Higher Kindended Types in F# Part III - Concept Emulator

I hope you liked the previous parts of this series

Are you ready to see how you can do those damned Higher Kinded Types in F#?
Before we start please be aware that as HKTs are not a native concept to F# our solution is clumsy and far from Haskell’s elegance. It requires quite a bit of plumbing and it is leaky as it will show on call sites.

You can read about the general idea in the paper “Lightweight higher-kinded polymorphism”.

In F# we will have to introduce one or multiple explicit types for so we can defunctionalize. We will call this kind of type Brand(s) as in the paper.

type Brand<'a, 'b> (value:obj) =
    member self.Apply() : obj = value   

That’s it. No really! On a very basic level thats all that is needed. Lets move on with our business domain.

type Decision<'a> = Decision of 'a
type InProgress<'a> = {value: 'a; decision: Decision<'a>}
type Finished<'a> = {value: 'a; initial: 'a; timestamp: DateTime}

Nothing new here sofar but now we have to create some marker classes that mirror our states in our business domain.

type InProgress private  () = class end
type Finished private () = class end

And 2 helper functions to make live a bit easier on the call site. This helpers will be repsonsible to project and inject our real values from the Brand. Accourdingly they are named prj and inj.

module INP =
    let inline inj (value: InProgress<'A>) : Brand<InProgress, 'A> = new Brand<_,_>(value)
    let inline prj (value: Brand<InProgress, 'A>) : InProgress<'A> = value.Apply() :?> _

module FIN =
    let inline inj (value: Finished<'A>) : Brand<Finished, 'A> = new Brand<_,_>(value)
    let inline prj (value: Brand<Finished, 'A>) : Finished<'A> = value.Apply() :?> _

So for every type that we want to push into the HKT domain we create on the one hand a marker class and a set of inj and prj functions to make working with the brands easier on the call site. Armed with these tools we can already harvest much of what HKTs has to offer. Let’ see the code in action:

let x: Brand<InProgress, int> = {value = 1, Decision 2} |> INP.inj
let y: InProgress<int> = x |> INP.prj

Now this is already cool. The call site ceremony isn’t exactly nice looking but using F#s pipe operator acceptable.

How about we write transition function from Brand<InProgress, 'a> to Brand<Finished, 'a>?

let applyDecision (Decision d) v = d

let toFinishedStructure ({value = v; decisions = d}: InProgress<'a>) : Finished<'a> =
    {value = applyDecision d v; initial = v; timestamp = DateTime.Now}

let toFinishedBrand (x: Brand<InProgress, 'a>) : Brand<Finished, 'a> = 
    |> INP.prj
    |> toFinishedPure
    |> FIN.inj

let a = {InProgress.value = 1; decision = Decision 2} |> INP.inj
let b = toFinishedBrand a

Wow! That ain’t so bad. As I said not as elegant in Haskell but still… If you want HKT and you work with homogeneous sets of those what you have seen so far will carry you quite a long way.

But what if we have a heterogeneous set of HKTs like in our LineItem example?

type LineItem<'a> = {
    articleID: Brand<'a, string>
    units:     Brand<'a, int>
    amount:    Brand<'a, float>
let someLineItemProgress: LineItem<InProgress> = someCreateFn()

The easiest way would be to simply define a map function and call it with some transition function

let mapLineItem f li = {
        articleID   = li.articleID |> f
        units       = li.units |> f
        amount      = li.amount |> f

mapLineItem toFinishedBrand someLineItemProgress

Only that doesnt work and gives us an error

let mapLineItem f li = {
    articleID   = li.articleID |> f
    units       = li.units |> f
                                Type mismatch. Expecting a
                                'Brand<'a,int> -> 'b'    
                                but given a
                                'Brand<'a,int> -> 'd'    
                                The type 'string' does not match the type 'int'
    amount      = li.amount |> f

This is because the type inferrence algorithm looks first at the application of articleID = li.articleID |> f and infers the type of f to be (Brand<'a,string> -> Brand<'c,string>). Only that is not true. The type of f should be (Brand<'a, 'x> -> Brand<'c,'x>). We would need a way to inline first class function parameters: let mapLineItem (inline f) li = . Sadly that is not possible in F#. In Haskell can do this only by using the {-# LANGUAGE RankNTypes #-} compiler directive. Which means: “allow polymorphic functions as first class paramters”. Haskell expresses this via function signatures similar to this one map :: forall f g. (forall a. f a -> g a) -> R f -> R g.

Can we get that into F# nevertheless? Yes - but now it’s getting ugly. Let’s start with some helper function and a member method on our to type to be mapped over.

let createSafe (parent:obj) (prop:Lazy<'t>): 't = if not <| isNull parent then prop.Force() else Unchecked.defaultof<'t>

type LineItem<'a> = 
    member inline self.Imap(invoker, s:LineItem<'a>): LineItem<'b> =
            articleID = invoker $ (createSafe s (lazy(s.articleID)))
            units =     invoker $ (createSafe s (lazy(s.units))) 
            amount =    invoker $ (createSafe s (lazy(s.amount))) 

First we create a helper function for lazily creating properties on any object if it is null. This is particular helpful if you will create new objects of deeply nested HKTs. And an inlined (member) function imap that is used to map over our type. Why imap you ask? Because its not really a map function. At least not from a signature point of view. Only now we are ready to implement the element mapping. And wow this is getting really, really heavy! Oh F#!

type ToFinish = ToFinish with 
    static member inline ($) (ToFinish, b) = toFinishedBrand b

let inline imap (invoker: ^I) (s:^S) : ^R = (^S: (member Imap: ^I -> ^S -> ^R) (s, invoker, s))

let inline toFinished li = imap ToFinish li

So we need a type with one single static member that is also inlined where we implement our mapping. puuuh! The imapfunction is only a helper (and can be reused) to create the final toFinish wrapper function so this whole mess can be easily used on the call site.

And finally let’s use that all

let ili : LineItem<InProgress> = 
        articleID = {InProgress.value = "0815"; decision = Decision "0815"} |> INP.inj
        units =     {InProgress.value = 4; decision = Decision 5} |> INP.inj
        amount =    {InProgress.value = 9.99; decision = Decision 9.9} |> INP.inj

let fli : LineItem<Finished> = toFinished ili

I’d say that looks good. At least on the call site! I hope you enjoyed the show.

Follow me to the 4th part of this series for a roundup and FAQ

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Written on February 1, 2017