Author Archives: Ashley Yakeley

Sixteen Unusual Things About Pinafore

Pinafore is an interpreted purely functional language using Algebraic Subtyping that explores structuring information and composing graphical user interfaces.

It has a number of unusual features, some fairly trivial, some more significant, some occasionally found in other programming languages, and some I believe are entirely unique to the language.

Haskellish Things

As an interpreted language written in Haskell, Pinafore programs use the Haskell run-time system. In addition, Pinafore borrows a lot of features from Haskell. As a baseline, if a feature is also part of Haskell, I don’t count it as an Unusual Thing. These include, for example:

  • A type system that extends Hindley-Milner
  • Type inference
  • User-defined data types with constructors
  • Constructor pattern matching
  • Separation of pure functions from effects
  • Laziness
  • Garbage collection
  • Threads
  • “do” notation, for monads

Pinafore’s syntax is somewhat different. Here’s a quick cheat-sheet of equivalent Haskell and Pinafore syntax and types:

v :: Tv: T
\x y -> x + yfn x, y => x + y
x & f = f xx >- f = f x
case x ofx >- match
h : th :: t
(P, Q)P *: Q
Either P QP +: Q

1. Algebraic Subtyping

Algebraic Subtyping is a fairly recent type system that extends Hindley-Milner with subtyping in a way that is decidable and has principality. To achieve this, it distinguishes positive and negative types, and makes use of union and intersection types.

As far as I know, Pinafore is the only non-academic implementation of Algebraic Subtyping.

Why did I pick this type system? Part of the purpose of Pinafore is to represent information in an intuitive manner, and people naturally think in terms of subtypes, e.g. “every dog is an animal”, “every apple is a fruit” and so forth. And Hindley-Milner-based approaches are much cleaner mathematically, and therefore easier to reason about, than the kind of two-level type systems you find in languages such as Java, C#, C++ and so forth.

A subtype relation between types P and Q, written P <: Q, and pronounced “P is a subtype of Q“, simply means “every P is a Q“, or “accept a P where a Q is expected”. It implies a conversion function from P to Q. As an example, Pinafore has three numeric types Integer, Rational, and Number, with the intuitive subtype relations Integer <: Rational <: Number. So you can pass an Integer to a function expecting a Rational, and Pinafore will perform the conversion implicitly.

It’s important to note that in this type system, and in Pinafore, conversion functions do not need to be injective, nor does a retraction function Q -> Maybe P need to exist. It’s perfectly OK for type conversion to lose information. The only constraint is that, at any given point in your program, the system of subtype relations that are in scope must be consistent. This essentially means that “subtype diagrams commute”, that is, if there is more than one subtype “path” for the same types, it must imply the same conversion function.

Algebraic Subtyping, like Hindley-Milner, has type variables, and types can be constructed from type constructors that have parameters. In Algebraic Subtyping, however, each type parameter of each type constructor must be either covariant or contravariant. In cases such as a “reference cell” for which the type parameter would not naturally be either, a contravariant/covariant pair of type parameters is used instead. Pinafore has a special syntax for such pairs to make them easier to work with. The “optics” section of the library makes particular use of these, corresponding to similar pairs of parameters in various optics types in Haskell and Scala.

Algebraic subtyping can give types to expressions that Hindley-Milner cannot. For example:

pinafore> :type fn x => x x
: (a -> b & a) -> b
pinafore> :type fn f => (fn x => f (x x)) (fn x => f (x x))
: (a -> a) -> a

Pinafore’s implementation of Algebraic Subtyping includes equirecursive types, which are necessary as the principal types of certain expressions. They are, however, not particularly used. Pinafore omits the record types defined in the paper.

A quick note on terminology, for those familiar with this type system: a positive type is a type that can appear in positive position, likewise a negative type, while an ambipolar type is one that is both positive and negative, and a concrete type is a type with no free type variables.

2. General Subtype Relations

Given two types P and Q already in scope, Pinafore lets you declare a subtype relation P <: Q simply by providing the conversion function of type P -> Q.

As far as I know, there is no other programming language that implements this.

Here’s an example, defining a subtype relation from Map a to the function type Entity -> Maybe a:

subtype Map a <: Entity -> Maybe a =
  fn m, e => lookup.Map e m;

The Map type and function type are already defined, and here we declare a subtype relation. So now we can treat maps as functions.

Of course, since Pinafore separates pure functions from effects, these conversion functions are all pure functions without effects.

To maintain consistency, Pinafore will reject any subtype declaration if it would create a subtype “diamond” (two different subtype paths from the same types), since it cannot prove that the diamond commutes. But if you want to override this, because you want Pinafore to blindly accept your claim that the two paths are the same conversion, then you can add the trustme flag to the declaration.

3. Greatest Dynamic Supertype

Subtype conversion functions do not need to have retractions. But, some of them do! Pinafore provides a general mechanism for working with them, the greatest dynamic supertype.

Here’s how it works. Every (concrete ambipolar) type T has a greatest dynamic supertype D(T), with these properties:

  • T <: D(T)supertype
  • D(D(T)) = D(T)greatest
  • The retraction function is defined: check @T: D(T) -> Maybe Tdynamic

(Note that check is a “special form” that requires specifying a type, rather than an ordinary binding.)

So for most T, D(T) = T, which is not very interesting. However, for example, the GDS of all literal types (numeric types, Text, date & time types, etc.) is Literal, which internally stores (“dynamically”) the run-time type. So if you have Literal, you can inspect it with check @Rational to obtain the Rational if it represents one.

Alternatively, there’s a pattern-matching form :? that does the same thing as check:

    t :? Rational => Just t;
    _ => Nothing

If you define a subtype within a datatype definition, then that will give the appropriate GDS relationship:

let rec

    datatype List +a of
        subtype datatype List1 of
            Cons a (List a);


This defines two type constructors, List (representing lists), and List1 (representing non-empty lists), both of which have one covariant parameter, with List1 a <: List a, and D(List1 a) = List a.

4. No Top Level

A main program is an expression. All declarations, including name bindings, type declarations, and subtype relation declarations, are within let or let rec declarators.

It’s perfectly OK for values to escape the scope of their types. They typically can’t be used at that point, but I don’t believe that that’s unsound. Internally, each type is assigned an ID, so there’s no possibility of aliasing a different type with the same name.

5. “Backwards” Name Qualification

Name qualification in Pinafore runs in the order specific-to-general. For example, the name a within namespace N within namespace M is referred to, fully qualified, as a.N.M. (note dot at end). This is the reverse of how it’s done in most other languages such as C++ (which would have ::M::N::a) or Java (M.N.a).

So why? It’s simply easier to read, especially as the capitalisation of the first letter of the name is significant. Note that the data constructors of a type are placed in the “companion namespace” of the type, and this too is easier to read with this order.

Pinafore does not have anything like Haskell’s type classes, and there is no overloading, so namespaces play a bigger role in distinguishing equivalent functions for different types.

datatype Expression +a of
    Closed a;
    Open Text (Expression (Value -> a));

evaluate: Expression a -> Action a =
    Closed.Expression a => pure.Action a;
    Open.Expression name _ => fail $ "undefined: " <>.Text name;

Perhaps this is a matter of taste, but I find Open.Expression, pure.Action, and <>.Text to be easier to read than Expression.Open, Action.pure, and Text.<>.

6. Record Constructors

Data type definitions look like this:

datatype D of
    Mk1 T1 T2 T3;
    Mk2 T4;

Mk1.D, Mk2.D, etc. are data constructors, and T1, T2, etc. are types. These types must be concrete (i.e. monomorphic, no free type variables), and ambipolar.

Why is this? Consider the types of these expressions:

Mk2.D: T4 -> D

fn Mk2.D t => t: D -> T4

As you can see, T4 appears in both negative and positive positions, and these are the only ways to use the Mk2.D constructor to construct and examine values of D.

Record constructors are a different kind of constructor, that allow you to store any polymorphic positive types. They look like this:

datatype Record of
    Mk of
        f: (a -> a) -> a -> a;
        g: Integer;
        h: Action a -> Action (Action a);

Mk.Record is a record constructor, and when matched as a pattern, it brings its members f, g, h into scope.

fn Mk.Record => f: Record -> (a -> a) -> a -> a

To use it to construct a Record, you can do this:

Mk of
    f = fn x, a => x (x a);
    g = 17;
    h = map.Action pure;

7. Expose Declarations

Pinafore gives you fine-grained control of which names to expose out of a declaration. Here’s an example:

    a = 1;

        p = 2;
        q = 7;
        r = p * q;
    in expose p, r;

in a + p + r

The expose-declaration (letin expose p, r) exposes p and r, but not q, within the bindings of the outside let– expression.

This sort of thing is useful for creating “subset” and “quotient” types of a given type. For example, consider the Text type, for representing strings of text. Lower-case text is a subset of text, where only some text (all lower case) can be constructed. Case-insensitive text is a quotient of text, where text-case cannot be distinguished. We can create Pinafore for both of these types by carefully exposing certain functions and hiding others:


    datatype LowerCaseText of Mk Text end;
    subtype LowerCaseText <: Text = fn Mk.LowerCaseText t => t;

    datatype InsensitiveText of Mk Text end;
    subtype Text <: InsensitiveText = Mk.InsensitiveText;

    myToLowerCase: InsensitiveText -> LowerCaseText =
        fn Mk.InsensitiveText t => Mk.LowerCaseText $ toLowerCase t;

in expose LowerCaseText, InsensitiveText, myToLowerCase;

Note that the two types we have defined are exposed, but not their data constructors. (Subtype relations are always exposed.) Thus, we can examine a value of type LowerCaseText, since it’s a subtype of Text, but we can’t create one except with our myToLowerCase function. Likewise, we can create a value of type InsensitiveText, since it’s a supertype of Text, but we can’t examine it except with our myToLowerCase function.

