Compelling Higher Kinded Types and Type Classes in F#

There is no parameterized module in F#, however, as the result of the existance of some other power infrastructures, it becomes much easier for F# to express higher abstractions tersely.

The secret of the F#’s conciseness comes from the following 2 parts.

  • Active Patterns help to avoid write inj and prj manually.

    If you’ve never heard of inj and prj before, there is a prerequisite article for you to figure out what they are and some necessary knowledge about Higher Kined Types and Type Classes.

    If you cannot get along with that prerequisite article, you might need to try Haskell language to learn about Functor and Monad to get some related basic knowledge.

  • Static Resolved Type Parameters help to support resolution for implicit arguments.

    That’s the way to achieve something similar to modular implicits in OCaml.

FYI, this repo provides an actual implementation.

Active Pattern

  • Temporary enumerations

let (|IsWhatIWant|NotWhatIWant|) x =
    if x > 10 then IsWhatIWant
    else NotWhatIWant

let checkNumber():
    match 10 with
    | IsWhatIWant -> failwith "fatal!"
    | _ -> ()
  • Temporary recognizers

let (|Dec|) x =
    if x > 0 then Some <| x - 1
    else None

let test(): int option =
    match 1 with
    | Dec x -> x
  • Simulating datatypes

let (|Just|Nothing|) (a: 'a option) =
    match a with
    | Some a -> Just a
    | None a -> Nothing

let Just (a: 'a): 'a option = Some a
let Nothing<'a>: 'a option = None

// Now you can use `Just` and `Nothing` to replace
// constructors `Some` and `None` of type `'a option`.

let bind (m:'a option) (k: 'a -> 'b option):
    match m with
    | Nothing -> Nothing
    | Just x  -> k x

Above snippets are about some use cases of Active Patterns, which might not be exhaustive but sufficent for today’s topic:

Allow to inject extra logics when destructuring data,

subsequently, which is the key to implement LHKT in F#.

Statically Resolved Type Parameters

In the language reference of F#, statically resolved type parameters are not well documented, but you might want to check it firstly.

Here I’m to bring more cases to tell you how to take advantage of it.

The first case is about structural typing.

In structural typing, an element is considered to be compatible with another if,
for each feature within the second element’s type, a corresponding and identical feature exists in the first element’s type.[3]

So let’s imagine a case that we want to make some functions for all objects/types that have a common specific behavior.

Let’s think about the sound of animals, which could be taken as the behavior.

Firstly, let’s define the enumerate some sounds:

type Sound = Ooooh | Uhhh | Ahhh | QAQ | Woke | Wa | Ding | Miao

Then we define some intuitive active patterns to make natural-language-like notations:

let (|LessThan|_|) param value =
    if value > param then Some ()
    else None

// 1 is less than 20
assert (match 1 with LessThan 20 -> true | _ -> false)
type CatSpecies = ScottishFold | RussianBlue | MeoGirl
type Cat = {
    weight   : int
    species  : CatSpecies
} with
    static member sound: Cat -> Sound =
        function
        | {weight = LessThan 20; species = ScottishFold} -> Miao
        | {weight = LessThan 40; species = RussianBlue} -> Uhhh
        | {weight = LessThan 100; species = RussianBlue} -> Wa
        | {weight = LessThan 100} -> QAQ
        | {species = MeoGirl} -> Ding

type Color = Black | Brown | White
type Scale = {height : int, thickness : int}
type Dog = {
    color   : Color
    scale   : Scale
} with
    static member sound : Dog -> Sound =
        function
        | {height = LessThan 30; thickness = LessThan 10} -> Wa
        | {height = LessThan 30} -> Woke
        | {height = LessThan 50} -> Ooooh
        | _ -> failwith "a fat wolf detected"

Now I want to a function, which takes an animal as input, and output its sound.

For we didn’t use the discriminated union/ADT to represent animals, how can we make this polymorphic function?

Here you are:

let sound(a : ^a when ^a: (static member sound: ^a -> ^b)) =
    (^a : (static member sound: ^a -> ^b) a)

assert sound {weight = 100; species = MeoGirl} = Ding
assert sound {height = 40; thickness = 15} = Ooooh

Another case could be more formal and quite related to our topic.

Firstly, to represent higher kinded types, we use

type hkt<'K, 'T> = interface end

Here, the hkt<'K, 'T> is something that emulates the application of type constructor 'K on 'T, and the kind ascription of 'K is(at least) * -> * (e.g., a concrete type List<int> is the application of type constructor List on type int, so semantically it could be written as hkt<List, int>).

