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Created July 7, 2019 20:47
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Method for Emulating Higher-Kinded Types in Rust

Method for Emulating Higher-Kinded Types in Rust


I've been fiddling about with an idea lately, looking at how higher-kinded types can be represented in such a way that we can reason with them in Rust here and now, without having to wait a couple years for what would be a significant change to the language and compiler.

There have been multiple discussions on introducing higher-ranked polymorphism into Rust, using Haskell-style Higher-Kinded Types (HKTs) or Scala-looking Generalised Associated Types (GATs). The benefit of higher-ranked polymorphism is to allow higher-level, richer abstractions and pattern expression than just the rank-1 polymorphism we have today.

As an example, currently we can express this type:

fn example<A,B>(x:Vec<A>, f:Fn(A)->B) -> Vec<B>;

but we can't express the more generic

fn example2<
	forall A.M<A>:Container, //imaginary syntax
	>(x:M<A>, f:Fn(A)->B) -> M<B>;

as while we can express forall A.Vec<A> there is no way to currently express the typeforall M.forall A.M<A> (note: the nesting a forall within another forall type is what gives us our rank-2-ness of the type).

Haskell solves this problem with Higher-Kinded Types, where not only can we express the type List parameterised with Int as List Int :: *, we can also express and reason about List on its own: List :: * -> *. Current Rust doesn't allow this, as all generic types must be fully instantiated (i.e. Vec<i32> exists, but Vec does not).

The GAT approach suggests solving this by allowing generic types within traits, as in

trait VecGAT {
    type Applied<A> = Vec<A>;

so now we can express the unapplied Vec by our VecGAT, and (Vec) i32 is expressed by VecGAT::Applied<i32>. This is a pretty Rust-y solution, and is easy enough to understand, but has the drawback of only really existing on paper, for the time being at least. There is an excellent article on implementing Monads in Rust with GATs here - if only we could wait for Rust to have GATs implemented and syntax bikeshedded and semantics satiated.

But alas, this is still a proposal a ways off, and we have navel-gazing to do today.

First attempts

Rust already has a pretty solid constraint system in its traits and associated types if you know how to (ab)use them. As a teen I spent my evenings contorting the Scala type system revising, so I had a pop at expressing a representation for unapplied generic types:

struct forall_t; //The 'unapplied' type
//Expressing M<A> as distinct M<forall_t> and A types
struct Lift<MA,A>(PhantomData<MA>,PhantomData<A>);
trait Lower<MA:Unapply> {
    fn (lifted:Lift<
        <MA as Unapply>::unapplied_t,
        <MA as Unapply>::param_t>
    ) -> Self;

plus an isomorphism betweenM<A> and Lift<M<forall_t>,A>.

This proved hard to use, not least because the Lift type has zero size. I would employ the wonderfully unsafe core::mem::transmute between Box<M<A>> and Box<Lift<M<forall_t>,A>>, resulting in segfaults once optimisations were turned on. It's also pretty difficult to properly relay the meaning of Isomorphism to the compiler when we are working with a new representation, as our assumptions about things like to . fro = fro . to = id have to be proven from scratch to make anything useful typecheck.

However, this first failed approach was not entirely fruitless - I ended up writing a trait Unapply which would come in handy later.

trait Unapply {
    type unapplied_t;
    type param_t;
impl<A> Unapply for Vec<A> {
     type unapplied_t = Vec<forall_t>;
     type param_t = A;

For a given type M:Unapply, I could almost close the isomorphism loop through my lifting and unlifting traits: M<A> --[M<A> as Unapply]--> (M<forall_t>, A) --[M<A> as Lower]-->M<A>. This loop worked when I had a concrete M<A> as my starting point to go to (M<forall_t>,A), but the inference required inferring M<A>:Lower, which I couldn't do from (M<forall_t>,A)

Breakthrough #1

So, currently Rust can't express a trait accessible via MyTrait::GenericTy<A>. But what about moving the generic parameter left once, to get MyTrait<A>::GenericTy?

trait ReplaceWith<A> {
    type result_t;
impl<A,B> ReplaceWith<B> for Vec<A> {
    type result_t = Vec<B>;

Now we get <Vec<A> as ReplaceWith<B>>::result_t == Vec<B>. While the syntax is unwieldy, this works to emulate GATs/HKTs: what might ideally be represented as Ty::GAT<A> in a future syntax can today be expressed as <Ty as GAT_Trait<A>>::result_t.

Breakthrough 2

Now we have simple GATs expressible, let's get to working on our representations of higher-kinded types.

