In my previous post, I talked over the basics of how unification works and showed how that “mathematical version” winds up being expressed in chalk. I want to go a bit further now and extend that base system to cover associated types. These turn out to be a pretty non-trival extension.

What is an associated type?

If you’re not a Rust programmer, you may not be familiar with the term “associated type” (although many langages have equivalents). The basic idea is that traits can have type members associated with them. I find the most intuitive example to be the Iterator trait, which has an associated type Item. This type corresponds to kind of elements that are produced by the iterator:

trait Iterator {
    type Item;
    
    fn next(&mut self) -> Option<Self::Item>;
}

As you can see in the next() method, to reference an associated type, you use a kind of path – that is, when you write Self::Item, it means “the kind of Item that the iterator type Self produces”. I often refer to this as an associated type projection, since one is “projecting out”1 the type Item.

Let’s look at an impl to make this more concrete. Consider the type std::vec::IntoIter<T>, which is one of the iterators associated with a vector (specifically, the iterator you get when you invoke vec.into_iter()). In that case, the elements yielded up by the iterator are of type T, so we have an impl like:

impl<T> Iterator for IntoIter<T> {
    type Item = T;
    fn next(&mut self) -> Option<T> { ... }
}

This means that if we have the type IntoIter<i32>::Item, that is equivalent to the type i32. We usually call this process of converting an associated trait projection (IntoIter<i32>::Item) into the type found in the impl normalizing the type.

In fact, this IntoIter<i32>::Item is a kind of shorthand; in particular, it didn’t explicitly state what trait the type Item is defined in (it’s always possible that IntoIter<i32> implements more than one trait that define an associated type called Item). To make things fully explicit, then, one can use a fully qualified path like this:

<IntoIter<i32> as Iterator>::Item
 ^^^^^^^^^^^^^    ^^^^^^^^   ^^^^
 |                |          |
 |                |          Associated type name
 |                Trait
 Self type

I’ll use these fully qualified paths from here on out to avoid confusion.

Integrating associated types into our type system

In this post, we will extend our notion of types to include associated type projections:

T = ?X               // type variables
  | N<T1, ..., Tn>   // "applicative" types
  | P                // "projection" types   (new in this post)
P = <T as Trait>::X

Projection types are quite different from the existing “applicative” types that we saw before. The reason is that they introduce a kind of “alias” into the equality relationship. With just applicative types, we could always make progress at each step: that is, no matter what two types were being equated, we could always break the problem down into simpler subproblems (or else error out). For example, if we had Vec<?T> = Vec<i32>, we knew that this could only be true if ?T == i32.

With associated type projections, this is not always true. Sometimes we just can’t make progress. Imagine, for example, this scenario:

<?X as Iterator>::Item = i32

Here we know that ?X is some kind of iterator that yields up i32 elements: but we have no way of knowing which iterator it is, there are many possibilities. Similarly, imagine this:

<?X as Iterator>::Item = <T as Iterator>::Item

Here we know that ?X and T are both iterators that yield up the same sort of items. But this doesn’t tell us anything about the relationship between ?X and T.

Normalization constraints

To handle associated types, the basic idea is that we will introduce normalization constraints, in addition to just having equality constraints. A normalization constraint is written like this:

<IntoIter<i32> as Iterator>::Item ==> ?X   

This constraint says that the associated type projection <IntoIter<i32> as Iterator>::Item, when normalized, should be equal to ?X (a type variable). As we will see in more detail in a bit, we’re going to then go and solve those normalizations, which would eventually allow us to conclude that ?X = i32.

(We could use the Rust syntax IntoIter<i32>: Iterator<Item=?X> for this sort of constraint as well, but I’ve found it to be more confusing overall.)

Processing a normalization constraint is very simple to processing a standard trait constraint. In fact, in chalk, they are literally the same code. If you recall from my first Chalk post, we can lower impls into a series of clauses that express the trait that is being implemented along with the values of its associated types. In this case, if we look at the impl of Iterator for the IntoIter type:

impl<T> Iterator for IntoIter<T> {
    type Item = T;
    fn next(&mut self) -> Option<T> { ... }
}

We can translate this impl into a series of clauses sort of like this (here, I’ll use the notation I was using in my first post):

// Define that `IntoIter<T>` implements `Iterator`,
// if `T` is `Sized` (the sized requirement is
// implicit in Rust syntax.)
Iterator(IntoIter<T>) :- Sized(T).

