead of time.

Background - linuxkit-nix

The need for a Linux builder wasn’t a showstopper for me because there was already prior art for bootstrapping a Linux builder on macOS, which was the linuxkit-nixproject. So what I could have done was:

  • Launch a (non-NixOS) linuxkit VM on macOS for use as a Linux builder
  • Use the linuxkit builder to build the desired NixOS qemu VM
  • Run that NixOS qemu VM on macOS

However, I was curious if I could use a NixOS VM for the first step, too! In other words:

  • Launch a cached NixOS qemu VM on macOS for use as a Linux builder
  • Use the qemu builder to build the desired (non-builder) NixOS qemu VM
  • Run that NixOS qemu VM on macOS

The only difference between the two approaches is the first step: instead of using linuxkit to create the Linux builder we use qemu to create a NixOS builder. This works because the qemu builder’s NixOS configuration doesn’t need to change so can build and cache the NixOS qemu builder ahead of time.

There were a few reasons I took interest in this approach:

  • linuxkit-nix appears to not work on aarch64-darwin (i.e. Apple Silicon)

    This seems like it is potentially fixable, but I wasn’t yet ready to volunteer to do that work.

  • It’s easier to customize a NixOS builder

    linuxkit-nix doesn’t use NixOS for the builder and instead creates a bespoke builder for this purpose. This means that you can’t use the NixOS module system to more easily customize the behavior of the builder.

  • The qemu-based solution is simpler than linuxkit-nix

    I think the easiest way to explain this is for me to link to the macos-builder.nixNixOS module, which has the entirety of the code that I contributed, which is significantly simpler than linuxkit-nix.

    The main reason that the qemu-based solution is simpler than linuxkit-nix is because it is reusing more infrastructure that has already been upstreamed into Nixpkgs (most notably, NixOS and qemu VMs).

  • linuxkit-nix appears to be unmaintained

    There was a nascent attempt to upstream linuxkit-nix into Nixpkgs, but that stalled because it seems like linuxkit-nix appears to have been abandoned a couple of years ago.

    I could have restored that effort, but personally I was fine with using the simpler qemu-based approach. I haven’t given up on the idea of reviving linuxkit-nix, but it’s not on my immediate roadmap.

There is one notable downside to using qemu over linuxkit, which is that qemu is supposed to be slower than linuxkit

Note: I have not actually verified this claim since I can’t run linuxkit-nix on my M1 Mac, but this is purportedly the reason that the authors of linuxkit-nix did not opt to use qemu for their approach according to this PR description.

qemu performance hasn’t been an issue for me (yet), but that could change, especially if I try to make use of this at work, where performance could potentially matter more.


As I mentioned above, the long-term goal for all of this is to run NixOS VMs on macOS. There are two main reasons I’m interested in this:

  • I’m working on a NixOS book

    … and I wanted macOS users to be able to test-drive example NixOS configurations on their local machine without requiring them to own and operate a separate Linux machine.

  • I’m interested in running NixOS tests on macOS

    … primarily for work-related reasons. At work developers have to install postgres on their development machines for integration testing, and it would be much nicer if we could restructure our integration tests as NixOS tests (which run inside of qemu VMs instead of running on the host).

    However, at the time of this writing this would still require additional work which is in progress on this draft pull request.

by Gabriella Gonzalez ( at December 30, 2022 02:47 PM

December 23, 2022

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.


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 compChoicescompChoices and decompose which are essential 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 = (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


In order to implement compChoices and decompose, 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)

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

It is also useful to calculate a list of the 4 tile edges of a completed half-tile piece clockwise from the origin of the tile. (This is useful for colour filling a tile)

tileEdges:: Piece -> [V2 Double]
tileEdges (LD v) = pieceEdges (RD v) ++ map negated (reverse (pieceEdges (LD v)))
tileEdges (RD v) = tileEdges (LD v)
tileEdges (LK v) = pieceEdges (LK v) ++ map negated (reverse (pieceEdges (RK v)))
tileEdges (RK v) = tileEdges (LK v)

To fill whole tiles with colours, darts with dcol and kites with kcol we can use leftFillDK. This uses only the left pieces to identify the whole tile and ignores right pieces so that a tile is not filled twice.

leftFillDK:: Colour Double -> Colour Double -> Piece -> Diagram B
leftFillDK dcol kcol c =
  case c of (LD _) -> (strokeLoop $ glueLine $ fromOffsets $ tileEdges c)
                       # fc dcol
            (LK _) -> (strokeLoop $ glueLine $ fromOffsets $ tileEdges c)
                        # fc kcol
            _      -> mempty

To fill half tiles separately, we can use fillPiece which fills without drawing edges of a half tile.

fillPiece:: Colour Double -> Piece -> Diagram B
fillPiece col piece = drawJPiece piece # fc col # lw none

For an alternative fill operation  we can use fillDK which fills darts and kites with given colours and draws the edges with drawPiece.

fillDK:: Colour Double -> Colour Double -> Piece -> Diagram B
fillDK dcol kcol piece = drawPiece piece <> fillPiece col piece where
    col = case piece of (LD _) -> dcol
           (RD _) -> dcol
           (LK _) -> kcol
           (RK _) -> kcol

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 a piece  and rotate a piece by an angle. (Positive rotations are in the anticlockwise direction.)

scale:: Double -> Piece -> Piece
rotate :: Angle Double -> Piece -> Piece


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 cd for drawing the pieces, we use

patchWith cd patch = position $ fmap (viewLoc . mapLoc cd) 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.

The common special case drawPatch uses drawPiece on each piece

drawPatch = patchWith drawPiece

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 =  penta [rkite `at` origin, lkite `at` origin]
star = penta [rdart `at` origin, ldart `at` origin]

This figure shows some example patches, drawn with drawPatch 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 compChoices and decompPatch.

To understand these processes, take a look at the following figure.


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 a composed patch arising from the smaller scale patch. Conversely, the larger kites and darts decompose to the smaller scale ones.


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.


