Associated type constructors, part 4: Unifying ATC and HKT

9 November 2016

This post is a continuation of my posts discussing the topic of associated type constructors (ATC) and higher-kinded types (HKT):

  1. The first post focused on introducing the basic idea of ATC, as well as introducing some background material.
  2. The second post showed how we can use ATC to model HKT, via the “family” pattern.
  3. The third post did some exploration into what it would mean to support HKT directly in the language, instead of modeling them via the family pattern.
  4. This post considers what it might mean if we had both ATC and HKT in the language: in particular, whether those two concepts can be unified, and at what cost.

Unifying HKT and ATC

So far we have seen “associated-type constructors” and “higher-kinded types” as two distinct concepts. The question is, would it make sense to try and unify these two, and what would that even mean?

Consider this trait definition:

trait Iterable {
    type Iter<'a>: Iterator<Item=Self::Item>;
    type Item;
    fn iter<'a>(&'a self) -> Self::Iter<'a>;

In the ATC world-view, this trait definition would mean that you can now specify a type like the following

<T as Iterable>::Iter<'a>

Depending on what the type T and lifetime 'a are, this might get “normalized”. Normalization basically means to expand an associated type reference using the types given in the appropriate impl. For example, we might have an impl like the following:

impl<A> Iterable for Vec<A> {
    type Item = A;
    type Iter<'a> = std::vec::Iter<'a, A>;

    fn iter<'a>(&'a self) -> Self::Iter<'a> {

In that case, <Vec<Foo> as Iterable>::Iter<'x> could be normalized to std::vec::Iter<'x, Foo>. This is basically exactly the same way that associated type normalization works now, except that we have additional type/lifetime parameters that are placed on the associated item itself, rather than having all the parameters come from the trait reference.

Associated type constructors as functions

Another way to view an ATC is as a kind of function, where the normalization process plays the role of evaluating the function when applied to various arguments. In that light, <Vec<Foo> as Iterable>::Iter could be viewed as a “type function” with a signature like lifetime -> type; that is, a function which, given a type and a lifetime, produces a type:

<Vec<Foo> as Iterable>::Iter<'x>
^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^
function                     argument

When I write it this way, it’s natural to ask how such a function is related to a higher-kinded type. After all, lifetime -> type could also be a kind, right? So perhaps we should think of <Vec<Foo> as Iterable>::Iter as a type of kind lifetime -> type? What would that mean?

Limitations on what can be used in an ATC declaration

Well, in the last post, we saw that, in order to ensure that inference is tractable, HKT in Haskell comes with pretty strict limitations on the kinds of “type functions” we can support. Whatever we chose to adopt in Rust, it would imply that we need similar limitations on ATC values that can be treated as higher-kinded.

That wouldn’t affect the impl of Iterable for Vec<A> that we saw earlier. But imagine that we wanted Range<i32>, which is the type produced by 0..22, to act as an Iterable. Now, ranges like 0..22 are already iterable – so the type of an iterator could just be Self, and iter() can effectively just be clone(). So you might think you could just write:

impl<u32> Iterable for Range<u32>
    type Item = u32;
    type Iter<'a> = Range<u32>;
    //              ^^^^ doesn't use `'a'` at all
    fn iter(&self) -> Range<u32> {

However, this impl would be illegal, because Range<u32> doesn’t use the parameter 'a. Presuming we adopted the rule I suggested in the previous post, every value for Iter<'a> would have to use the 'a exactly once, as the first lifetime argument. So Foo<'a, u32> would be ok, as would &'a Bar, but Baz<'static, 'a> would not.

Working around this limitation with newtypes

You could work around this limitation above by introducing a newtype. Something like this:

struct RangeIter<'a> {
    range: Range<u32>,
    dummy: PhantomData<&'a ()>,
    //                  ^^ need to use `'a` somewhere

We can then implement Iterator for RangeIter<'a> and just proxy next() on to But this is kind of a drag.