// Define that the `Item` for `IntoIter<T>`
// is `T` itself (but only if `IntoIter<T>`
// implements `Iterator`).
IteratorItem(IntoIter<T>, T) :- Iterator(IntoIter<T>).

So, to solve the normalization constraint <IntoIter<i32> as Iterator>::Item ==> ?X, we translate that into the goal IteratorItem(IntoIter<i32>, ?X), and we try to prove that goal by searching the applicable clauses. I sort of sketched out the procedure in my first blog post, but I’ll present it in a bit more detail here. The first step is to “instantiate” the clause by replacing the variables (T, in this case) with fresh type variables. This gives us a clause like:

IteratorItem(IntoIter<?T>, ?T) :- Iterator(IntoIter<?T>).

Then we can unify the arguments of the clause with our goals, leading to two unification equalities, and combine that with the conditions of the clause itself, leading to three things we must prove:

IntoIter<?T> = IntoIter<i32>
?T = ?X
Iterator(IntoIter<?T)

Now we can recursively try to prove those things. To prove the equalities, we apply the unification procedure we’ve been looking at. Processing the first equation, we can simplify because we have two uses of IntoIter on both sides, so the type arguments must be equal:

?T = i32 // changed this
?T = ?X
Iterator(IntoIter<?T>)

From there, we can deduce the value of ?T and do some substitutions:

i32 = ?X
Iterator(IntoIter<i32>)

We can now unify ?X with i32, leaving us with:

Iterator(IntoIter<i32>)

We can apply the clause Iterator(IntoIter<T>) :- Sized(T) using the same procedure now, giving us two fresh goals:

IntoIter<i32> = IntoIter<?T>
Sized<?T>

The first unification will yield (eventually):

Sized<i32>

And we can prove this because this is a built-in rule for Rust (that is, that i32 is sized).

Unification as just another goal to prove

As you can see in the walk through in the previous section, in a lot of ways, unification is “just another goal to prove”. That is, the basic way that chalk functions is that it has a goal it is trying to prove and, at each step, it tries to simplify that goal into subgoals. Often this takes place by consulting the clauses that we derived from impls (or that are builtin), but in the case of equality goals, the subgoals are constructed by the builtin unification algorithm.

In the previous post, I gave various pointers into the implementation showing how the unification code looks “for real”. I want to extend that explanation now to cover associated types.

The way I presented things in the previous section, unification flattens its subgoals into the master list of goals. But in fact, for efficiency, the unification procedure will typically eagerly process its own subgoals. So e.g. when we transform IntoIter<i32> = IntoIter<?T>, we actually just invoke the code to equate their arguments immediately.

The one exception to this is normalization goals. In that case, we push the goals into a separate list that is returned to the caller. The reason for this is that, sometimes, we can’t make progress on one of those goals immediately (e.g., if it has unresolved type variables, a situation we’ve not discussed in detail yet). The caller can throw it onto a list of pending goals and come back to it later.

Here are the various cases of interest that we’ve covered so far

Fallback for projection

Thus far we showed how projection proceeds in the “successful” case, where we manage to normalize a projection type into a simpler type (in this case, <IntoIter<i32> as Iterator>::Item into i32). But sometimes we want to work with generics we can’t normalize the projection any further. For example, consider this simple function, which extracts the first item from a non-empty iterator (it panics if the iterator is empty):

fn first<I: Iterator>(iter: I) -> I::Item {
    iter.next().expect("iterator should not be empty")
}

What’s interesting here is that we don’t know what I::Item is. So imagine we are given a normalization constraint like this one:

<I as Iterator>::Item ==> ?X

What type should we use for ?X here? What chalk opts to do in cases like this is to construct a sort a special “applicative” type representing the associated item projection. I will write it as <Iterator::Item><I>, for now, but there is no real Rust syntax for this. It basically represents “a projection that we could not normalize further”. You could consider it as a separate item in the grammar for types, except that it’s not really semantically different from a projection; it’s just a way for us to guide the chalk solver.

The way I think of it, there are two rules for proving that a projection type is equal. The first one is that we can prove it via normalization, as we’ve already seen:

IteratorItem(T, X)
-------------------------
<T as Iterator>::Item = X

The second is that we can prove it just by having all the inputs be equal:

T = U
---------------------------------------------
<T as Iterator>::Item = <U as Iterator>::Item

We’d prefer to use the normalization route, because it is more flexible (i.e., it’s sufficient for T and U to be equal, but not necessary). But if we can definitively show that the normalization route is impossible (i.e., we have no clauses that we can use to normalize), then we we opt for this more restrictive route. The special “applicative” type is a way for chalk to record (internally) that for this projection, it opted for the more restrictive route, because the first one was impossible.