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 drawPatch $ 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 leftFillDK

sun6 = suns!!6
filledSun6 = patchWith (leftFillDK red blue) sun6 # lw ultraThin

The earlier figure illustrating larger kites and darts emphasised from the smaller ones is also sun6 but this time drawn with

experimentFig = patchWith experiment sun6 # lw thin

where pieces are drawn with

experiment:: Piece -> Diagram B
experiment pc = emph pc <> (drawJPiece pc # dashingN [0.002,0.002] 0
                            # lw ultraThin)
  where emph pc = case pc of
   -- emphasise join edge of darts in red
          (LD v) -> (strokeLine . fromOffsets) [v] # lc red
          (RD v) -> (strokeLine . fromOffsets) [v] # lc red 
   -- emphasise long edges for kites
          (LK v) -> (strokeLine . fromOffsets) [rotate (ttangle 1) v]
          (RK v) -> (strokeLine . fromOffsets) [rotate (ttangle 9) v]

Compose Choices

You might expect composition 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 composition 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 make further composition choices. 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' <- compChoices lp
    compNChoices (n-1) lp'

This figure illustrates 5 consecutive choices for composing 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 $ fmap drawPatch [[ld,lk'], multiDecomp 5 [lk']] where 
-- two separate patches
       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 = filledSun6
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 easily 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 December 23, 2022 01:03 PM

December 21, 2022

Ken T Takusagawa

[duartbli] Digit strings easy to count their length

It is easy to count the length of the following digit strings:


Strings like these can serve as example input when it is useful to communicate the length of the string but the content of the string is not important, for example, an example block of data encoded in base 10 (future post vkhdrcsg).  Each string documents its own length.

For strings of lengths 20-29, we write the length-10 string above, then ten 2s, then up to 9 digits:


You have to trust that there are ten 2s.  Strings of length 30-39 follow the pattern:


For expository purposes only, we introduce the shorthand (3x10) to indicate ten 3s.  So the above strings of length 30-39 we express as


This shorthand will become especially useful when we explain longer strings.

The pattern easily extends up to strings of length 99.

Strings of length 10-19 are not (1x10)12345... because 10 and 11 would be difficult to distinguish:

1111111111 11111111111

Replacing long strings of 1s with ascending digits will also become useful later in creating a "ladder".

The first idea for 100 is (1x100), but that has same difficulty as 10-11 for strings of length 101.  Next we try (1x10)(2x10)(3x10)(4x10)(5x10)(6x10)(7x10)(8x10)(9x10)(0x10).  We then replace the initial (1x10) with 1234567890 as we did for 10-19, for reasons we will explain shortly, yielding


For strings of length 101-199, we simply append the strings of length 1-99 to the right of the above string.

The strings of length 200-299 has the string of length 100 from above, then (2x100), then the string of length 1-99.  You have to trust that there are 100 2s.  For example, the string of length 200:


Then we can similarly construct strings of length 300-999 as we did for 30-99.

The string of length 1000 is


One can see the recursive structure. The string of length 1000 has the string of length 100 at its beginning, which in turn has the string of length 10 at its beginning.  This "ladder" of more and more stretched out versions of 1234567890 lets you determine what power of 10 you are dealing with by counting the number of recursions.  If we were not to do this, it would be difficult to distinguish between (say) one thousand and ten thousand 1s.  Once we know what power of 10 we are at -- what rung of the ladder we are at -- we can assume that the following long strings of 2s, 3s, etc. each have the same length.

Here is the string of length 1023, illustrating how nothing interesting happens when the length has an internal zero digit:


The string of length 2000 is


There will be multiple ladders, one for each nonzero digit in the length of a string.  The string 1234567890 indicates that the previous digit in the length is complete and a new ladder is beginning.  Here is the string of length 2345:


If possible, text figures (lowercase numerals) should be used to make it easy to spot the beginning of ladders and where digits change.

Haskell source to construct these strings in any base.  Here are the two key routines:

-- expand by 10
dogrow :: Integer -> Char -> String;
dogrow base '1' = List.genericTake (base-1) positive_digits ++ "0";
dogrow base c = List.genericReplicate base c;

-- process little-endian input one digit at a time
growasdigits :: Integer -> [Integer] -> String;
growasdigits _base [] = "";
growasdigits base (h:t) = (growasdigits base t & concatMap (dogrow base)) ++ List.genericTake h positive_digits;

Future work: parser to verify correctness of a string.

Below is the construction in base 4 instead of base 10, showing strings of length 1 through 70.  The first column is string lengths expressed in base 4.


by Unknown ( at December 21, 2022 12:08 AM

Matt Parsons

Haddock Performance

I was recently made aware that haddock hasn’t been working, at all, on the Mecury code base. I decided to investigate. Watching htop, haddock slowly accumulated memory, until it exploded in use and invoked the OOM killer.

My laptop has 64GB of RAM. What.

I rebooted, and tried again. With no other programs running, haddock was able to complete. I enabled -v2 and --optghc=-ddump-timings, which printed out GHC timing information and Haddock time/memory information. With these flags, I could see that HTML generation alone was allocating 800GB of RAM.

I decided to look at the source code and see if there were any low hanging fruit. Fortunately, there was!

Don’t use WriterT for logging

This section culminated in this PR #1543, which I’ll reference. At time of writing, it has been merged.

The first thing that jumped out at me is that haddock used WriterT for logging. Even worse, it used WriterT [String]. This is maybe the slowest possible logging system you can imagine.

At least, use the CPS Writer

WriterT has a big problem with space leaks. Even the strict WriterT has this issue. The only WriterT that can avoid it is the CPS variant, newly available in mtl-2.3 in Control.Monad.Writer.CPS. This is documented in Infinite Negative Utility’s post “Writer Monads and Space Leaks”, which references two posts from Gabriella Gonzalez to the mailing list in 2012 and 2013.