If any problem, refer to this prerequisite article.

Now, we’re to make some type constructors to substitute the type variable 'K, like List, Array, Maybe / Option, etc. These type constructors(also types) can have some common features, e.g., could be applied with a map function,

type 'a Maybe = 'a Option
let map_maybe : ('a -> 'b) -> 'a Maybe -> 'b Maybe = ..
let map_list  : ('a -> 'b) -> 'a List  -> 'b List  = ..
let map_array : ('a -> 'b) -> 'a Array -> 'b Array = ..

which could extracted as a useful concept, the functor. The functor is a Type Class, which is introduced to make abstractions on types(including the higher kinded types).

So, how can we emulate the functor, in another words, how can we assign the expected features to the corresponding type constructors (List, Array, etc.)?

I’d like to directly give a solution below that was figured out on my own:

[<AbstractClass>]
type functor<'F>() =
    abstract member fmap<'a, 'b> :
        ('a -> 'b) -> hkt<'F, 'a> -> hkt<'F, 'b>

Now you might have lots of questions about above codes,

  1. How can it help with implementing type classes?

  2. Why we have to use AbstractClass instead of ML Style records, or interfaces that’re more lightweighted?

  3. Why we should use such a shape functor<'F> instead of straightforward functor ?

I’ll answer above questions in the following sections, but in terms of this section’s topic, I should answer the first one here.

Firstly we need some infrastructures:

let inline wrap<'o, ^f, 'a when ^f : (static member wrap : 'o -> hkt<'f, 'a>)>
    (o: 'o) : hkt< ^f, 'a> =
    (^f : (static member wrap : 'o -> hkt<'f, 'a>) o)

let inline unwrap<'o, ^f, 'a when ^f : (static member unwrap : hkt<'f, 'a> -> 'o)>
    (f : hkt< ^f, 'a>) : 'o =
    (^f : (static member unwrap : hkt<'f, 'a> -> 'o) f)

[<GeneralizableValue>]
let getsig<'a> :'a = failwith "" // the implementation would be given subsequently.

Then we make an instance of functor for List.

type mkList<'L>() =

    inherit functor<mkList<'L>>()

        static member wrap<'a> (x : List<'a>): hkt<mkList<'L>, 'a> =
            {wrap = x} :> _
        static member unwrap<'a> (x : hkt<mkList<'L>, 'a>): List<'a> =
            (x :?> _).wrap

        override member __.fmap<'a, 'b> (f : 'a -> 'b) (m : hkt<mkList<'L>, 'a>) : hkt<mkList<'L>, 'b> =
            List.map f <| unwrap m |> unwrap

and listData<'L, 'a> =
    {wrap : List<'a>}
    interface hkt<mkList<'L>, 'a>

let fmap<'a, 'b, 'F when 'F :> functor<'F>> :
    ('a -> 'b) -> hkt<'F, 'a> -> hkt<'F, 'b> =
    getsig<'F>.fmap

Now we can use fmap on instances of hkt<mkList<'L>, 'a> where 'L <: mkList<'L>.

type MyList() =
    inherit mkList<MyList>()
    // you can interface other type classes here.
type MyList = mkList<MyList>

assert [2, 3, 4] = (
        let lst : hkt<MyList, _> = wrap [1, 2, 3]
        in unwrap <| fmap (fun x -> x + 1) lst
)

Also the Maybe functor:

type mkMaybe<'M> =

    inherit functor<mkMaybe<'M>>()

        static member wrap<'a> (x : Option<'a>): hkt<mkMaybe<'M>, 'a> =
            {wrap = x} :> _
        static member unwrap<'a> (x : hkt<mkMaybe<'M>, 'a>): Option<'a> =
            (x :?> _).wrap

        override member __.fmap<'a, 'b> (f : 'a -> 'b) (m : hkt<mkMaybe<'M>, 'a>) : hkt<mkMaybe<'M>, 'b> =
            let m = unwrap m
            wrap <|
            match m with
            | Some m -> Some <| f m
            | None -> None


type MyMaybe() =
    inherit mkMaybe<MyMaybe>()
    // you can interface other type classes here.

type MyMaybe = mkMaybe<MyMaybe>


assert Some 2 = (
    let m : hkt<MyMaybe, _> = wrap <| Some 1
    in unwrap <| fmap (fun x -> x + 1) m)