An alternative: give users the choice

For a long time, I had assumed that if we were going to introduce HKT, we would do so by letting users define the kinds more explicitly. So, for example, if we wanted the member Iter to be of kind lifetime -> type, we might declare that explicitly. Using the <_> and <'_> notation I was using in earlier posts, that might look like this:

trait Iterable {
    type Iter<'_>;

Now the trait has declared that impls must supply a valid, partially applied struct/enum name as the value for Iter.

I’ve somewhat soured on this idea, for a variety of reasons. One big one is that we are forcing trait users to mak this choice up front, when it may not be obvious whether a HKT or an ATC is the better fit. And of course it’s a complexity cost: now there are two things to understand.

Finally, now that I realize that HKT is going to require bounds, not having names for things means it’s hard to see how we’re going to declare those bounds. In fact, even the Iterable trait probably has some bounds; you can’t just use any old lifetime for the iterator. So really the trait probably includes a condition that Self: 'iter, meaning that the iterable thing must outlive the duration of the iteration:

trait Iterable {
    type Iter<'iter>: Iterator<Item=Self::Item>
        where Self: 'iter; // <-- bound I was missing before
    type Item;
    fn iter<'iter>(&'iter self) -> Self::Iter<'iter>;

Why focus on associated items?

You might wonder why I said that we should consider <T as Iterable>::Iter to have type lifetime -> type rather than saying that Iterable::Iter would be something of kind type -> lifetime -> type. In other words, what about the input types to the trait itself?

It turns out that this idea doesn’t really make sense. First off, it would naturally affect existing associated types. So Iterator::Item, for example, would be something of kind type -> type, where the argument is the type of the iterator. <Range<u32> as Iterator>::Item would be the syntax for applying Iterator::Item to Range<u32>. Since we can write generic functions with higher-kinded parameters like fn foo<I<_>>(), that means that I here might be Iterator::Item, and hence I<Range<u32>> would be equivalent to <Range<u32> as Iterator>::Item.

But remember that, to make inference tractable, we want to know that ?X<Foo> = ?Y<Foo> if and only if ?X = ?Y. That means that we could not allow <Range<u32> as Iterator>::Item to normalize to the same thing as <Range<u32> as SomeOtherTrait>::Foo. You can see that this doesn’t even remotely resemble associated types as we know them, which are just plain one-way functions.


This is kind of the “capstone” post for the series that I set out to write. I’ve tried to give an overview of what associated type constructors are; the ways that they can model higher-kinded patterns; what higher-kinded types are; and now what it might mean if we tried to combine the two ideas.

I hope to continue this series a bit further, though, and in particular to try and explore some case studies and further thoughts. If you’re interested in the topic, I strongly encourage you to hop over to the internals thread and take a look. There have been a lot of insightful comments there.

That said, currently my thinking is this:

  • Associated type constructors are a natural extension to the language. They “fit right in” syntactically with associated types.
  • Despite that, ATC would represent a huge step up in expressiveness, and open the door to richer traits. This could be particularly important for many libraries, such as futures.
    • I know that Rayon had to bend over backwards in some places because we lack any way to express an “iterable-like” pattern.
  • Higher-kinded types as expressed in Haskell are not very suitable for Rust:
    • they don’t cover bounds, which we need;
    • the limitation to “partially applied” struct/enum names is not a natural fit, even if we loosen it somewhat.
  • Moreover, adding HKT to the language would be a big complexity jump:
    • to use Rust, you already have to understand associated types, and ATC is not much more;
    • but adding to that rules and associated syntax for HKT feels like a lot to ask.

So currently I lean towards accepting ATC with no restrictions and modeling HKT using families. That said, I agree that the potential to feel like a lot of “boilerplate”. I sort of suspect that, in practice, HKT would require a fair amount of its own boilerplate (i.e, to abstract away bounds and so forth), and/or not be suitable for Rust, but perhaps further exploration of example use-cases will be instructive in this regard.


Please leave comments on this internals thread.