Don’t use [String] or String

Beyond just leaking space, the String format for log messages is extremely inefficient. This is equal to a [Char], which builds up as a big thunk in memory until the whole WriterT computation can complete. Each Char takes 2 words of memory, and a [Char] takes (1 + 3N) words + 2N where N is the number of characters. Or, ~5 words per character. On a 64 bit machine, each word is 8 bytes, so each character costs 40 bytes. A UTF-8 encoded ByteString will use 1 to 4 bytes per character. Using a ByteString would make the representation much more compact, but these things get concatenated a bunch, and a Builder is the appropriate choice for an O(1) append.

Switching to CPS.WriterT [Builder] instead of WriterT [String] helps, but we’re not done yet. [] is a bad choice for a WriterT, since tell will mappend the lists. mappend on lists can result in bad performance if it isn’t associated correctly - (((a ++ b) ++ c) ++ d) ++ e is accidentally quadratic, since we’ll traverse over the a list for every single ++ call. A “difference list” has much faster appends, since it can associate things correctly regardless of how you construct it.

To make it easier to use the API, I created an ErrorMessages type:

newtype ErrorMessages = ErrorMessages { unErrorMessages :: [Builder] -> [Builder] }
    deriving newtype (Semigroup, Monoid)

runErrMsgM :: ErrMsgM a -> (a, ErrorMessages)
runErrMsgM = runWriter . unErrMsgM

singleMessage :: Builder -> ErrorMessages
singleMessage m = ErrorMessages $ (m :)

errorMessagesToList :: ErrorMessages -> [Builder]
errorMessagesToList messages = unErrorMessages messages []

Don’t use nub

There were a few places that called nub. nub is O(n^2) on a linked list - it must search the entire list for every element in the list to verify uniqueness.

That last one is especially gnarly. We’re doing an O(n^2) job, leaking space along the way, and finally we accumulate the big list- only to do an O(n^2) traversal over it to delete duplicates.

Fortunately, each call site of nub can be replaced with the easy ordNub:

ordNub :: Ord a => [a] -> [a]
ordNub = Set.toList . Set.fromList

This also sorts the list, which may not be desired. A more cumbersome implementation does this:

ordNub :: Ord a => [a] -> [a]
ordNub = go Set.empty
    go seen [] = []
    go seen (x:xs) 
        | Set.member x seen = 
            go seen xs
        | otherwise = 
            x : go (Set.insert x seen) xs


This small change resulted in a huge improvement in performance for my test case. Running haddock on the persistent-test library, I observed a 30% improvement in the time to generate documenation with total memory use 4% better. Allocations went from 42GB to 25GB.

I didn’t bother profiling to determine this as a hot-spot - it’s always wrong to use WriterT as a logger. A further performance improvement would be to remove WriterT entirely and simply output the messages directly. Then instead of retaining a big difference list of log messages, you can just print them right then and there.

xhtml and [String]

This section is represented by this haddock PR and this xhtml PR.

haddock uses a library xhtml for generating the HTML. This library is old - the initial copyright is 1999. xhtml predates ByteString entirely, which has an earliest copyright of 2003.

Anyway, we have a similar problem. The Html type is defined like this:

newtype Html = Html { getHtmlElements :: [HtmlElement] }

data HtmlElement
      = HtmlString String
      | HtmlTag {
              markupTag      :: String,
              markupAttrs    :: [HtmlAttr],
              markupContent  :: Html

-- | Attributes with name and value.
data HtmlAttr = HtmlAttr String String

The xhtml library uses ++ on lists all over the place. The renderHtml' function uses ShowS, fortunately - this is difference list of Char, so we probably won’t be seeing pessimal performance.

Like the above PR to remove WriterT [String] and replace it with a difference list of Builder, I did that to xhtml. All explicit lists are now difference lists, and all String are replaced with Builder.

The performance results are impressive:

  Haddock Head xhtml Builder Absolute Difference Relative Change
HTML allocations 1134 MB 1141 MB +7 MB 0.6% worse
HTML time: 380 ms 198 ms -182 ms 47.9% improvement
Total Memory: 554 MB 466 MB -88 MB 15.9% improvement
Total Allocated: 16.0 GB 16.0 GB 0 No change
Max residency: 238 MB 195 MB -43 MB 18.1% improvement
Total Time: 10.88 s 6.526s s -4.354 s 40% improvement

Avoiding [] and String halves the time to render HTML, and results in a 40% overall improvement in the time to run haddock. While we don’t allocate any fewer memory during HTML generation, we’re using 16% less total memory and maximum residency is down by 18%.


Haskell performance doesn’t have to be hard. If you avoid common footguns like WriterT, [], String, nub, etc. then your code will probably be pretty quick.

Picking the low hanging fruit is usually worthwhile, even if you haven’t spent the effort determining the real problem. Profiling shows that haddock spends an enormous amount of time generating object code - a necessary step for any module that has TemplateHaskell enabled. With GHC 9.6, we’ll be able to pass -fprefer-byte-code, which will use the much faster byte code representation instead of object code.

Even in HTML generation, profiling indicates that we spend the majority of time doing fixChar - the process of escaping a character into an HTML appropriate Builder. We also spend a bunch of time regenerating HTML for re-exports - the HTML documentation for a datatype, function, type class, etc is generated fresh for every module that exports it.

Even if HTML were perfectly optimized, Haddock’s current design creates a huge [Interface], where each Interface is a module that you are generating documentation for. This [Interface] must be retained in memory, because it is passed to each “component” of the documentation build. Refactoring haddock to stream these interfaces isn’t obvious, since some doc building steps require summary of the entire [Interface] in order to proceed.

Figuring out a fix for the “real problems” would have been much more difficult than these easy fixes, which have still made a huge difference in overall perforamnce.

December 21, 2022 12:00 AM

December 19, 2022

Gabriella Gonzalez

Nixpkgs support for incremental Haskell builds


The context for this post is that at work I recently implemented Nix ecosystem support for “incrementally” building Haskell packages. By “incrementally” I mean that these Nix builds only need to build what changed since the last full build of the package so that the package doesn’t need to be built from scratch every time.

The pull requests implementing this feature have not yet been approved or merged at the time of this writing, but I figured that I would explain the motivation, design, results, and limitations of this work to hopefully persuade people that this work should be merged.