8. Soft Exceptions

stop: Action None;

onStop: Action a -> Action a -> Action a;

In Pinafore, any action can stop. For example, attempting to change a model that happens to be immutable will stop. Trying to retrieve missing/null data will stop. Or you can just call stop. This is essentially a kind of exception, which can be caught, if you want, with onStop.

Stops are “soft” in the sense that they are intended to control flow rather than indicate an error or exceptional condition. Default handlers (e.g. for button presses, or the main program) will catch and silently discard the stop without raising any kind of further error or complaint. The idea is, the user clicks the button or whatever, and if the action fails, by default nothing else happens.

As such, stop can be used much like break in C, for breaking out of iteration rather than indicating a problem. For example:

tryStop_ $ forever $ do
    m <- read.Source someinput;
    m >- match
        Item.ItemOrEnd x => someaction x;
        End.ItemOrEnd => stop;

9. Live Models

The purpose of Pinafore is to represent information, and compose user interfaces (using GTK) for it. If you’re familiar with the “model/view/controller” approach to user interface, Pinafore separates “models” of various types from “view/controller”. These combine to make a UI element (GTK widget) that can be laid out in a window.

A model represents some information than can change and be changed. It might be a file, or information in storage (see below), or simply pure memory. Change goes in both directions: the UI “controller” can change the model in response to user input (by calling functions on it), and the model can change the UI “view” (the UI subscribes to updates from the model). Multiple widgets can be constructed from the same model, or from composed models; and changes from one widget will cause asynchronous updates to other widgets, to keep them all consistent.

It is not enough to represent the type of some piece of information, one must also represent the characteristic ways that that information changes. For example, something simple like a flag or a number changes as a whole, that is, the entire value is replaced. But something like a list or a set or text changes in a more complicated manner, with, for example, insertions and deletions. To represent changeable information properly, the various types of changes must be represented.

Pinafore has a number of model types to represent changeable information, each with its own type for changes and updates. The simplest is WholeModel T for any given type T. This simply represents information of type T that can be retrieved, set and updated as a whole value, with no further structure. Values of WholeModel can be composed in various ways, including applicatively (see below).

By contrast, ListModel T represents a list of elements each of type T, where individual elements can be retrieved, inserted, deleted, and changed. The type of updates from the model to the UI specifies insertion/deletion/change, so the UI view can react cleanly to the semantic of the update. You can also pick out an element of a ListModel T to get a WholeModel T, which will track that element as other elements are inserted and deleted around it.

A TextModel tracks insertions/deletions/replacements of text. A SetModel T tracks additions and deletions of members of type T. A FiniteSetModel T is a SetModel T that also tracks its list of members. It is possible to compose some of these model types in various set-theoretic ways (unions, intersections, Cartesian sums and products).

10. Declarative User Interface

Pinafore uses GTK to provide user interface, however, all UI elements (“widgets”) are created declaratively by connecting them to models. Thereafter, changes to the UI reflect changes to the models.

open.Window gtk (300,400) {"My Window"} $ vertical.Widget
        button.Widget {"<"} {do d <- get r; r := pred.Date d; end},
        layoutGrow.Widget $ label.Widget {encode (unixAsText.Date "%A, %B %e, %Y") %r},
        button.Widget {">"} {do d <- get r; r := succ.Date d; end}
    layoutGrow.Widget $ textArea.Widget (fromWhole.TextModel $ dateNotesOf !$ r)

In this example, the widgets are simply declared in a static structure, and don’t need to be referred to again. Any aspect of a widget that can change is connected to a model: when the model updates, the widget will change. When the user interacts with the widget, it will pass those changes to the model.

But what if some part of the widget structure itself should change dynamically, in response to user actions? For that there’s the dynamic widget:

dynamic.Widget: WholeModel +Widget -> Widget

Simply create a whole-model of how the widget structure should change in response to other models, and pass it to dynamic.Widget as your widget.

11. Declarative Drawing

Pinafore provides bindings to Cairo for drawing functions, which can be used to construct your own GTK UI widgets. Rather than using an imperative style as Cairo usually provides, Pinafore’s Cairo bindings are declarative: the basic Cairo type is Drawing, and visual elements of type Drawing can be appropriately composed and modified, without having to work with any kind of mutable context state.

Here’s a complete Pinafore program that creates a window with a rotating red triangle on top of a navy disc:

import "pinafore-gnome", "pinafore-media" in
with GTK., Cairo., Drawing.Cairo., Number. in
    pointer: Drawing None =
    with Path., Colour. in
    source red $ fill $ concat [moveTo (-0.1,0), lineTo (0.1,0), lineTo (0,1), close];

    disc: Drawing None =
    with Path., Colour. in
    source navy $ fill $ arc (0,0) 1 0 (2*pi);
newClock (Seconds 0.1) >>= fn now =>
run.GTK $ fn gtk =>
open.Window gtk (400,400) {"Drawing"} $
    fn (w,h) =>
    let angle = fromTime.LocalTime 0 %now >- fn DateAndTime _ (SinceMidnight (Seconds ss)) => ss / 10 in
    translate (w/2,h/2) $ scale (w/2,h/2) $ concat [disc, rotate angle pointer]

You can see the drawing code in the following lines. This creates a closed triangular path, and fills it with red:

source red $ fill $ concat [moveTo (-0.1,0), lineTo (0.1,0), lineTo (0,1), close]

This creates a complete circle and fills it with navy:

source navy $ fill $ arc (0,0) 1 0 (2*pi)

This layers the rotated pointer over the disc, and scales and translates it to the window’s co-ordinate system:

translate (w/2,h/2) $ scale (w/2,h/2) $ concat [disc, rotate angle pointer]

Since the drawing is converted to a GTK widget, one can also add click actions for particular drawing elements.

12. Lifecycles

Many actions require some later “closing” or “clean-up”, for example, opening a GUI window means it will be closed again, or starting an asynchronous task means waiting for it to complete, and so forth. These are sometimes called resources.

Pinafore has a general mechanism for working with resources, called lifecycles. The basic idea is that any resource “opened” in a lifecycle will be “closed” at the end of that lifecycle.

These are the functions provided:

run.Lifecycle: Action a -> Action a;

onClose.Lifecycle: Action Unit -> Action Unit;

closer.Lifecycle: Action a -> Action (a *: Action Unit);

Calling run on an action will run it in a new lifecycle. Thus, if a GUI window is opened in that action, it will be closed at the end of the run call.

Calling onClose will create a “closing action” for your own resource. Closing actions will be run (in reverse order of creation) at the end of the lifecycle.

Calling closer on some action that opens resources gives the result of that action together with its (idempotent) “closer”. Calling the closer allows you to close the resources at any time, before the end of the lifecycle, allowing closing of resources in any order.

A Pinafore program is itself run in a lifecycle, that will run closers before exiting, so all resources will be cleaned up.

13. Applicative Notation

Pinafore does not have type classes, and all type variables have kind *, so there is no equivalent to Haskell’s Applicative class. Nevertheless, certain type constructors are essentially applicative, and Pinafore has a special notation for composing them applicatively. These include Action, Maybe, Task (see below), List, WholeModel, among others, though you can define your own, if you define the correct functions in a namespace.

pinafore> {.List %[3,4,5] + %[10,20]}
[13, 23, 14, 24, 15, 25]

In this example, two lists are combined applicatively. The braces { .. } indicate applicative notation, and .List is the namespace from which applicative operations pure, map, ap, etc. are obtained. This is how it gets converted:

{.List %[3,4,5] + %[10,20]}

ap.List (ap.List (pure.List (fn var1, var2 => var1 + var2)) [3,4,5]) [10,20]

Applicative notation is particularly useful for working with whole-models. Models created in this way are immutable, however, their updates are propagated. The WholeModel namespace is the default for applicative notation if the namespace is omitted.

import "pinafore-gnome" in
run.GTK $ fn gtk =>
    now <- newClock $ Seconds 1;
    open.Window.GTK gtk (400,400) {"Clock"} $
        label.Widget.GTK {"The time is " <>.Text show %now};

In this example, now is a model of a clock that updates every second with the current time. It has type WholeModel +Time. The notation {"The time is " <>.Text show %now} of type WholeModel +Text creates a model of text that updates every second. Running this program will create a window with a label widget with text that updates every second.

Note that the title of the window, {"Clock"}, is also a whole-model, although in this case of constant text.

14. Composable Asynchronous Tasks

Pinafore uses a locking system to make GTK user interface code thread-safe. When a live model (see above) updates, subscriptions to that update happen in a new thread, which may call GTK. Functions to set and change models are also atomic and thread-safe.

As part of this design, Pinafore makes wide use of threads rather than relying on the kind of yielding state machines found in C#. A task in Pinafore is simply anything that can be waited upon for completion with a result value.

async.Task: Action a -> Action (Task a)

await.Task: Task a -> Action a

check.Task: Task a -> Action (Maybe a)

Pinafore does have a function async, but in this design it simply starts a task running in a new thread. Tasks started in this way complete in the current lifecycle, that is, calling async creates a lifecycle-closing action that waits for the thread to complete.

Tasks can be composed applicatively using pure, map, **, etc., or using applicative notation. Waiting on such a task simply waits for all its component tasks to complete, and then combines the result values.

15. Robust Storable Data

Pinafore has a built-in storage system, storing data in a SQLite file in a directory in your home directory.

Pinafore storage is designed to be robust with regards to changes in data schema. New constructors can be added to storable data types without affecting existing data in storage. Constructors can also be removed: such data is simply considered null or missing, and the usual retrieval function will “stop” (see above).

Only certain types are storable. But Pinafore does not store values of types as such. You cannot, for example, retrieve all values of a type. Rather, Pinafore stores subject-property-object triples. The subject and object of each triple are values of storable types. Properties are declared in code. You can retrieve the object of a given subject and property. You can also retrieve the set of subjects for a given property and object. Pinafore stores knowledge rather than items.

    opentype Person;
    name: Property Person Text = property @Person @Text !"person-name" store;
    dave: Person = point.OpenEntity @Person !"my friend dave";
in name !$ {dave} := "Dave"

In this example, opentype Person creates a new open entity type. Open entity types are purely abstract points: they are represented and stored as 256 random or hashed bits and have no further internal structure or meaning. Instead, they are used as subjects and objects of properties.