Firstly, a given type must be liftable to be supported in our ad-hoc representation system. I'll call this Unplug, as we are conceptually operating on some M<A> to separate the M<_> from the A.

trait Unplug {
    type F; //The representation type of the higher-kinded type
    type A; //The parameter type
impl<A> Unplug for Vec<A> {
    type F=Vec<forall_t>; //All unapplied Vecs are represented by
    type A=A;

And to re-combine a split application (to re-plug an unplugged type):

trait Plug<A> {
    type result_t;
//This is identical to the ReplaceWith trait shown above
impl<A,B> Plug<A> for Vec<B> {
    type result_t = Vec<A>;

so now we can take a Vec<A>, split it into separate Vec<_> and A types, and re-apply a new type to Vec<_> to get a Vec<B> if we want to.

This is pretty close to complete for the basics of our representation of higher-kinded types. We can now plug and unplug parameters from HKTs like Option<_> and Vec<_>, so long as we provide the instances to tell the compiler how.

Next up is our wrapper, so that we can hold values with some type M<T> while still being able to reason about the M<_> and the T separately. Whereas before I tried some pointer/transmute/phantom reference shenanigans, I eventually found that I could just store the underlying value (woops), now that I had a way to express its type given the unplugged types available to the wrapper.

pub  struct  Concrete<M:Unplug+Plug<A>,A> {
    pub unwrap:<M as Plug<A>>::result_t
//Conceptually equivalent to the GAT-syntax
//pub struct Concrete<M,A>{unwrap: M::Plug<A>}

(Concrete is a holdover name from when I tried juggling lifted pointers and phantoms, but fits as 'the real value represented by our constructed type (M<forall_t>,A)'). This struct has some rather nice properties, once you help the type inference out a bit. To this end, I created a helper function to guide the inference:

impl<M:Unplug+Plug<A>,A> Concrete<M,A> {
    fn of<MA:Unplug<F=M,A=A>+Plug<A>>(x:MA) -> Self
    where M:Plug<A, result_t = MA> {
        Concrete { unwrap: x }

The associated types in the signature are used to close the loop of plug . unplug and unplug . plug, so the compiler recognises that we are working on the same plug (M<_>,A) == MA and unplug MA == (M<_>,A). A lot of this feels like working in a weird Prolog-Idris hybrid at times. The type-inference hinting out of the way, let's take a quick look at use:

let myVec = vec![1,2,3i32];
let conc = Concrete::of(myVec);
let myVec = conc.unwrap;

This should be a zero-cost abstraction! Woohoo!

Putting it to work

Now we can express higher-kinded types, what do we get?

Well, for our demonstration, I will cook up everyone's favourite burrito* - the Monad! (*this is the last time a monad will be described as a burrito).

In Haskell, the typeclass hierarchy for a Monad goes Functor f => Applicative f => Monad f. For familiarity's sake, I'll start with Functor:

class Functor f where
    fmap :: (a -> b) -> f a -> f b

is written in our representation as

pub trait Functor: Unplug+Plug<<Self as Unplug>::A> {
    //Self is conceptually our haskell-ese "f a"
    fn map<B, F>(f:F, s:Self) -> <Self as Plug<B>>::result_t
        F:FnMut(<Self as Unplug>::A) -> B

//Example impl for a represented Vec
impl<A> Functor for Concrete<Vec<forall_t>,A> {
    //remember, Self ~ (Vec<_>, A) ~ "f a"
    fn map<B,F>(f:F, s:Self) -> <Self as Plug<B>>::result_t
        F:FnMut(<Self as Unplug>::A) -> B 

To show functor-level polymorphism in action, here's a simple compose-then-map function:

fn  functor_test<F:Functor,A,B,C>(
    fun:impl  Fn(A)->B,
    fun2:impl  Fn(B)->C
) -> <F as Plug<C>>::result_t
    <F as Unplug>::F: Plug<A> + Plug<B> + Plug<C>
    let cmp =  |x|fun2(fun(x));
    Functor::map(cmp, functor)

Like magic, not a single type annotation in sight (in the function body). The function signature is a bit unwieldy again, as we need to convince the compiler that our HKT's Plug and Unplug traits behave in the way that we would expect application and abstraction in true HKTs to behave (closing our loops again).