If you’re not interested in the design then you can skip straight to the Demo section below.


I work on Mercury’s Backend Development User Experience team and we support developers contributing to a large Haskell monolith consisting of 3000+ modules. That may seem like a lot but the vast majority of these modules are small and the whole codebase takes ~14 minutes to compile in CI if we disable optimizations (although we still build with optimizations enabled for deployment).

In my experience, that’s pretty good for a Haskell project of this size, thanks not only to the work of our team but also other teams who also contribute to improving the development experience. In fact, the pioneering work for this “incremental builds” feature actually originated from two engineers outside our team.

First, Harry Garroodimproved GHC’s change detection algorithm so that GHC would use the hash of the file to detect changes instead of using the timestamp. In this post he explains how you can make use of this to implement incremental builds for traditional CI services (e.g. GitHub actions) where each build reuses the intermediate build products from the prior build instead of building from scratch.

That alone would not be enough for us to use this at work since we use Nix where this sort of build impurity doesn’t fly. However, Harry and Jade Lovelace prototyped using this feature in Nixpkgs so that Nix builds of Haskell packages could also reuse intermediate build products from prior builds to save work. You can find their prototype here.

The basic idea behind the prototype Nixpkgs integration is that you split a Haskell package build into two separate builds:

  • A “full build” that builds the Haskell package from scratch

    This full build exports its intermediate build products (i.e. the dist directory) which can then be reused by:

  • An “incremental build” that only builds what changed since the full build

    This incremental build imports the intermediate build products from the corresponding full build so that it doesn’t have to build the package from scratch.

So you might wonder: if that was already implemented then what work still remained for me to do?


The main issue with the initial Nixpkgs integration is that it does not provide any support for selecting which Git revision to use as the basis for the full build. The existing solutions require some out-of-band process to automatically select and lock the appropriate git revision to use for the older (full) build.

Non-solution #0: Rolling rebuilds

The first non-solution is for each revision to always reuse the build products from the previous revision. This doesn’t work well with Nix because it would create an increasingly-long chain of dependent derivations; in order to build the most recent revision you’d have to build all preceding revisions.

The dilemma here is that Nix is forcing us to confront something that other build tools gloss over: if you’re always reusing build products from the last build then you can’t accurately reproduce the most recent build from scratch without reproducing all prior builds. You’ve essentially “contaminated” the current build with all prior builds by doing things in this way.

So what we really want is something more like this:

Periodically do a full build from scratch and then make each incremental build relative to the last full rebuild.

That’s much more compatible with Nix because then we only need to do two builds of our project if we rebuild things from scratch, instead of one build for every revision in our project’s history.

There’s also another issue with rolling rebuilds when you’re not using Nix, which is that most naïve attempts to do this don’t ensure that the starting build products came from the parent commit. You can end up with contamination of build products across branches if you’re not careful, which further complicates reproducibility.

Non-solution #1: Lockfile

Okay, so suppose you periodically do a full build of the project from scratch and then each incremental build is relative to the last full build. You would need to do a full rebuild frequently enough so that the incremental builds stay quick. If you wait too long in between full rebuilds then the project will evolve to the point where the incremental builds can no longer reuse most of the build products from the last full build and in the extreme case the incremental builds degenerate into full builds if they can’t reuse any old build products.

For example, at our work we currently do a full build of our large package once a day, so we need some way to update the full build to point to the last revision from the preceding day.

One existing approach to solving this involved using Nix flakes to manage the git revision for the older build. The idea is that you periodically run nix flake update to update the revision used for the full build and you might even automate this process by having some recurring cron job generate a pull request or commit to bump this revision on the main development branch. You don’t have to use flakes for this purpose, but flakes are probably the most ergonomic solution along these lines.

However, there are a few issues with this approach:

  • It only works well for short-lived pull requests

    In other words, if you update the revision used for the full build once a day then typically only pull requests that are less than a day old will benefit from incremental builds.

    Specifically, what we’d really like is “branch-local” incremental builds. In other words if a longer-lived development branch were to deposit a few commits a day we’d like there to be a full rebuild once a day on that branch so that incremental builds against the tip of that development branch remain snappy.

  • It pollutes the git history

    If you bump the lockfile, say, once per day then that’s one junk commit that you’ve added to your git history every day.

  • It’s difficult to open source any useful automation around this

    If the solution requires out-of-band machinery (e.g. some recurring cron job) to bump the lockfile you can’t provide a great user experience for open source projects. It only really works well for proprietary projects that can tolerate that complexity.

That last point was the most important one for me. Generally, when I design something (even something intended for internal, proprietary use) I try to design it in such a way that it works well in an open source context, too. In my experience, doing things in this way tends to improve the design, quality, and user experience of software that I build.

In particular, I wanted a solution where all the automation could be implemented entirely within the Nix language. However, this is not possible in Nix’s present form!

Non-solution #2: Rollback derivation

So what I really wanted was a Nix function (which I will call “truncate”) that would take any git repository and roll it back in time to the last commit before some repeating time boundary (where the time boundary might be, say, an hour, or day, or week). For simplicity, let’s just say that the desired time interval is one day so I want to roll back the repository to the last revision from the day before.

If I had such a truncate function then it would be easy to automatically select which revision to use for the full build. I would:

  • extract the source git repository from the current Haskell package build

  • truncate that git repository to the last revision from the day before

  • Use that “truncated” revision as the source for the full build

  • Use that full build as the input to the current (incremental) build

Then if I built multiple revisions for the same day they would all share the same full build since they would all get “truncated” to the same revision from the previous day.

However, there isn’t a great way to implement this truncate function in Nix. To see why, consider the following (wrong) solution:

  • extract the source git repository from the current Haskell package build

    Let’s call the derivation for this git repository “src

  • create a new Nix derivation (“src2”) that rolls back src

    In other words, this would be a trivial Nix derivation that begins from src and runs something like:

    $ git checkout $(git rev-list -1 --before '1 day ago' HEAD)

    … and stores that as the result

  • Use src2 as the input to the full build

Do you see the problem with that approach?