In the next line, property @Person @Text !"person-name" store defines a Text property of Person. Triples with this property will have a subject of type Person and an object of type Text. !"person-name" is an anchor, which is hashed and used to identify the property in storage. The binding property is a “special form”: something that takes (constant) type and anchor arguments. (The store argument represent storage, obtained from openDefault.Store.)

Properties can be composed and combined in various ways: for example, name ..Property father is a property for “father’s name”, while name **.Property father is a property for “name and father”.

The special form point.OpenEntity defines fixed values of open entity types, in this case a value of type Person identified in storage by the given anchor. Points can also be generated randomly.

The last line name !$ {dave} := "Dave" constructs a model, name !$ {dave}, and then sets it to the text. This model can also be passed to a user interface widget, which would be able to set it, and also be automatically updated when it changes.

Properties can also be used “in reverse”, for example, name !@ {"Dave"} will retrieve a FiniteSetModel of all points in storage with name set to "Dave".

You can create your own polymorphic storable data-types, however all types mentioned in constructors must be storable, and all type parameters must be covariant. Each constructor is given an anchor to identify it in storage. For example:

let rec
    datatype storable Tree +a of
        Mk a (List (Tree a)) !"Mk.Tree";

Since open entity types represent abstract points without further structure, subtype relationships between them can be declared arbitrarily without providing any conversion functions, or worrying about “diamonds”. For example:

opentype Named;
name = property @Named @Text !"name" store;

opentype Living;
birthdate = property @Living @Date !"birthdate" store;
deathdate = property @Living @Date !"deathdate" store;

opentype Person;
subtype Person <: Named;
subtype Person <: Living;

Since open entity types represent simple points, they don’t carry type information. For example, if you have an value of type Living, you cannot determine whether or not it is a Person. To store type information, you can use dynamic entity types instead:

dynamictype Human = !"Human";
dynamictype Cat = !"Cat";
dynamictype Mammal = Human | Cat;
dynamictype Bird = !"Bird";
dynamictype Animal = Mammal | Bird;

Like open entity types, dynamic entity types are storable, but each value internally stores an anchor representing its “dynamic type” or “runtime type”. The greatest dynamic supertype of all dynamic entity types is DynamicEntity, so it’s easy to check a dynamic type at run-time:

checkMammal: Animal -> Maybe Mammal =
    m :? Mammal => Just m;
    _ => Nothing;

16. Undo Handling

Being able to undo and redo actions is an essential part of user interface. Pinafore provides functions to create “undo handlers” and connect them to storage or to specific models. Changes to these things will be recorded to a queue, and can then be undone and redone.

new.UndoHandler: Action UndoHandler

handleStore.UndoHandler: UndoHandler -> Store -> Action Store
handleWholeModel.UndoHandler: UndoHandler -> WholeModel a -> WholeModel a
handleTextModel.UndoHandler : UndoHandler -> TextModel -> TextModel
# etc., for all model types

queueUndo.UndoHandler : UndoHandler -> Action Boolean
queueRedo.UndoHandler : UndoHandler -> Action Boolean

Future Directions

Everything above is available in the latest release of Pinafore, version 0.4. But I have lots more ideas and plans for future versions.

On the language side:

  • Predicate types: types which need to satisfy a user-supplied predicate to construct. Thus you could create a “type of prime numbers”.
  • Existential types: a record constructor could declare a type name, to be instantiated with a type when the value is constructed.

More general feature ideas:

  • Graphs and charts, especially along the lines of The Grammar of Graphics.
  • Easy connection to external data sources.
  • More composability of user-interface widgets, such as variable lists and grids of a given widget type.

Learn More

Pinafore 0.4 is available as a Debian package for 64-bit Linux, that will work on Ubuntu 22.04 (and derivatives), Debian 12, and later. There is also a Nix flake for 64-bit Linux. Go to

— Ashley Yakeley

Pinafore 0.4

I’ve just released version 0.4 of Pinafore, representing more than a year’s worth of full-time work since the last release.

I’ll make another post shortly explaining what Pinafore is all about and what makes it unusual, but in the mean time, this is the release notes for the new version:

  * Install
    - Debian package works on:
      . Ubuntu 22.04 LTS "jammy"
      . Debian 12 "bookworm"
    - add Nix flake
  * Language
    - overhaul of declarations
      . separate namespaces from modules,
        with "namespace" declarations and "with" and "import" declarators
      . both non-recursive ("let") and recursive ("let rec") declarators
      . "expose" declarations
      . allow declaration documentation with #| and {#| #} comments
    - datatype declarations
      . "closedtype" now "datatype storable"
      . can now have parameters
      . can now have subtypes
      . record constructors/patterns for datatypes
      . allow recursive types in datatypes
    - can declare arbitrary subtype relations
    - import lists
    - syntax
      . allow defintion of new operators
      . changed recursive type syntax from "rec v. T" to "rec v, T"
      . type names (+:), (*:), List, Unit
      . tuple constructor/pattern (,,) etc.
      . type signatures now attach to bindings, not stand-alone
      . separate syntax for static ":" and dynamic ":?" pattern typing
      . new syntax for function expressions: fn, match, =>
      . new syntax for datatype definitions
      . generalised "{}" and "do" syntax to any namespace
    - reject rather than mutate uninvertible type signatures
    - allow polymorphic recursion with type signatures
  * Interactive
    - :doc to retrieve name documentation
  * Library
    - Std
      . Literal type now byte array rather than text
      . Literal types now have GDS Literal
      . add Showable type for showing, show replacing toText
      . add min/max/lesser/greater functions
      . add List1 type for non-empty lists, subtype of List
      . rename "Ref" types and functions to "Model"
      . add TextModel type & associated functions, use for uiTextArea
    - new Task module
      . add Task type & associated functions
    - new Stream module
      . add sinks & sources
    - new Env module
      . move invocation-type stuff here
      . add stdin, stdout, stderr
    - new Eval module
      . move evaluate here
    - new Colour module
      . add Colour & AlphaColour types, etc.
    - new GIO module
      . add GIO File type and functions
    - new Cairo module
      . add Cairo-based functions for creating drawings
    - new Image module
      . add Image, HasMetadata, PNGImage, JPEGImage types, etc.
    - GTK
      . rename (from "UI")
      . explicit control over context
      . menu bar is just ordinary element
      . element for Image
  * Storage
    - Anchors now 256 bit, hash using BLAKE3
    - Store literals as binary rather than as text
    - Embed smaller literals directly in the anchor
  * Fixes
    - fix defect in lexical scoping

— Ashley Yakeley

Monadology 0.1

Monadology is intended as a collection of the best ideas in monad-related classes and types, with a focus on correctness and elegance, and theoretical understanding, rather than practical performance. I am interested in hearing further ideas, so at least initially expect a lot of change version-to-version.

Re-exported Transformers

Monadology is built on the existing transformers package. It re-exports most of it. (It does not re-export ListT).


This general-purpose “result” monad represents either success or failure, of any type. This sort of thing is so useful it could have been in base, but it isn’t.

data Result e a
    = SuccessResult a
    | FailureResult e

Of course, it’s isomorphic to Either. But whereas Either has a more general-purpose “symmetrical” feel, Result is more intelligible to the reader as a monad.


Monadology makes two separate approaches to exceptions: one type and many types. For example, for the IO monad, there are many different exception types that can be both thrown and caught. But there is also the one type SomeException that represents all the possible exceptions.

Many Types

For the many-types approach, Monadology simply provides MonadThrow and MonadCatch classes, along with various functions:

class Monad m => MonadThrow e m where
    throw :: forall a. e -> m a

class MonadThrow e m => MonadCatch e m where
    catch :: forall a. m a -> (e -> m a) -> m a

One Type

In principle, every monad m has a single type of all the exceptions it can throw and catch. For this approach, this type is named Exc m:

class Monad m => MonadException m where
    type Exc m :: Type
    throwExc :: Exc m -> m a
    catchExc :: m a -> (Exc m -> m a) -> m a

type Exc Identity = Void
type Exc ((->) r) = Void
type Exc Maybe = ()
type Exc (Result e) = e
type Exc (ExceptT e m) = Either e (Exc m)
type Exc (StateT s m) = Exc m
type Exc IO = SomeException

Functions such as finally and bracket, that make no reference to any particular exception type, make use of this to ensure that they work for all exceptions that can be thrown.

Composing Monads

You can compose two functors to get a functor. And you can compose two applicative functors to get an applicative functor. But, famously, you cannot compose two monads to get a monad.

At least, you cannot in general. But you can, of course, in certain cases. And we can capture the most useful cases by specifying the constraints we need on one of the monads so as to leave the other unconstrained.

Inner Monad

MonadInner is just the right constraint on the inner monad so as to compose with any outer monad to get a monad.

class (Traversable m, Monad m) => MonadInner m where
    retrieveInner :: forall a. m a -> Result (m Void) a

newtype ComposeInner inner outer a = MkComposeInner (outer (inner a))

instance (MonadInner inner, Monad outer) => Monad (ComposeInner inner outer)
instance MonadInner inner => MonadTrans (ComposeInner inner)

Essentially, inner a must be isomorphic to Either P (Q,a) for some P, Q. If you examine the structure of the WriterT, ExceptT, and MaybeT monad transformers, you’ll see that they are cases of this composition pattern.

Outer Monad

MonadOuter is just the right constraint on the outer monad so as to compose with any inner monad to get a monad.

newtype WExtract m = MkWExtract (forall a. m a -> a)

class Monad m => MonadOuter m where
    getExtract :: m (WExtract m)

newtype ComposeOuter outer inner a = MkComposeOuter (outer (inner a))

instance (MonadOuter outer, Monad inner) => Monad (ComposeOuter outer inner)
instance MonadOuter outer => MonadTrans (ComposeOuter outer)

Essentially, outer a must be isomorphic to P -> a for some P. If you examine the structure of the ReaderT monad transformer, you’ll see that it’s a case of this composition pattern.