As for Applicative, here's one I made earlier:

pub trait Applicative: Functor {
    fn pure(s:<Self as Unplug>::A) -> Self;
    fn app<B, F>(
        f:<Self as Plug<F>>::result_t, //M<F>
        s:Self                         //M<A>
    ) -> <Self as Plug<B>>::result_t   //M<B>
        F:FnMut(<Self as Unplug>::A) -> B + Clone,
        Self:Plug<F> + Plug<B> + Unplug,
        <Self as Plug<F>>::result_t:
            Unplug<F=<Self as Unplug>::F,A=F> +
            Plug<F> +
        <Self as Unplug>::F:Plug<F>

This one took me a little while to get typechecking. Thankfully, most of the type-astronomy is done in the trait, leaving the impl easier to understand and/or implement.

impl<A:Clone> Applicative for Concrete<Vec<forall_t>,A> {
    fn pure(a:A) -> Self {
    fn app<B, F>(
        fs:<Self as Plug<F>>::result_t,
    ) -> <Self as Plug<B>>::result_t
        F:FnMut(<Self as Unplug>::A) -> B + Clone,
        <Self as Plug<F>>::result_t: Clone,
        let flat:Vec<B> =

Finally, to round things off, we define a true Monad in Rust, complete with type inference and checking (provided you have a suitable signature to hint to the compiler)

pub trait Monad : Applicative {
    fn bind<F,B>(f:F, s:Self) -> <Self as Plug<B>>::result_t
        F:FnMut(<Self as Unplug>::A) ->
            <Self as Plug<B>>::result_t + Clone

impl<A:Clone> Monad for Concrete<Vec<forall_t>,A> {
    fn bind<F,B>(f:F, s:Self) -> 
        <Self as Plug<B>>::result_t       //M<A> -> M<B>
        F:FnMut(<Self as Unplug>::A)
            -> <Self as Plug<B>>::result_t + Clone //A -> M<B>
        let res:Vec<B> =

Wrapping Up

In conclusion, I've presented a method for emulating Higher-Kinded Types/Generic Associated Types in current Rust via casting generic traits down by the type parameter, turning the idealised/future syntactic MyTrait::someTy<T> into <MyTrait as SomeTy<T>>::result_t with an appropriate trait-based structure to aid the compiler in typechecking. I have then shown a simple example of how to use this technique to create Functors, Applicatives and Monads within Rust, such that we don't lose any type safety, yet will successfully compile, build and run on current toolchains.

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burjui commented Jul 11, 2019

impl<A> Functor for Concrete<Vec<forall_t>,A> {
        ^^^^^^^ the trait `Plug<_>` is not implemented for `Concrete<std::vec::Vec<forall_t>, A>`

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ricochet1k commented Jul 17, 2019

impl<F, A> Unplug for Concrete<F, A>
    F: Unplug + Plug<A>,
    type F = F;
    type A = A;

impl<F, A, B> Plug<B> for Concrete<F, A>
    F: Unplug + Plug<A> + Plug<B>,
    type result_t = Concrete<F, B>;

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burjui commented Jul 17, 2019

Right, I should have put more effort into it. Though P should be replaced with result_t.

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Whoops yeah, fixed.

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ricochet1k commented Jul 18, 2019

Why do you impl Functor, etc. on Concrete<Vec<forall_t>, T> instead of just Vec<T>? That seems to work just as well with a lot less mess.

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burjui commented Jul 18, 2019

It works indeed, I even managed to make Option monad work. It was a pain though, and I still do not fully understand everything, so there may be some caveats that I am missing.

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Emulating HKTs with GATs on nightly gives us a nice looking (basic) Monad:


trait Monad {
    type Start;
    type End<B>: Monad;
    fn bind<B>(self, f: impl Fn(Self::Start) -> Self::End<B>) -> Self::End<B>;

It can be implemented for Option<T> like so:

impl<A> Monad for Option<A> {
    type Start = A;
    type End<B> = Option<B>;
    fn bind<B>(self, f: impl Fn(A) -> Option<B>) -> Option<B> {

fn main() {
    let monad = Some(1).bind(|x| Some(x * 2));
    println!("{:?}", monad); // Some(2)

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amosonn commented Dec 16, 2020

The sample method doesn't check the Functor so thoroughly, because it doesn't actually run it twice, only once. I would say a better check (with a heavier signature) is:

fn it_compiles<F: Functor, F2: Functor, F3: Functor, A, B, C>(functor:F, fun:impl Fn(A)->B, fun2:impl Fn(B)->C) -> F3 where
    F: Plug<A> + Plug<B, result_t=F2> + Unplug<A=A>,
    <F as Unplug>::F: Plug<A> + Plug<B> + Plug<C>,
    F2: Plug<B> + Plug<C, result_t=F3> + Unplug<A=B>,
    <F2 as Unplug>::F: Plug<B> + Plug<C>,
    F3: Plug<C>
    let x = Functor::map(fun, functor);
    Functor::map(fun2, x)

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amosonn commented Dec 16, 2020

And if the Functor::map is updated to be fn map(self, f: F) then the code can even be written as

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8573 commented Dec 9, 2021

A variant of this technique from the latest TWIR:

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