The above wrong solution doesn’t allow multiple incremental builds from the same day to share the same full build from the prior day. This is because src2 depends on src and since each incremental build has a different src repository then each also have a different src2 derivation and therefore a different full build. That in turn defeats the purpose of incremental builds if we have to do a new full rebuild for each incremental build.

For this to work we would need a way to roll back a gitrepository to an older revision that less sensitive to the current revision.

Non-solution #3: Plain fetchGit

The builtins.fetchGit utility almost does what we want! This primitive function lets you fetch a gitrepository at evaluation time, like this:

nix-repl> builtins.fetchGit { url = ~/proj/turtle; revision = "837f52d2101368bc075d382774460a717904d2ab"; }
{ lastModified = 1655501878; lastModifiedDate = "20220617213758"; narHash = "sha256-Ic4N2gzm0hYsPCynkzETJv7lpAWO1KM+FO+r3ov60y0="; outPath = "/nix/store/ygznanxv6rmbxw5gkgk7axfxazhsa93z-source"; rev = "837f52d2101368bc075d382774460a717904d2ab"; revCount = 566; shortRev = "837f52d"; submodules = false; }

The above result is the same no matter what revision I currently have checked out at ~/proj/turtle because Nix’s fetchGit function produces a content-addressed derivation. In other words, if two invocations of fetchGit generate the same final repository state then they share the same outPath. This is exactly the behavior we want: we need the source repository for the full build to be content-addressed so that multiple incremental builds can share the same full build.

However, the problem is that I don’t exactly know which revision I want. What I really want to be able to say is “get me the last revision from the day before this other revision”. fetchGit does not expose any way to do something like that.

That brings us to the actual solution:


The solution I went with was the following two pull requests:

  • Add optional date argument to builtins.fetchGit

    This amends builtins.fetchGit to allow a date specification, which can either be a relative date (e.g. 1 day ago) or an absolute date (e.g. 2020-01-01T00:00:00 or a Unix timestamp like 1671388622). Basically, this argument accepts anything git accepts as a date specification (which is a lot since git is pretty flexible in this regard).

    The cool thing about this change is that it doesn’t compromise the purity of builtins.fetchGit. If a given fetchGit specification was pure then adding a date specification preserves that purity.

  • Add haskell.lib.incremental utility

    This pull request actually does two separate things:

    • This polishes and upstreams the prototype support for incremental builds

      In other words, this upstreams Harry and Jade’s work to split a Haskell build into two builds: a full build and incremental build

    • This uses the fetchGit patch to automate the full build selection

      There’s a new pkgs.haskell.lib.incremental utility which uses builtins.fetchGit to automatically update the full build for you and it has all the desired behaviors (including branch-local incrementalism).

    I could have split this into two separate pull request (and I still might) but for internal testing purposes it was easier to do everything on one branch. I’m waiting for a decision on the other pull request before deciding whether or not to split up this branch.


I’ll use my turtle package as the running example for the demo. If you clone the gabriella/incremental branch of my turtle repository:

$ git clone --branch gabriella/incremental \
$ cd turtle

… you’ll find the following default.nix file making use of the Nixpkgs support for incremental Haskell builds:

{ interval ? 24 * 60 * 60 }:

nixpkgs = builtins.fetchTarball {
url = "";
sha256 = "1k3swii3absl154154lmk6zjw11vzzqx8skaiw1250armgfyv9v8";

# We need GHC 9.4 or newer for this feature to work
compiler ="ghc94";

overlay = self: super: {
haskell = super.haskell // {
packages = super.haskell.packages // {
"${compiler}" =
super.haskell.packages."${compiler}".override (old: {
overrides =
(old.overrides or (_: _: { }))
[ (self.haskell.lib.packageSourceOverrides {
turtle = ./.;

(hself: hsuper: {
turtle-incremental =
{ inherit interval;

makePreviousBuild =
truncate: (import (truncate ./.) { }).turtle;

pkgs = import nixpkgs { config = { }; overlays = [ overlay ]; };

{ inherit (pkgs.haskell.packages."${compiler}")

However, that alone is not enough to make use of incremental builds. If you attempt to build that (at the time of this writing) you’ll get an error message like this:

$ nix build --file ./default.nix turtle-incremental
error: evaluation aborted with the following error message:
'pkgs.haskell.lib.incremental requires Nix version 2.12.0pre20221128_32c182b or
(use '--show-trace' to show detailed location information)

The Nixpkgs support for incremental builds depends on a matching change to the Nix interpreter, so you actually have to run:

$ nix run github:Gabriella439/nix/gabriella/fetchGit -- \
build --file ./default.nix turtle-incremental

… or if you don’t yet have flakes enabled, then use this pedantically complete command:

$ nix --option extra-experimental-features 'nix-command flakes' \
run github:Gabriella439/nix/gabriella/fetchGit -- \
build --file ./default.nix turtle-incremental

… and that will definitely work.

Once the build is complete you can inspect the logs and you should see something like the following buildPhase:

$ nix log ./result

@nix { "action": "setPhase", "phase": "buildPhase" }
Preprocessing library for turtle-1.6.1..
Building library for turtle-1.6.1..
Preprocessing test suite 'regression-broken-pipe' for turtle-1.6.1..
Building test suite 'regression-broken-pipe' for turtle-1.6.1..
[2 of 2] Linking dist/build/regression-broken-pipe/regression-broken-pipe [Libr>
Preprocessing test suite 'regression-masking-exception' for turtle-1.6.1..
Building test suite 'regression-masking-exception' for turtle-1.6.1..
[2 of 2] Linking dist/build/regression-masking-exception/regression-masking-exc>
Preprocessing test suite 'tests' for turtle-1.6.1..
Building test suite 'tests' for turtle-1.6.1..
[2 of 2] Linking dist/build/tests/tests [Library changed]
Preprocessing test suite 'system-filepath-tests' for turtle-1.6.1..
Building test suite 'system-filepath-tests' for turtle-1.6.1..
[2 of 2] Linking dist/build/system-filepath-tests/system-filepath-tests [Librar>
Preprocessing test suite 'cptree' for turtle-1.6.1..
Building test suite 'cptree' for turtle-1.6.1..
[2 of 2] Linking dist/build/cptree/cptree [Library changed]

This is shows that the incremental builds are indeed working. We still have to re-link some executables (for reasons that are still not clear to me), but none of the Haskell modules needed to be rebuilt since nothing has changed (yet) since the last rebuild.