LifecycleT is a monad transformer for managing the closing of opened resources, such as file handles, database sessions, GUI windows, and the like. You can think of it as a conceptually simpler version of ResourceT.

The actual code is slightly different in the contents of the MVar, but it basically looks like this:

newtype LifecycleT m a = MkLifecycleT (MVar (IO ()) -> m a)

runLifecycle :: (MonadException m, MonadTunnelIO m) => LifecycleT m a -> m a

lifecycleOnClose :: MonadAskUnliftIO m => m () -> LifecycleT m ()

type Lifecycle = LifecycleT IO -- the most common usage

That MVar simply stores all the “close” operations to be run at the end of each “lifecycle” when called by runLifecycle, in reverse order of their opening. You can add your own close operations with lifecycleOnClose.

Of course you may be thinking, what if I want to close things in a different order? For example, GUI windows get closed when the close box is clicked, not in the reverse order of opening.

For this you want to get a closer function:

lifecycleGetCloser :: MonadIO m => LifecycleT m a -> LifecycleT m (a, IO ())

For example,

newGUIWindow :: Lifecycle Window

makeMyWindow :: Lifecycle Window
makeMyWindow = do
    (window,closer) <- lifecycleGetCloser newGUIWindow
    lift $ onCloseBoxClicked window closer
    return window

Here, closer is an idempotent operation that will call the closer of newGUIWindow, that is, to close the window. Subsequent calls do nothing. It also gets called at the end of the lifecycle, to ensure that the window is eventually closed if it hasn’t been already.

Also, you may come across certain functions that make use of the “with” pattern, to manage opening and closing. Here are a couple from the base library:

withFile :: FilePath -> IOMode -> (Handle -> IO r) -> IO r

withBinaryFile :: FilePath -> IOMode -> (Handle -> IO r) -> IO r

Monadology is capable of “unpicking” this pattern coroutine-style, and converting it to a Lifecycle:

lifecycleWith ::  (forall r. (a -> IO r) -> IO r) -> Lifecycle a

fileHandle :: FilePath -> IOMode -> Lifecycle Handle
fileHandle path mode = lifecycleWith $ withFile path mode


Speaking of coroutines, Monadology has a class for that.

class Monad m => MonadCoroutine m where
    coroutineSuspend :: ((p -> m q) -> m r) -> CoroutineT p q m r

This is experimental, as the only useful instances I’ve come across are monads based on IO, which supports coroutines by using threads.

The CoroutineT transformer is a special case of the StepT transformer, which is for step-by-step execution.

Transitive Constraints

For many transformers, certain constraints on a monad are transitive to the transformed monad. For example:

Monad m => Monad (ReaderT r m)
(MonadPlus m, Monoid w) => MonadPlus (WriterT w m)
MonadIO m => MonadIO (ExceptT m)

Monadology has a class for this:

class TransConstraint c t where
    hasTransConstraint :: forall m. c m => Dict (c (t m))

instance TransConstraint Monad (ReaderT r)
instance Monoid w => TransConstraint MonadPlus (WriterT w)
instance TransConstraint MonadIO ExceptT

Why not just use GHC’s QuantifiedConstraints extension? Because GHC has issues satisfying quantified constraints. So there’s an explicit class instead.

Tunnelling, Hoisting and Commuting

Tunnelling allows you to manipulate monads underneath a transformer. Each tunnellable transformer is associated with a tunnel monad, that represents the “effect” of the transformer.

type p --> q = forall a. p a -> q a

class (MonadTrans t, TransConstraint Monad t) => MonadTransHoist t where
    hoist :: forall m1 m2. (Monad m1, Monad m2) =>
        (m1 --> m2) -> t m1 --> t m2

class (MonadTransHoist t, MonadInner (Tunnel t)) => MonadTransTunnel t where
    type Tunnel t :: Type -> Type
    tunnel :: forall m r. Monad m =>
        ((forall m1 a. Monad m1 => t m1 a -> m1 (Tunnel t a)) -> m (Tunnel t r)) -> t m r

Tunnel monads are, curiously enough, always instances of the aforementioned MonadInner. For example:

type Tunnel (ReaderT s) = Identity
type Tunnel (WriterT w) = (,) w
type Tunnel (StateT s) = (,) (Endo s)
type Tunnel MaybeT = Maybe
type Tunnel (ExceptT e) = Either e
type Tunnel (ComposeInner inner) = inner
type Tunnel (ComposeOuter outer) = Identity

(This is essentially a correction and generalisation of MonadTransControl.)

It’s straightforward to derive hoisting from tunnelling, which is why MonadTransHoist is a superclass of MonadTransTunnel. And furthermore, you can commute two transformers in a stack, if you can commute their tunnel monads (which you always can).

commuteTWith :: (MonadTransTunnel ta, MonadTransTunnel tb, Monad m) =>
    (forall r. Tunnel tb (Tunnel ta r) -> Tunnel ta (Tunnel tb r)) ->
    ta (tb m) --> tb (ta m)

commuteInner :: (MonadInner m, Applicative f) => m (f a) -> f (m a)

commuteT :: (MonadTransTunnel ta, MonadTransTunnel tb, Monad m) =>
    ta (tb m) --> tb (ta m)
commuteT = commuteTWith commuteInner


Monadology has two classes for unlifting transformers.

type Unlift c t = forall m. c m => t m --> m
newtype WUnlift c t = MkWUnlift (Unlift c t)

class (...) => MonadTransUnlift t where
    -- | lift with an unlifting function that accounts for the transformer's effects (using MVars where necessary)
    liftWithUnlift :: forall m r. MonadIO m =>
        (Unlift MonadTunnelIOInner t -> m r) -> t m r
   -- | return an unlifting function that discards the transformer's effects (such as state change or output)
    getDiscardingUnlift :: forall m. Monad m =>
        t m (WUnlift MonadTunnelIOInner t)

-- | A transformer that has no effects (such as state change or output)
class MonadTransUnlift t => MonadTransAskUnlift t where
    askUnlift :: forall m. Monad m => t m (WUnlift Monad t)

Only ReaderT (and IdentityT) and the like can be instances of the more restrictive MonadTransAskUnlift.

However, MonadTransUnlift also has instances for StateT and WriterT. These allow correct unlifting without discarding effects (though another function is provided if you want discarding). How is this possible? Magic! MVars! Unlifting StateT simply holds the state in an MVar. Unlifting WriterT uses an MVar to collect effects at the end of each unlift.

Using MVars also makes everything thread-safe. Here’s an example:

longComputation1 :: IO ()
longComputation2 :: IO ()

ex :: StateT Int IO ()
ex = liftWithUnlift $ \unlift -> do
    a <- async $ do
        unlift $ modify succ
    unlift $ modify succ
    wait a

Here, longComputation1 and longComputation2 can run in parallel, in different threads. But unlift forces synchronisation, meaning that the modify statements never overlap. Instead, state is passed from one to the other. So ex is guaranteed to add two to its state.

As mentioned earlier, the tunnel monads of transformers in MonadTransTunnel are all instances of MonadInner. But if the transformer is an instance of MonadTransUnlift, its tunnel monad will be an instance of the stricter class MonadExtract. And if the transformer is an instance of MonadTransAskUnlift, then its tunnel monad will be an instance of MonadIdentity, monads equivalent to the identity monad.

The Same, but Monads Relative to IO

Often a monad can be understood as some transformer over IO. In such a case, we might want to know the properties of that transfomer.

Monadology provides classes for such monads, that mirror classes for transformers:


Composing and Stacking Transformers

The ComposeT transformer allows you to compose monad transformers (unlike composing monads, there is no restriction on this). Generally speaking, if t1 and t2 both have some property, then ComposeT t1 t2 will have it too.

The StackT transformer allows you to deal with whole stacks of transformers, parameterized by a list of their types:

type TransKind = (Type -> Type) -> (Type -> Type)

type StackT :: [TransKind] -> TransKind
newtype StackT tt m a = MkStackT (ApplyStack tt m a)

type ApplyStack :: forall k. [k -> k] -> k -> k
type family ApplyStack f a where
    ApplyStack '[] a = a
    ApplyStack (t ': tt) a = t (ApplyStack tt a)

Monad Data

The concepts of “reader”, “writer”, and “state” monads each imply a kind of data: readers have parameters, writers have products, and states have references. And pretty much any monad has exceptions. So, why not make that data explicit, so we can manipulate it directly?

data Param m a = MkParam
    { paramAsk :: m a
    , paramWith :: a -> m --> m

readerParam :: Monad m => Param (ReaderT r m) r

data Ref m a = MkRef
    { refGet :: m a
    , refPut :: a -> m ()

stateRef :: Monad m => Ref (StateT s m) s

data Prod m a = MkProd
    { prodTell :: a -> m ()
    , prodListen :: forall r. m r -> m (r, a)

writerProd :: Monad m => Prod (WriterT w m) w

data Exn m e = MkExn
    { exnThrow :: forall a. e -> m a
    , exnCatch :: forall a. m a -> (e -> m a) -> m a

allExn :: forall m. MonadException m => Exn m (Exc m)
someExn :: forall e m. MonadCatch e m => Exn m e

Parameters and references can be mapped by lenses. Not so much products, though there is one thing we can do with them.

mapParam :: Functor m => Lens' a b -> Param m a -> Param m b

mapRef :: Monad m => Lens' a b -> Ref m a -> Ref m b

foldProd :: (Applicative f, Foldable f, Applicative m) => Prod m a -> Prod m (f a)

Of course, other monads have their own references:

ioRef :: IORef a -> Ref IO a

stRef :: STRef s a -> Ref (ST s) a 

Odd Stuff

ReaderStateT and TransformT are odd things that I make use of elsewhere, but don’t really understand. Both of them convert Params into Refs.

newtype WRaised f m = MkWRaised (forall a. f a -> m a)
type ReaderStateT f m = StateT (WRaised f m) m
readerStateParamRef :: Monad m => Param f a -> Ref (ReaderStateT f m) a

newtype TransformT m a = MkTransformT (forall r. (a -> m r) -> m r)
transformParamRef :: Monad m => Param m a -> Ref (TransformT m) a

Not Included

  • ListT. This does not transform every monad to a monad, so is not a monad transformer.
  • Any notion of a “base” monad. While every transformer stack must logically have some base monad, the concept is non-parametric as transformed monads cannot be base monads.
  • Lifted “batteries” functions. Just use lift.
  • An effect system.