Something deserved to be mentioned here is, we are capable of simplifying this via active patterns:

let Just<'M, 'a> (a: 'a) : hkt<mkMaybe<'M>, 'a> = wrap <| Some a

[<GeneralizableValue>]
let Nothing<'M, 'a> : hkt<mkMaybe<'M>, 'a> = wrap <| None

let (|Just|Nothing|) (m: hkt<mkMaybe<'M>, 'a>) =
    let s: 'a Option = unwrap m
    match s with
    | Some m -> Just m
    | None    -> Nothing

assert Some 2 = (
    let m : hkt<MyMaybe, _> = Just 1
    in unwrap <| fmap (fun x -> x + 1) m
)
assert 2 = match Just 2 with
           | Just a -> a
           | _ -> failwith ""

Now we have introduced how to use active patterns and static resolved type parameters to implement type classes, but there are some problem remained here need to be answered.

The getsig Function and Implicits

The getsig function makes it more convenient for F# to achieve type classes than OCaml(without modular implicits[4]).

With a glance of the polymorphic map in OCaml, which is implemented in this article:

type 'a mappable_impl = (module Mappable with type t = 'a)

let map (type t) (m: t mappable_impl) (f: 'a -> 'b) (a: ('a, t) app) =
    let module M = (val m : Mappable with type t = t) in
    M.map f a


let lst_mapped : (int, ListApp.t) app = map (module MapList) (fun x -> x + 1) lst_data_hkt
let arr_mapped : (int, ArrayApp.t) app = map (module MapArray) (fun x -> x + 1) arr_data_hkt

We could find that the explicit argument m : t mappable_impl of map is redundant. In fact, it could be inferred through the latter argument a : ('a, t) app. If there is a way in OCaml to automatically create an instance typed t mappable_impl from the given argument a : ('a, t) app, then it could reach the presence of F#.

Finally, I’d present an implementation of getsig here:

open System
open System.Reflection

let private ts  = Array.zeroCreate<Type> 0

[<GeneralizableValue>]
let getsig<'a> =
    let t = typeof<'a>
    let f = t.GetConstructor(
                BindingFlags.Instance ||| BindingFlags.Public,
                null,
                CallingConventions.HasThis,
                ts,
                null)
    let o = f.Invoke([||])
    o :?> 'a

Why AbstractClass?

This is required for we need

  1. Default implementations of type classes

  2. Inheritances among type classes(such as Monad <: Applicative <: Functor)

These two features are necessary, if you have any problem about this point, please check Functor.fs, Applicative.fs, and Monad.fs.

So the questions above could be answered here:

Q:Why cannot we use records instead of abstract classes?

A:Records(in F#) conflict against type classes’ Inheritances.

Q:Why cannot we always use interfaces instead of abstract classes?

A:Currently there is no support for interfaces with default methods in F#. I’ve heard that C# 8 has supported this feature, but it hasn’t been introduced into F# yet.

Moveover, there is a related story about Traits, which is a concept equivalent to type classes. Type classes could be implemented more smoothly via interfaces with default methods, which is leveraged by Scala language[5].

Why Functor<’F> instead of the straightforward Functor

You might have thought of this:

[<AbstractClass>]
type functor() =
    abstract member fmap<'a, 'b, 'F <: functor> :
        ('a -> 'b) -> hkt<'F, 'a> -> hkt<'F, 'b>

There is something wrong with above codes.

Firstly, it mixes up inheritances among type classes with making instances of type classes.

If we have List <: functor, and Applicative <: functor, isn’t it saying Applicative is in the same category as the List?

Furthermore, static resolved type parameters cannot tell you which interface/abstract class your type has extended with:

type mkList() =
    inherit Functor()
    // ...

let fmap<'a, 'b, 'F when 'F :> functor> :
    ('a -> 'b) -> hkt<'F, 'a> -> hkt<'F, 'b> =
    getsig<'F>.fmap

The getsig used above is impossible to implement in F#.

'F when 'F :> functor doesn’t work with instance resolution, for the F# compiler won’t make assumptions that 'F has the abstract methods of functor, but 'F when 'F :> functor<'F> is okay.

Therefore, we should use functor<'F>.

References and Further Reading

[1] Haskell ViewPatterns: https://ghc.haskell.org/trac/ghc/wiki/ViewPatterns

[2] Haskell ViewPatterns vs F# Active Patterns: https://mail.haskell.org/pipermail/haskell-cafe/2009-January/053643.html

[3] Structural-Typing: https://en.wikipedia.org/wiki/Structural_type_system#Description

[4] OCaml Modular Implicits : https://discuss.ocaml.org/t/modular-implicits/144

[5] Scala compiles traits to interfaces with default methods: https://github.com/scala/scala/pull/5003