Now let’s test that by making a small whitespace change to one of the Turtle modules:

$ echo >> src/Turtle/Prelude.hs 

Then if we rebuild the package we’ll see the following build phase:

$ nix --option extra-experimental-features 'nix-command flakes' \
run github:Gabriella439/nix/gabriella/fetchGit -- \
build --file ./default.nix --print-build-logs

turtle> building
turtle> Preprocessing library for turtle-1.6.1..
turtle> Building library for turtle-1.6.1..
turtle> [ 7 of 10] Compiling Turtle.Prelude ( src/Turtle/Prelude.hs, dist/build/Turtle/Prelude.o, dist/build/Turtle/Prelude.dyn_o ) [Source file changed]
turtle> src/Turtle/Prelude.hs:319:1: warning: [-Wunused-imports]
turtle> The import of ‘Data.Monoid’ is redundant
turtle> except perhaps to import instances from ‘Data.Monoid’
turtle> To import instances alone, use: import Data.Monoid()
turtle> |
turtle> 319 | import Data.Monoid ((<>))
turtle> | ^^^^^^^^^^^^^^^^^^^^^^^^^
turtle> Preprocessing test suite 'regression-broken-pipe' for turtle-1.6.1..
turtle> Building test suite 'regression-broken-pipe' for turtle-1.6.1..
turtle> [2 of 2] Linking dist/build/regression-broken-pipe/regression-broken-pipe [Library changed]
turtle> Preprocessing test suite 'regression-masking-exception' for turtle-1.6.1..
turtle> Building test suite 'regression-masking-exception' for turtle-1.6.1..
turtle> [2 of 2] Linking dist/build/regression-masking-exception/regression-masking-exception [Library changed]
turtle> Preprocessing test suite 'tests' for turtle-1.6.1..
turtle> Building test suite 'tests' for turtle-1.6.1..
turtle> [2 of 2] Linking dist/build/tests/tests [Library changed]
turtle> Preprocessing test suite 'system-filepath-tests' for turtle-1.6.1..
turtle> Building test suite 'system-filepath-tests' for turtle-1.6.1..
turtle> [2 of 2] Linking dist/build/system-filepath-tests/system-filepath-tests [Library changed]
turtle> Preprocessing test suite 'cptree' for turtle-1.6.1..
turtle> Building test suite 'cptree' for turtle-1.6.1..
turtle> [2 of 2] Linking dist/build/cptree/cptree [Library changed]

Our package only built the “diff” (the Turtle.Preludemodule we just changed)!


For the turtle package the speed-up is not a huge deal because the package doesn’t take long time to compile, but the benefit for our main project at work is dramatic!

As I mentioned in the introduction, our work project normally takes ~14 minutes to build and after this change builds can be as fast as ~3.5 minutes. In fact, they could even be faster except for the presence of a Paths_* module that is rebuilt each time and triggers a large number of gratuitous downstream rebuilds (we’re working on fixing that).


There is one major issue with this work, which is that it does not work well with flakes.

Specifically, if you try to turn the above default.nixinto the equivalent flake the build will fail because Nix’s flake mechanism will copy the project into the /nix/store but without the .git history, so builtins.fetchGitwill fail to to fetch the current repository’s history necessary to truncate the build to the previous day.

I believe this can be fixed with a change to flakes to support something like a ?shallow=false or ?allRefs=true addendum to git URLs, but I have not implemented that, yet.

by Gabriella Gonzalez ( at December 19, 2022 03:34 PM

December 17, 2022

Gil Mizrahi

sqlite-easy: A primitive yet easy to use sqlite library

sqlite-easy is a new database library for interacting with SQLite3 databases.

It provides the ability to create database resource pools, run SQL statements and queries, wrap statements in simple transactions, and perform database migrations.

It is built on top of sqlite-direct, migrant, and resource-pool.

I would advise against using it for anything serious, but it might be useful for fun hobby projects.

You can try it now in ghci:

➜  cabal repl --build-depends sqlite-easy
λ> :set -XOverloadedStrings
λ> import Database.Sqlite.Easy
λ> withDb ":memory:" $ run "select 1 + 1"
[[SQLInteger 2]]
λ> pool <- createSqlitePool "/tmp/mysqliteasydb.bin"
λ> withPool pool $ run "create table t (x text)"
λ> :{
λ| withPool pool $
λ|   transaction $ do
λ|     run "insert into t values ('a'),('b'),('c')"
λ|     run "insert into t values ('b'),('a'),('b')"
λ| :}
λ> withPool pool $ runWith "select x, count(x) from t where x <> ? group by x" [SQLText "c"]
[[SQLText "a",SQLInteger 2],[SQLText "b",SQLInteger 3]]

I created this library because I wanted something that is plug and play and doesn't require a lot of thinking. Because of these requirements I decided to:

  • Provide an easy API for common cases, and try to avoid pitfalls like SQL injection by default
  • Include solutions for migrations and resource-pooling - no need to search for libraries or read other docs
  • Avoid config files, Template Haskell, type classes and type level programming - if you know some basic Haskell (and SQL), you should be fine
  • Avoid customization or less common features - small API, less to learn
  • Just use SQL for interacting with the database - no restrictions, but also no validation
  • No external dependencies - SQLite3 bundled and embedded into the executable via sqlite-direct

None of the solutions I was familiar ticked all of the boxes, and after writing the same code a few times for different projects I decided to just make it a library and publish it.

See also How does it compare to sqlite-simple?

How to use sqlite-easy

The docs explain it better, but in essence there are 3 important steps:

1. Connect to a database

Two ways to connect to a SQLite3 database:

  1. withDb - takes a connection string and actions to run, and runs them, producing a result.
  2. Creating a database resource pool using createSqlitePool and then run actions using withPool like with withDb.