And also…

I have substantially expanded, cleaned up and reorganised witness, my package for type witnesses, which Monadology makes use of. I have also published type-rig, which provides the Summable and Productable classes used for monad data.

— Ashley Yakeley

The Fate of Bitcoin

This is largely in response to Nassim Taleb’s paper Bitcoin, Currencies, and Fragility. I don’t know much about finance or economics, this is merely my best guess and I may have made some elementary errors, or missed some possibilities.

The relentless volatility of Bitcoin and other cryptocurrencies continues to astonish me. As far as I can figure out, the volatility of an asset cannot be synthesized: you cannot construct a pure asset (that is, with no liability) with volatility higher than the assets used to construct it. So the volatility must be “natural” in some sense. One can try to chalk it up to “investor irrationality”, but that’s unhelpful, because guesses about the long future are inevitably irrational. Bitcoin is all about the long future, and the future is not only open, it’s unguessably open, and induction always fails eventually (at least, it always has in the past…).

In my experience, owning cryptocurrency is like owning an insane, incomprehensible fragment of the future, the true contemplation of which will fill your head with terrifying visions, visions that gleefully mock your sense of order and justice, of which “Lambo” is only the most well-known. As the price increases, your sanity is traded for wealth. The price should not be this high. As it falls to what seems like a more defensible level, you may get it back again, for the moment. Is Bitcoin in a bubble? If so, why has each “bubble” been succeeded by a bigger bubble?

But maybe the present behaviour is temporary. What is the fate of the value of Bitcoin, the scenaro that best describes its behaviour in the long term, over a thousand years perhaps, or ten thousand years, or as time stretches to infinity?

True Zero

As Taleb points out, gold has value as jewellery. For example, if it’s worth $10 to you to be able to show off your gold necklace at a party, then whatever the necklace is worth at the end of the night, it’s worth $10 more to you at the beginning. Thus, while on the face of it gold seems like useless soft metal, it has a “bling” value preventing its price falling to zero.

Not so Bitcoin. Owning Bitcoin provides no revenue or value over time, so comparison to gold is inapt. Furthermore, Bitcoin has absorbing barriers: the accumulating probability that Bitcoin will fail catastrophically, becoming worthless. Since over infinite time this will eventually happen, and Bitcoin yields no other value, then logically this means Bitcoin is worth zero now.

But as far as I can tell, Taleb is wrong.

Firstly, perhaps surprisingly, Bitcoin does have a very tiny amount of “bling” value. This isn’t obvious, because Taleb fails to distinguish between “close to zero” and “exactly zero”. Let us suppose that the market capitalisation of Bitcoin falls to $1000. I think this could be reasonably, if loosely, described as “Bitcoin falling to zero”. But there are plenty of people who would buy all the Bitcoin at this price, just so they can have the bling value of “being the person who owns all the Bitcoin”.

Secondly, Bitcoin does not actually seem to have absorbing barriers. For something to be an absorbing barrier, it must have all of these properties:

  1. There is a non-zero probability of reaching the barrier, that accumulates to certainty over time.
  2. The asset is unusable at the barrier.
  3. There is exactly zero probability of escaping the barrier.

Taleb gives three examples of absorbing barriers, but they all miss one of the properties:

  • Miners become extinct (but there is a non-zero probability that mining starts again).
  • The technology becomes obsolete (but Bitcoin would still be usable).
  • Future generations lose interest in it (same).

To this we might add:

  • Most major governments ban it (but there is a non-zero probability that they un-ban it again).
  • There is a fatal flaw in the Bitcoin protocol that, for example, makes it possible to obtain private keys, or to trivially mint infinite coins. This has properties 2 and 3, but the probability of this is fixed and does not accumulate over time to certainty.
  • All copies of the blockchain are lost. The probability of this per unit time is so low, and thus the time horizon is so long, perhaps millions of years, that It’s not clear to me that predictions about human destiny are good enough to rule out the probability that this will never happen.

Stable Low Value

OK, so Bitcoin won’t eventually reach true zero. What happens if almost everyone loses interest in it and it only has value to hobbyists and collectors? Surely it would stabilise at some low market cap, corresponding to its “hobbyist value”? Surely no more than this is the “true” or “intrinsic” value of Bitcoin?

But without its characteristic volatility, and since Bitcoin has limited supply and is easy to trade and store, it becomes valuable as a hedge against inflation, which would cause its price to rise. So stable low value is not a fate either.

Why aren’t Beanie Babies® a hedge against inflation? Because Beanie Babies are expensive to store and trade in sufficient quantities, and their supply is not limited, among other reasons. By contrast, the supply of Bitcoin is limited, holding large amounts of Bitcoin long-term is effectively free, and trading fees are constant regardless of the size of the trade.

Bounded Volatility

What if Bitcoin is volatile enough to discourage hedging against inflation, but otherwise remains generally of low value? In this case, in the long term, people would eventually discover the bounds of its value, and buy when it was relatively low, and sell when it was relatively high, causing the price to stabilise.

“Stable” High Value

If Bitcoin becomes established as some kind of hedge against inflation, everyone who wants such a thing will buy it. Perhaps major governments will obtain reserves, and the price may more-or-less stabilise relative to the world economy, at least within one order of magnitude. This seems to be the fate that a lot of Bitcoin advocates, particularly “Bitcoin maximalists”, predict. This seems possible to me, but it’s not the only possibility.

Unbounded Volatility

It’s also possible that Bitcoin remains deeply volatile, with swings of value that not only cannot be predicted, but even defy attempts at any probabilistic modelling. Bitcoin would then be useless as any kind of hedge, and might continue to be fairly useless as a currency. Indeed, Bitcoin could remain almost completely useless, and yet still have arbitrarily high swings of value.

This fate for Bitcoin arguably reflects the fate of the world at large, which is also deeply unknowable. Why is there something instead of nothing? Why is Bitcoin worth something instead of nothing?

— Ashley Yakeley

Pinafore 0.3

I had more planned for Pinafore 0.3, but two things happened: I found a bug in the type-checker that absolutely had to be fixed, and took me awhile to figure out. And last week I started full-time work, so I wanted to get a version out now, since I won’t have much time to work on Pinafore for about the next year.


List References

The most important new feature is a new kind of reference type. Pinafore 0.2 had WholeRef, SetRef, and FiniteSetRef. 0.3 adds ListRef, which UI.listTable uses directly. ListRef values can track items more accurately than using a WholeRef of a list. For example, you can obtain a WholeRef for a given item in the ListRef, which will keep track of the position of the item in the ListRef as other items are inserted and removed.

In future releases, ListRef will be the model for other UI elements. For example, a “grid” UI element is naturally a Cartesian product of a list of rows and a list of columns. Operations such as inserting a new row or column thus become straightforward insertions and deletions to two ListRefs.

Subsumption Expressions

A new kind of expression, expr: type, subsumes the expression to the type. So for example, 3 has type Integer, while 3: Number has type Number. Pretty straightforward.

Module-Qualified Names

You can now write, e.g., UI.listTable to refer to the listTable symbol in the UI module. Note that this isn’t just using the namespace, it also “does” the import. In other words, it’s essentially the same as let import UI in listTable.

Why, you might ask? Well, Pinafore doesn’t follow Haskell’s example of putting all the import statements at top level, because Pinafore has no “top level”. Since this means one can always bury import statements deep inside expressions, it’s simpler to just do the import with the module qualification.

Better Type Simplification

Formerly, the type simplifier eliminated one-sided type-variables. A type variable is one-sided if it only appears in positive position, or else only appears in negative position.

This is now more general: the type simplifier now eliminates all fully-constrained type variables. Here’s an example:

a & Integer -> a | Number

gets simplified to

Integer -> Number

(because Integer <: Number).

The idea is that the constraint a <: T is equivalent to the equation a = a & T. This is a fundamental principle of Algebraic Subtyping that comes from lattice theory, and making the substitution is how type-checking deals with constraints. However, one can also reverse this substitution, to extract constraints. In this case you’d extract the constraints a <: Integer and a :> Number. Note that because Integer <: Number, a is “fully constrained”: Pinafore can specialise it either to Integer or to Number, or indeed any type “in between” (such as Rational), without losing any generality. So that’s exactly what it does.

As it happens, one-sided type-variables are a special case of this. For example, the type a | Text yields the constraints a <: None and a :> Text. And obviously None <: Text, so a is fully constrained, and the type can be simplified to Text.

Other Library Changes

  • UI stuff is now in a separate UI module (which is actually built in a separate Haskell package).
  • The UI type has been renamed Element, with the expectation that it will usually be referred to as UI.Element.
  • There are some new UI elements for doing CSS styling.
  • There are some new functions and some name changes in the Std library. It’s still early for the Pinafore project as a whole, so expect a certain amount of incompatibility between versions.

Next Milestone

My plans for 0.4 are basically everything I dropped from 0.3:

  • Graphics, such as how to display images and diagrams in UI elements (earlier awkward Haskell library issues have now been sorted)
  • Files, which are the easiest way of handling large blobs of data such as images, video, etc.
  • New reference type for text


Version 0.3 of Pinafore is available from Github as a Debian package. There’s also a syntax-highlighting extension for Visual Studio Code.

The Pinafore website has all the documentation.

— Ashley Yakeley

Pinafore 0.2

Pinafore was originally motivated by my desire to organise and structure (i.e., type) various bits of information relating to my life. This is why, for instance, the example Pinafore code in the documentation relates to contacts and events. After I released Pinafore 0.1, I spent some time actually trying to do this, and ran into a number of problems. 0.2 was originally going to be a bigger, longer, milestone, but instead I made fixing these problems a priority. As a result, Pinafore 0.2 is much more pleasant and usable than Pinafore 0.1.


These are the main new features.

Module System

There are now two kinds of Pinafore file, scripts and modules. Scripts are typically UNIX executable files with no filename extension and #!/usr/bin/pinafore at the top. A script contains a single expression.

Modules are located in one of a number of known places with filenames that end with .pinafore. The contents of a module file generally looks like let <bindings> in export <names>.

Recalling that there is no “top level” in Pinafore, both scripts and modules can import modules into any scope with an import statement inside a let expression.

Anyway, it all works the way you’d expect. Here’s an example.

Dynamic Supertypes

In algebraic subtyping, every subtype relation P <: Q implies an “upcasting” conversion function of type P -> Q. The overall system of conversion functions needs to be consistent, of course, but apart from that, there’s no restriction on what these functions do. In particular, they do not need to be injective, and retraction functions (“downcasting”, of type Q -> Maybe P) do not need to exist.

However, for some subtype relations, a retraction function is available. Pinafore now provides a general mechanism for making use of them.

Every (ambipolar) type T has a greatest dynamic supertype D(T), with these properties:

  • T <: D(T) (“supertype”)
  • D(D(T)) = D(T) (“greatest”)
  • There exists a special form check @T: D(T) -> Maybe T (“dynamic”)

There’s now a new kind of pattern that can be used to match according to the retraction function: (pat: T) has type D(T) and matches pat as type T.

There are actually two special forms, check and coerce. They can be defined in terms of the pattern, more-or-less like this:

check @T: D(T) -> Maybe T;
check d = case d of
    (t:T) -> Just t;
    _ -> Nothing;

coerce @T: D(T) -> T;
coerce d = case d of
    (t:T) -> t;
    _ -> error "coercion from <D(T)> to <T> failed";

So what are the greatest dynamic supertypes of various types? In most cases, D(T) = T, not very interesting. We do have D(Integer) = D (Rational) = Number. But the real motivator is dynamic entity types, see below. For these, D(T) = DynamicEntity.

For the future, it actually wouldn’t be hard to introduce predicate types. Given a type T and a function of type T -> Boolean, one would be able to define a subtype P of T consisting of those values that satisfied the function. This would be an obvious choice for the dynamic supertype mechanism (with D(P) = D(T), of course). However, for the time being I don’t have a strong motivation for predicate types, so Pinafore 0.2 does not provide them.

Dynamic Entity Types

So generally, Pinafore erases types. There is no information about entity types in storage, nor about which values have which types. This is helpful for reasoning about storage, and for refactoring your schema. However, sometimes you do want store some kind of type information. For example, you might want to implement something like this:

Human <: Animal
Dog <: Animal
dateOfBirth: Animal ~> Date
animalUI: Animal -> UI

In this case, you do want to store type information for each Animal: you want to know whether it’s a Human or a Dog.

This is how you’d do this with dynamic entity types:

dynamictype Human = !"mytype.Human";
dynamictype Dog = !"mytype.Dog";
dynamictype Animal = Human | Dog;
dateOfBirth = property @Animal @Date !"Animal.dateOfBirth";
animalUI animal = case animal of
    (human: Human) -> humanUI human;
    (dog: Dog) -> dogUI dog;

All dynamic entity types are subtypes of DynamicEntity, which internally encodes a pair of anchors, one for type and one for “value”. A dynamic entity type simply represents a set of type anchors: in the example, Human and Dog are each one anchor (as given), and Animal is both of them.

A dynamic entity type is concrete if it represents only one type anchor. Given a concrete dynamic entity type, one can generate new values dynamically or statically with the newDynamicEntity and dynamicEntity special forms.

Comparing the type system of Pinafore to that of, say, Java, there seems to be a certain sense in which dynamic entity types resemble Java’s classes, and open entity types resemble interfaces. I’m not sure how far this intuition goes, though.

Generalised Open Entity Subtypes

Previously, subtype relations could be declared with subtype P <: Q, with P and Q both open entity types. This is now generalised so that P can be other (ambipolar) entity types.

But this gives rise to a problem: we cannot have (or at least make use of) more than one subtype relation for a given “ground type constructor” if their parameters are incompatible. Consider this:

opentype P;
opentype Q;
subtype Maybe Integer <: P;
subtype Maybe Boolean <: Q;
f: Entity -> ()
f _ = ()
g x = f (Just x)

Here the ground type constructor is Maybe, so what is the type of g? There are two different subtype “paths” to get from Maybe a to Entity, so we want something like g: (Integer | Boolean) -> (). But that is not allowed, as Integer | Boolean is not a negative type.

The solution for the time being is simply to disallow subtype relations on parametered types (such as Maybe Integer). This isn’t perfect, but overall an improvement.

Command-Line Arguments & UNIX Environment

If you make a script, you may wish to pass arguments to it when invoking from the command line. If so, scriptArguments is what you want. There’s also scriptName, environment, and getEnv.

Next Milestone

My 0.3 plans are still quite fluid, but the main points are:

  • Graphics, such as how to display images and diagrams in UI elements (involves some awkward Haskell library issues)
  • Files, which are the easiest way of handling large blobs of data such as images, video, etc.
  • New reference types for text and for lists.


Version 0.2 of Pinafore is available from Github as a Debian package. There’s also a syntax-highlighting extension for Visual Studio Code.

The Pinafore website has all the documentation.

— Ashley Yakeley

Whole Haskell Use of GADTs

For some reason GADTs are a particular sticking point for many advocates of Simple Haskell. But sometimes GADTs are the natural form of expressing a concept.

By way of example, here’s a snippet of API I’ve borrowed from JuicyPixels:

data Image a = Image {
    imageWidth :: !Int,
    imageHeight :: !Int,
    imageData :: Vector (PixelBaseComponent a)

data DynamicImage =
    ImageY8 (Image Pixel8) |
    ImageY16 (Image Pixel16) |
    ImageY32 (Image Pixel32) |
    ImageYF (Image PixelF) |
    ImageYA8 (Image PixelYA8) |
    ImageYA16 (Image PixelYA16) |
    ImageRGB8 (Image PixelRGB8) |
    ImageRGB16 (Image PixelRGB16) |
    ImageRGBF (Image PixelRGBF) |
    ImageRGBA8 (Image PixelRGBA8) |
    ImageRGBA16 (Image PixelRGBA16) |
    ImageYCbCr8 (Image PixelYCbCr8) |
    ImageCMYK8 (Image PixelCMYK8) |
    ImageCMYK16 (Image PixelCMYK16)

promoteImage :: ColorConvertible a b => Image a -> Image b 

JuicyPixels was originally released in 2012, and there’s always a good presumption against changing an API, so we can’t blame the developers for not making use of GADTs. But if we were starting a project like this from scratch today, how would we write such code, using the whole language?

The way DynamicImage has been defined makes it a bit difficult to work with. For example, if you want to get the width or height of a dynamic image, you’ll have to write cases for each constructor. And the convertibility from one pixel type to another is encoded at the type level (with ColorConvertible), but there’s no clean way to do it at the value level, if we wanted to provide that functionality dynamically.

We can make this code more expressive, and more elegant, by using a GADT that represents the pixel type of a dynamic image:

data PixelType a where
    PixelType8 :: PixelType Pixel8
    PixelType16 :: PixelType Pixel16
    PixelType32 :: PixelType Pixel32
    PixelTypeF :: PixelType PixelF
    PixelTypeYA8 :: PixelType PixelYA8
    PixelTypeYA16 :: PixelType PixelYA16
    PixelTypeRGB8 :: PixelType PixelRGB8
    PixelTypeRGB16 :: PixelType PixelRGB16
    PixelTypeRGBF :: PixelType PixelRGBF
    PixelTypeRGBA8 :: PixelType PixelRGBA8
    PixelTypeRGBA16 :: PixelType PixelRGBA16
    PixelTypeYCbCr8 :: PixelType PixelYCbCr8
    PixelTypeCMYK8 :: PixelType PixelCMYK8
    PixelTypeCMYK16 :: PixelType PixelCMYK16

data DynamicImage = forall a. DynamicImage (PixelType a) (Image a)

Any code to get properties of the contained Image is now much simpler:

dynamicImageSize :: DynamicImage -> (Int, Int)
dynamicImageSize (DynamicImage _ img) = (imageWidth img, imageHeight img)

We can also express convertibility dynamically, if we want to add that:

canPromotePixel :: PixelType a -> PixelType b -> Maybe (Dict (ColorConvertible a b))
-- provided as appropriate

promoteDynamicImage :: DynamicImage -> PixelType b -> Maybe (Image b)
promoteDynamicImage (DynamicImage pta img) ptb = do
    Dict <- canPromotePixel pta ptb
    return $ promoteImage img

You can see how with Haskell’s fancy shiny GADT feature, the code is now more expressive, and easier to understand for anyone who knows the language.

GADTs are actually not a particularly difficult concept to learn, for someone who already understands rank-n types and existential quantification. And in many situations, they provide a huge benefit in expressibility.

— Ashley Yakeley

Whole Haskell is Best Haskell

The promise of Haskell over other languages is that it allows you to more cleanly and intuitively represent the application domain. This leads to more intelligible and maintainable code. But to take full advantage of what Haskell has to offer, you have to embrace the whole language. This means making use of any appropriate language feature, just as you would with any other language.

If you’re using Haskell in industry, it’s my belief that restricting your team to some “simple” subset of the language for complex problems will make your code more complex and more difficult for competent developers to understand. It will be more likely to have repetitive boilerplate, and more likely to require explicit error calls to handle “impossible” cases. It will be less comprehensible, and so less maintainable.