2. Run database migrations

Run migrations using the migrate function by defining a list of migration names, a setup function, and a teardown function that maps the name of the migration to the action to take.

3. Run statements and queries

We can define actions to run on the database using the functions run and runWith, These functions Take a SQL statement, and returns something of type SQLite [[SQLData]].

  • SQL is a type representing a SQL string. Use OverloadedStrings to construct values of this type.

  • SQLite is the type describing SQLite actions. SQLite actions can be composed using the monadic interface.

  • SQLData is a type representing a SQLite3 scalar value. SQLite3 has 5 possible value groups - integers, doubles, text strings, binary blobs and null. [[SQLData]] represents rows of columns of scalar values.

SQLite actions can also be wrapped in a transaction.


I've written two examples using this library:

  1. sqlite-easy-example-todo - a simple todo list application with persistent state
  2. learn-twain-bulletin-app the bulletin board from a previous blog post, but using sqlite-easy instead of stm
  3. jot-web - a webapp for writing down memos

The examples use The Handle Pattern.

December 17, 2022 12:00 AM

Abhinav Sarkar

Writing a Static Site Generator Using Shake

Static site generators (SSGs) are all rage these days as people realize that plain HTML websites are good enough for most cases. SSGs take raw data in various formats—often Markdown, JSON, and YAML—and process them to produce the static websites, which can then be hosted easily on any hosting provider, or on personal VPSes. In this post, we write a bespoke SSG using the Shake build system.

This post was originally published on


In the beginning, people coded websites by hand, painstakingly writing the HTML tags and CSS styles (and JavaScript code, if they were into DHTML). Many weblogs were crafted by the hands of passionate individuals, even before the word Blog came into being.

Over time, these websites grew in size and some clever people decided to separate the data for the websites from the presentation and layout. The data moved into databases and CGI scripts were invented to pull the data and create webpages out of them programmatically, on request. Thus began the age of Content management systems (CMS) like Drupal, and of course, blogging software like Wordpress and Blogspot.

Things eventually came full circle, as people realized that they don’t need the bloated and complicated mess of CMSes and blogging software, but at the same time appreciated the separation of presentation and data. Thus Static site generators were born1.

SSGs allow users to write blog articles and pages as plain data in various simple formats like Markdown or reStructuredText, and configurations in YAML, JSON or TOML, and process them to produce static websites in HTML/CSS/JS.

Most SSGs allow the user to operate in a default mode where you can follow the conventions of the SSG—like writing the blog articles in certain formats, and putting them in certain directories—and the SSG takes care of everything else. The user does not need to know any internals.

At the same time, most SSGs allow users to customize the output website by creating custom templates, and custom URLs. However, all SSGs limit what users can do with them. If you need to do something that goes against the grain of your SSG, you are stuck.

Build Systems

SSGs are used to create websites by transforming a set of input files (templates, content, and assets) into a set of output files (HTML, CSS, and JavaScript files). In this sense, SSGs can be seen as a type of build system, as they automate the process of building a website by following a set of rules and dependencies.

A build system is a tool for automating the process of building complex projects. Build systems are commonly used in software development to ensure that the correct sequence of steps is followed in order to produce a working version of the software. This typically involves compiling source code, linking libraries, and running tests to ensure that the software is correct. However, build systems can also be used for projects in other domains where a set of input files need to be transformed into a set of output files according to some rules.

Shake is a build system written in the Haskell. It is flexible and powerful enough for managing the build process of complex software projects like GHC, but at the same time, it is simple enough to be used to create an SSG2.


In Shake, build targets represent the files or outputs that need to be produced as part of the build process. These could be executable binaries, library files, or any other type of output that is required to complete the build. Build targets are declared in a build script, along with information about their dependencies. For example, if an executable binary depends on a particular library file, the build script would specify this dependency.

Once the build targets and their dependencies have been declared, Shake uses Rules to specify how those targets should be built. A rule typically consists of a pattern that matches one or more targets, along with a set of instructions—called build Actions by Shake—for building them. For example, a rule might specify that a certain type of source code file should be compiled using a particular compiler, with a certain set of flags. When Shake encounters a target that matches the pattern in a rule, it executes the instructions in the rule to build it.

By declaring dependencies between targets and defining rules to build them , Shake is able to figure out the correct order in which to build the targets 3. Shake also provides a number of features to help users customize and optimize their build process, such as support for parallel builds, on-demand rebuilding, and caching of intermediate results.

In this post, we use Shake to build an SSG by defining the build targets and rules for building the website. In addition, we use Pandoc to render Markdown content into HTML, and Mustache to render HTML templates.

Static Site Structure

The source of our website is arranged like this:

├── Site.hs
├── css
│   └── default.css
├── images
│   └── logo.png
├── posts
│   ├──
│   ├──
└── templates
    ├── archive.html
    ├── default.html
    ├── home.html
    ├── post-list.html
    └── post.html

Site.hs contains the Haskell code that we are going to write in this post. and are two static pages. The css and images directories contain assets for the website. The posts directory contains blog posts, names of which start with the post publication dates in YYYY-mm-dd format. Finally, the templates directory contains the Mustache templates for the website.

The blog posts start with YAML metadata sections that contain the title of the post, name of the author (optional) and a list of tags for the post. For example:

title: Welcome to my blog
author: Abhinav Sarkar
  - brag
  - note

Welcome to my new blog. I wrote the blog generator myself.


Pages are written in a similar fashion, but have only title in their YAML metadata.

After processing the input above, our SSG produces the following file structure:

├── about
│   └── index.html
├── archive
│   └── index.html
├── contact
│   └── index.html
├── css
│   └── default.css
├── images
│   └── logo.png
├── index.html
├── posts
│   ├── 2022-08-12-welcome
│   │   └── index.html
│   ├── 2022-10-07-hello-world
│   │   └── index.html
└── tags
    ├── brag
    │   └── index.html
    ├── note
    │   └── index.html
    └── programming
        └── index.html

The CSS and image assets are copied directly. One index.html file is generated for each page, post, and tag. Additionally, one file is generated for the archive of posts, and one for the home page.