Here are some claims from “The Simple Haskell Initiative”, which I believe are flawed:

Fancy Haskell is costly to teams because it usually takes more time to understand and limits the pool of people who can effectively contribute.

By taking this attitude, you are not only committing your code to mediocrity, but your team too. Whole, elegant, Haskell, firing on all language cylinders, takes less time for competent developers to understand. And you can raise less-experienced developers to the necessary competence specific to your project.

Things that have been around longer will be more well-tested and understood by a larger group of people. Prefer tried and true techniques over the latest shiny library or language feature. The more foundational something is in your tech stack, the more conservative you should be about adopting new versions or approaches to that thing.

These are two separate things.

For libraries, maturity is certainly a valid concern. But it is unrelated to the structure of the library’s API.

For language features, the GHC team has a high bar for releasing new extensions, often involving formal proof. The implementation of features considered “fancy”, such as GADTs, type families, and polykinds, is understood to be as sound as that of any other part of the language. They are unlikely to be a particular source of compiler defects.

If you adopt a new thing, how much of its complexity will spread throughout the rest of your codebase? You should be more hesitant to adopt something if its complexity is going to spread through a larger portion of your codebase.

Code written in whole Haskell is less complex than “simple” Haskell for the same complex application. That’s the whole point of it.

There is no one definition of what language features count as “Simple Haskell”. Michael Snoyman seeks to define a “boring” subset of Haskell, but his recommended set of “boring” language extensions is quite broad, including most extensions relating to classes and type families, and even PolyKinds. Boring Haskell, in practice, seems to be close to Whole Haskell.

Sam Halliday, by contrast, seeks a vastly more restricted language, rejecting GADTs, type families, multi-parameter classes, and apparently even rank-n types and existential quantification. Such restrictions lead to unnecessarily complicated code, in my view. Here’s a simple example of how GADTs and existential data quantification can improve code generality and intelligibility.

What you should be doing

  • Embrace the whole language.
  • Set a high quality bar for code within your team.
  • Mentor less-experienced developers.

Embrace the whole language. Pretty much every “fancy” feature of the whole GHC Haskell language has a productive purpose. That’s the point of all the academic research. As a competent Haskell developer, you should know when that purpose applies and how to make best use of each feature.

Note that language features are not exactly the same as language extensions. For example, the language gives you the option to either allow or prohibit “incoherent” use of class instances. In most cases, the best use of this language feature is to prohibit, as it can ensure a discipline that leads to more intelligibility and predictability.

Some language features are of relatively specialised use. Polykinded types are very useful for type-oriented applications (such as implementing a typed language interpreter). And, of course, some simpler or more straightforward projects might make use of relatively few language features.

Set a high quality bar for code within your team. Set code expectations as early as you can. Discuss ideas early, and make suggestions for design approaches. Haskell makes refactoring easier than many languages, take advantage of that as appropriate.

Mentor less-experienced developers. When you hire a Java, Python, or C++ developer, you can expect them to be fully competent in each of those languages. You can typically give them development ownership of progressively larger project features, and leave them to get on with it until code review. Given the current state of the Haskell job market, this may not be the case for Haskell developers.

If you hire junior developers who are not yet familiar with the whole language, you will need to mentor them. Take extra time with them to explain how features of the Haskell language work, how they are best used in general, and how you use them in your codebase.

If that sounds like an extra burden, bear in mind that not many developers make the effort to learn Haskell in the first place, and those that do are likely to have more aptitude to learn more about it. Invest in the people you hire. If you’re doing anything worthwhile, you’re in this for the long term.

… Why did you choose Haskell, anyway?

— Ashley Yakeley

Pinafore 0.1

We all generate a lot of information in our lives and in our work:

contacts, events, emails, tasks, photos, plans, budgets, financial records, media collections…

Computers are supposed to help us organise it all. So how’s that working out?

For the most part, we use application programs. Each application program works with information of a particular kind, with a polished and specific user interface. However, it fixes a particular schema for that information, and it is typically difficult to combine information from multiple applications.

Alternatively, we can work with information more loosely and flexibly in a spreadsheet. However, this provides a more limited user interface experience, and the very looseness makes it difficult to reason abstractly about the type and schema of the information.

Pinafore is an attempt to reimagine how computers represent and store information, and how users interact with it. It allows users to create their own schemas for information, and create their own interfaces to it.

This 0.1 release includes some of the major pieces: a type system for information, a language, a storage system, some composable user interface elements. Enough to get the gist of the project, and perhaps suggest some future possibilities. But much more needs to be done.


Pinafore is an interpreted language. Information is stored as predicate/subject/object triples in a database in your home directory, and the user interfaces are created with GTK+.

Pinafore generally resembles Haskell. It has a type system derived from Hindley-Milner, and features pattern-matching, lazy evaluation, and separation of pure functions from executable actions. There are some differences however:

  • There is no “top level”. A Pinafore file consists of a single expression. Type declarations, like bindings, are declared within let expressions.
  • Layout is not syntactically significant. Instead, lines are terminated by semicolons, and do and case expressions are terminated with the end keyword.
  • The colon is used for type signatures, while the double colon is used for list construction, the other way around from Haskell.
  • Line comments start with #, while (nestable) block comments are marked with {# and #}.
  • Only one equation is allowed for a function definition. Argument patterns can be matched with case expressions.

Of course, there are many features of Haskell that Pinafore lacks, and vice versa.

Type System

Pinafore is a strongly-typed language. The type system implements Stephen Dolan’s Algebraic Subtyping, which is an extension of Hindley-Milner to allow subtyping. This type system is decidable: if an expression has a type, Pinafore can always infer a principal type for it. Like Haskell, however, you can also add type signatures to definitions.

A subtype relation P <: Q is a relationship between two types, “P is a subtype of Q”, which simply means “every P is a Q”, or “allow a P where a Q is expected”. Of course, this implies an inclusion function P → Q that actually converts the P to the Q. These functions do not have to be injective, nor does there need to be any kind of reverse function Q → Maybe P, though these do exist in Pinafore in some cases.

Two types are equivalent if each is a subtype of the other.


The type system distinguishes positive and negative types. This is necessary, because certain type operations are only permitted with certain polarities:

  • A positive type is a type that can appear in a positive position. Think of this as the type of a value you’ve defined in your program. The type signature of a value is such a positive position.
  • A negative type is a type that can appear in a negative position. Think of this as the type of acceptance of values, such as the argument of a function type (that is itself positive).
  • An ambipolar type is a type that is both positive and negative. This includes simple types such as Text, Integer, and so forth.
  • If P and Q are positive types, then P | Q is a positive type. You can read this as “a P or a Q, not telling you which”. As you might expect:
    • P <: P | Q
    • Q <: P | Q
    • If P <: R and Q <: R, then P | Q <: R.
  • If P and Q are negative types, then P & Q is a negative type. Think of this as “must be both a P and a Q“. Likewise:
    • P & Q <: P
    • P & Q <: Q
    • If R <: P and R <: Q, then R <: P & Q.
  • None is a positive type, that is empty (and is a subtype of every type). None | P = P.
  • Any is a negative type, that accepts anything (and is a supertype of every type). Any & P = P.

Here are some examples of expressions with the principal types that Pinafore will infer. Note that Pinafore uses a single rather than double colon for type signatures:

\x -> 3: Any -> Integer
Nothing: Maybe None
\b -> if b then "hi" else 3: Boolean -> (Text | Integer)
\x -> x + textlength x: (Integer & Text) -> Integer

Type Constructors

Like Haskell, Pinafore has type constructors such as Maybe, [] (list), (,) (pair), -> (function) and so on. But all type parameters must be types (i.e. as if of Haskell’s kind *): in addition, each parameter must be either covariant or contravariant. This gives subtype relations. For example, suppose F is a type constructor with one argument:

  • If F is covariant in its argument, then,
    • F x has the same polarity as x.
    • P <: Q implies F P <: F Q.
  • If F is contravariant in its argument, then,
    • F x has the opposite polarity as x.
    • P <: Q implies F Q <: F P.

Of course, some types are, morally, neither contravariant nor covariant in their arguments. For these we use a pair of type parameters in a special syntax, one contravariant (marked with -) and one covariant (marked with +). For example, the WholeRef type constructor represents references with get and set operations:

WholeRef {-p,+q}
get: WholeRef {-p,+q} -> Action q
(:=): WholeRef {-p,+q} -> p -> Action ()

Pinafore has some abbreviations to make working with these a little easier, e.g. WholeRef T = WholeRef {-T,+T}, and WholeRef +T = WholeRef {-Any,+T}, etc.

Recursive Types

Pinafore has “equirecursive” types, written in the form rec v. T, where T is a type expression where v appears only covariantly. The key fact of recursive types is that they are equivalent to their unrolling. For example, these two types are equivalent:

rec a. (a, Maybe a)
(rec a. (a, Maybe a), Maybe (rec a. (a, Maybe a))

Recursive types are necessary for principality (that all typeable expressions have a principal type), though they’re not much used in practice.

Data Types

Pinafore allows you to create your own algebraic data types, like the data keyword in Haskell. Here’s an example:

datatype StopwatchState = StoppedState Duration | RunningState Time;

Typed Storage

Pinafore stores “knowledge” as relationships between entities of various types. The relationship types are called morphisms, which can be composed as the name suggests.

Types of entities are all subtypes of the Entity type. These include:

  • Literal types for small pieces of data, such as Integer, Number, Boolean etc.
  • Open entity types, that simply represent arbitrary points, declared with the opentype keyword. Values of open entity types can be declared statically (with an anchor) with the openEntity keyword, or generated at run-time with the newOpenEntity function.
  • Closed entity types, that have constructors, declared with the closedtype keyword. These are similar to data types, except that each constructor has an anchor, and the contained types must themselves be entity types.