With the input and output described, let’s get started with writing the generator.


We are going to write the program in a top-down fashion, starting with the main function. First come the extensions and imports. Other than imports from Shake, Pandoc and Mustache libraries, we also import from aeson, text, time and unordered-containers libraries4.

{-# LANGUAGE ApplicativeDo, DataKinds, DeriveGeneric #-}
{-# LANGUAGE DerivingVia, LambdaCase, TypeApplications #-}

module Main where

import Control.Monad (forM, void)
import Data.Aeson.Types (Result (..))
import Data.List (nub, sortOn)
import Data.Text (Text)
import Data.Time (UTCTime, defaultTimeLocale, formatTime, parseTimeM)
import Deriving.Aeson
import Deriving.Aeson.Stock (PrefixedSnake)
import Development.Shake (Action, Rules, (%>), (|%>), (~>))
import Development.Shake.FilePath ((<.>), (</>))
import Text.Pandoc (Block (Plain), Meta (..), MetaValue (..), Pandoc (..))
import qualified Data.Aeson.Types as A
import qualified Data.HashMap.Strict as HM
import qualified Data.Ord as Ord
import qualified Data.Text as T
import qualified Development.Shake as Shake
import qualified Development.Shake.FilePath as Shake
import qualified Text.Mustache as Mus
import qualified Text.Mustache.Compile as Mus
import qualified Text.Pandoc as Pandoc

The main function sets up the top-level Shake build targets, and lets Shake invoke the right one depending on the arguments passed at runtime.

main :: IO ()
main = Shake.shakeArgs Shake.shakeOptions $ do
  Shake.withTargetDocs "Build the site" $
    "build" ~> buildTargets
  Shake.withTargetDocs "Clean the built site" $
    "clean" ~> Shake.removeFilesAfter outputDir ["//*"]

  Shake.withoutTargets buildRules

outputDir :: String
outputDir = "_site"

There are two top-level build targets:

  1. build: generates the website.
  2. clean: deletes the generated website.

outputDir is the subdirectory in which the website is generated. Building the clean target deletes all files inside outputDir. The build target runs the buildTargets action that sets up the build targets for generating the site. The buildRules are also included in the Shake setup.

Build Targets

The buildTargets function sets up the build targets for the files to be generated by Shake.

buildTargets :: Action ()
buildTargets = do
  assetPaths <- Shake.getDirectoryFiles "" assetGlobs
  Shake.need $ map (outputDir </>) assetPaths

  Shake.need $ map indexHtmlOutputPath pagePaths

  postPaths <- Shake.getDirectoryFiles "" postGlobs
  Shake.need $ map indexHtmlOutputPath postPaths

  Shake.need $ map (outputDir </>) ["archive/index.html", "index.html"]

  posts <- forM postPaths readPost
    [ outputDir </> "tags" </> T.unpack tag </> "index.html"
      | post <- posts, tag <- postTags post ]

The Shake.need function registers one or more targets with Shake.

For assets, we just want them to be copied to the outputDir at the same path.

Page and post target paths in the outputDir are stripped of their extensions and appended with /index.html. So a post sourced from posts/ ends up at <outputDir>/posts/example/index.html.

We also register two composite targets for the post archive and the home page5.

The paths, globs and helper function are shown below:

assetGlobs :: [String]
assetGlobs = ["css/*.css", "images/*.png"]

pagePaths :: [String]
pagePaths = ["", ""]

postGlobs :: [String]
postGlobs = ["posts/*.md"]

indexHtmlOutputPath :: FilePath -> FilePath
indexHtmlOutputPath srcPath =
  outputDir </> Shake.dropExtension srcPath </> "index.html"

Now Shake knows what we want it to build. But how does it know how to build them? That’s what the build rules are for.

Build Rules

We have one build rule function for each build target type:

buildRules :: Rules ()
buildRules = do

Let’s start with the simplest one, the build rule for assets.


In Shake, the build rules are written with |%> or %> operators. The |%> operator takes a list of output globs or paths, and a function from target path to build action. When Shake.need is called with a file that matches a target glob, the corresponding build action is called with the target path.

assets :: Rules ()
assets = map (outputDir </>) assetGlobs |%> \target -> do
  let src = Shake.dropDirectory1 target
  Shake.copyFileChanged src target
  Shake.putInfo $ "Copied " <> target <> " from " <> src

In case of assets, we simply get the original source path by dropping the first directory from the target path (that is, outputDir), and copy the source file to the target path if the file has changed6.


Building pages is a bit more interesting. First, we write a data type to represent a page:

data Page = Page {pageTitle :: Text, pageContent :: Text}
  deriving (Show, Generic)
  deriving (ToJSON) via PrefixedSnake "page" Page

A page has a title and some text content. We also make Page data type JSON serializable so that it can be consumed by the Mustache library for filling templates.

Now, the code that builds pages:

pages :: Rules ()
pages = map indexHtmlOutputPath pagePaths |%> \target -> do
  let src = indexHtmlSourcePath target
  (meta, html) <- markdownToHtml src

  let page = Page (meta HM.! "title") html
  applyTemplateAndWrite "default.html" page target
  Shake.putInfo $ "Built " <> target <> " from " <> src

indexHtmlSourcePath :: FilePath -> FilePath
indexHtmlSourcePath =
    . (<.> "md")
    . Shake.dropTrailingPathSeparator
    . Shake.dropFileName

We get the source path from the target path by passing it through the indexHtmlSourcePath function. We read and render the source file by calling the markdownToHtml function. It returns the page YAML metadata as a FromJSON-able value (a HashMap in this case), and the page HTML text.

Next, we apply the Page data to the default.html template, and write it to the target path by calling the applyTemplateAndWrite function. This creates the HTML file for the page.

The default.html Mustache template can be seen below:

<!DOCTYPE html>
<html lang="en">
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta http-equiv="X-UA-Compatible" content="ie=edge">