Anchors are 256-bit values usually hashed from a literal string in your program. Pinafore erases types when storing information in its storage: it does not store the structure of types nor does it store which values have which types. Instead, it uses anchors to identify information in storage.

Open Entity Example

Here’s an example. Let us suppose to store two relationships concerning people:

  • The name of some person p is “James”.
  • The mother of p is some person q.

Firstly, we will need a type for people. This can be an open entity type: it has no information of its own besides identity: all information about people comes from morphisms.

opentype Person;

We also need properties for “name” and “mother”. We need to give these anchors, since this is what will identify them in storage, not the names of the language bindings we happen to use.

“Name” is a property from Person to Text, because the name of a person is text. We give it the anchor !"".

“Mother” is a property from Person to Person, because the mother of a person is a person. We give it the anchor !"myschema.mother".

Properties are morphisms, so the type of them is a morphism type, indicated by ~>. In fact, properties generate morphisms: you can compose morphisms together that are strings of properties. In this case, you can compose these two to get a morphism for “name of mother”.

name: Person ~> Text;
name = property @Person @Text !"";
mother: Person ~> Person;
mother = property @Person @Person !"myschema.mother";

Now we need entities p and q. These might be generated at run-time or obtained elsewhere, but here we’ll declare them statically. Again, it is the anchor that identifies them in storage, not the bindings p and q.

p: Person;
p = openEntity @Person !"someperson";
q: Person;
q = openEntity @Person !"otherperson";

Actually storing the relationships is an Action (similar to IO in Haskell). And like Haskell, Pinafore has do notation to make working with actions easier:

    name !$ {p} := "James";
    mother !$ {p} := q;

The !$ operator applies a morphism to reference to get another reference. Since p is an entity, not a reference, we must first convert it to a reference using “reference notation” {p}.

The := notation sets the value of a reference.

Here’s a typed breakdown of that first action:

p: Person
{p}: WholeRef +Person
name !$ {p}: WholeRef Text
name !$ {p} := "James": Action ()

Here’s what it looks like put altogether:

    opentype Person;
    name: Person ~> Text;
    name = property @Person @Text !"";
    mother: Person ~> Person;
    mother = property @Person @Person !"myschema.mother";
    p: Person;
    p = openEntity @Person !"someperson";
    q: Person;
    q = openEntity @Person !"otherperson";
in do
    name !$ {p} := "James";
    mother !$ {p} := q;

This is a complete Pinafore program. Running it will store those two relations in Pinafore’s persistent storage.

Of course, we can combine the morphisms in other ways:

# everyone who's mother's name is Kate
(name !. mother) !@ {"Kate"}

# the (name, mother) pair of p
(name !** mother) !$ {p}

It is important to note that Pinafore does not store entities per se, it stores relations between entities. One cannot, for example, retrieve all entities of type Person. Types of entities such as Person are erased for storage. However, one can retrieve all entities that have name “James”, or whose mother is some entity q.

Closed Entity Types

Closed entity types resemble data types, but they are all subtypes of Entity, and so can be stored. Every constructor of a closed entity type includes an anchor, to identify that constructor in storage. Here’s an example:

closedtype CelestialLocation
    = EquatorialLoc Number Number !"EquatorialLoc"
    | EclipticLoc Number Number !"EclipticLoc";


Any item of information retrieved from storage can be “unknown”, and Pinafore is robust with regards to what it happens to find in storage. This gives a certain amount of flexibility in modifying an existing “schema”, or system of entity types, without having to transform data in storage. For example, if you remove a property from your program, Pinafore will simply ignore that information in storage. If you add a property, Pinafore will initially find all values of that property to be “unknown”. Constructors can be added and removed from closed entity types. If Pinafore finds something it doesn’t recognise or cannot parse as it expects, it treats it as “unknown”.

Composable User Interface Models

In terms of the Model/View/Controller way of looking at user interface, the view and controller are represented by user interface elements (of type UI), while the model is represented by references (of type WholeRef, SetRef, and FiniteSetRef).

A reference represents the state of some thing. The user may wish to retrieve some part of that state, or make some change to it, or be notified when it changes. References are thus “live”: when connected to a UI element, the user can use the UI to change the reference, but also the reference can update the UI when its state changes.

  • Whole references (WholeRef) represent a single value (which might be “unknown”). There are operations for getting, setting, and deleting (making unknown).
  • Set references (SetRef) represent some arbitrary set or predicate of some type. There are operations for adding and removing members, and for checking membership of some value.
  • Finite set references (FiniteSetRef) are set references that have a finite number of members, that can be retrieved.

The various kinds of references can be composed in various ways, such as various set operations (union, intersection, Cartesian product and sum).

Pinafore’s “reference notation” makes working with whole references a little easier. For example, given two whole references to integers, we can create a new whole reference that is the sum of them:

p: WholeRef Integer;
q: WholeRef Integer;
pq: WholeRef +Integer;
pq = {%p + %q};

Whenever p or q updates, then pq also updates. Of course, pq is read-only: attempts to set it will fail.

User Interface Elements

User interface elements are things such as text areas, buttons, check boxes, and tables. They are constructed from the references they control, and can be composed by horizontal or vertical layout, and put in windows.


Version 0.1 of Pinafore is available from Github as a Debian package.

The Pinafore website has all the documentation.

— Ashley Yakeley

Forbidden Haskell Types

Sometimes, writing Haskell is like having an argument with the compiler. You give it your reasoning, and it checks it over for flaws. And if it thinks it finds one, it will tell you all about it. You then have to look over what it told you, and figure out exactly what its complaint is. Did I just express myself badly? Or am I actually wrong? Or, very occasionally, the compiler is just being petulant and you have to work around it. Of course, if you’re really upset about GHC not accepting your perfectly reasonably argued program, you can go tell its parents.

Haskell’s type system is pretty great, but one thing it doesn’t have, that some other type systems do have, is recursive types, by which I mean, types directly constructed from themselves. Recursive types are forbidden in Haskell.

Let’s say we want to create a recursive type T = Maybe T. The official introductory Haskell answer to this is that one cannot do this directly, but one can easily create a type T that contains Maybe T, sometimes called a “newtype wrapper”. Like this:

newtype T = MkT (Maybe T)

Simple enough. But can we do better than this? Can we create a type T that is actually equal to Maybe T? Can we create the forbidden recursive type?

Let’s start with won’t work:

type T = Maybe T
Example.hs:6:1: error:
    Cycle in type synonym declarations:
      Example.hs:6:1-16: type T = Maybe T

The type keyword creates an alias for a type, and aliases cannot be recursive as they must eventually refer to an actual type.

Oddly enough, GHC does accept this:

type family T where
    T = Maybe T

Type families are strange things. They’re not quite “real” types, but neither are they mere aliases for types, as they don’t have to be fully resolved. Indeed you can declare a type family in one module, use it all over as if it were a real type, and then define the actual instances of that family in another module.

Sadly, though, GHC won’t let us actually use the instance. It rejects the obvious proof that T and Maybe T are the same type:

type family T where
    T = Maybe T

proof :: T :~: Maybe T
proof = Refl
Example.hs:10:9: error:
    • Reduction stack overflow; size = 201
      When simplifying the following type: T

Why is this? It’s because GHC can’t just accept what it’s been told, and instead endlessly tries to “reduce” (that is, expand) T = Maybe T = Maybe (Maybe T) etc, giving up when it reaches a safety limit. You can think of it as a kind of intellectual black hole, or an “infohazard” for the compiler: if it ever starts examining the strange secrets of your recursive type, it will get lost in the infinite abyss, until a built-in emergency mechanism kicks it awake after it’s twisted in the vortex 200 times (you can change this number if you want, or even let it run forever).

It turns out we can construct a type T together with a proper (non-bottom) proof value of T :~: Maybe T, we just have to be discreet. The key is to make sure GHC never comes across the unfathomable constraint T ~ Maybe T, because it will immediately become ensnared in the contemplation of its endlessness.

Instead, we prepare a lemma that is both harmless and more general, and then carefully specialise it to get the terrifying conclusion we want, using the TypeApplications extension, thus avoiding the need for any kind of inference:

type family A p q where
    A () x = Maybe (A x ())

lemma :: forall x. (A () x) :~: Maybe (A x ())
lemma  = Refl

type T = A () ()

proof :: T :~: Maybe T
proof = lemma @()

Actually constructing and deconstructing values of T will require a similar circumspection. The trick is, we don’t let GHC infer types. We musn’t let GHC think, because thinking about the bad thing leads it to madness. Instead, we apply all types by hand. Here’s the full program, including all the necessary extensions:

{-# LANGUAGE RankNTypes, TypeApplications, TypeOperators,
    TypeFamilies, UndecidableInstances, AllowAmbiguousTypes #-}
module Main where
import Data.Type.Equality

type family A p q where
    A () x = Maybe (A x ())

lemma :: forall x. (A () x) :~: Maybe (A x ())
lemma  = Refl

type T = A () ()

proof :: T :~: Maybe T
proof = lemma @()

convert1 :: forall a b. a :~: b -> a -> b
convert1 Refl = id

convert2 :: forall a b. a :~: b -> b -> a
convert2 Refl = id

tconvert1 :: T -> Maybe T
tconvert1 = convert1 @T @(Maybe T) proof

tconvert2 :: Maybe T -> T
tconvert2 = convert2 @T @(Maybe T) proof

nothing :: T
nothing = tconvert2 $ Nothing @T

just :: T -> T
just x = tconvert2 $ Just @T x

count :: T -> Int
count t = case tconvert1 t of
    Nothing -> 0
    Just t' -> succ $ count t'

t3 :: T
t3 = just $ just $ just nothing

main :: IO ()
main = putStrLn $ show $ count t3

This program defines a truly recursive type T = Maybe T, constructs a value of it, and then deconstructs that value. (It does indeed correctly print “3”.) On the other hand, it’s much more complicated than just using a newtype like we’re supposed to…

— Ashley Yakeley