(In general, we’re starting to touch on Chalk’s proof search strategy, which is rather different from Prolog, but beyond the scope of this particular blog post.)

Some examples of the fallback in action

In the first() function we saw before, we will wind up computing the result type of next() as <I as Iterator>::Item. This will be returned, so at some point we will want to prove that this type is equal to the return type of the function (actually, we want to prove subtyping, but for this particular type those are the same thing, so I’ll gloss over that for now). This corresponds to a goal like the following (here I am using the notation I discussed in my first post for universal quantification etc):

forall<I> {
    if (Iterator(I)) {
        <I as Iterator>::Item = <I as Iterator>::Item
    }
}

Per the rules we gave earlier, we will process this constraint by introducing a fresh type variable and normalizing both sides to the same thing:

forall<I> {
    if (Iterator(I)) {
        exists<?T> {
            <I as Iterator>::Item ==> ?T,
            <I as Iterator>::Item ==> ?T,
        }
    }
}

In this case, both constraints will wind up resulting in ?T being the special applicative type <Iterator::Item><I>, so everything works out successfully.

Let’s briefly look at an illegal function and see what happens here. In this case, we have two iterator types (I and J) and we’ve used the wrong one in the return type:

fn first<I: Iterator, J: Iterator>(iter_i: I, iter_j: J) -> J::Item {
    iter_i.next().expect("iterator should not be empty")
}

This will result in a goal like:

forall<I, J> {
    if (Iterator(I), Iterator(J)) {
        <I as Iterator>::Item = <J as Iterator>::Item
    }
}

Which will again be normalized and transformed as follows:

forall<I, J> {
    if (Iterator(I), Iterator(J)) {
        exists<?T> {
            <I as Iterator>::Item ==> ?T,
            <J as Iterator>::Item ==> ?T,
        }
    }
}

Here, the difference is that normalizing <I as Iterator>::Item results in <Iterator::Item><I>, but normalizing <J as Iterator>::Item results in <Iterator::Item><J>. Since both of those are equated with ?T, we will ultimately wind up with a unification problem like:

forall<I, J> {
    if (Iterator(I), Iterator(J)) {
        <Iterator::Item><I> = <Iterator::Item><J>
    }
}

Following our usual rules, we can handle the equality of two applicative types by equating their arguments, so after that we get forall<I, J> I = J – and this clearly cannot be proven. So we get an error.

Termination, after a fashion

One final note, on termination. We do not, in general, guarantee termination of the unification process once associated types are involved. Rust’s trait matching is turing complete, after all. However, we do wish to ensure that our own unification algorithms don’t introduce problems of their own!

The non-projection parts of unification have a pretty clear argument for termination: each time we remove a constraint, we replace it with (at most) simpler constraints that were all embedded in the original constraint. So types keep getting smaller, and since they are not infinite, we must stop sometime.

This argument is not sufficient for projections. After all, we replace a constraint like <T as Iterator>::Item = U with an equivalent normalization constraint, where all the types are the same:

<T as Iterator>::Item ==> U

The argument for termination then is that normalization, if it terminates, will unify U with an applicative type. Moreover, we only instantiate type variables with normalized types. Now, these applicative types might be the special applicative types that Chalk uses internally (e.g., <IteratorItem><T>), but it’s an applicative type nontheless. When that applicative type is processed later, it will therefore be broken down into smaller pieces (per the prior argument). That’s the rough idea, anyway.

Contrast with rustc

I tend to call the normalization scheme that chalk uses lazy normalization. This is because we don’t normalize until we are actually equating a projection with some other type. In constrast, rustc uses an eager strategy, where we normalize types as soon as we “instantiate” them (e.g., when we took a clause and replaced its type parameters with fresh type variables).

The eager strategy has a number of downsides, not the least of which that it is very easy to forget to normalize something when you were supposed to (and sometimes you wind up with a mix of normalized and unnormalized things).

In rustc, we only have one way to represent projections (i.e., we don’t distinguish the “projection” and “applicative” version of <Iterator::Item><T>). The distinction between an unnormalized <T as Iterator>::Item and one that we failed to normalize further is made simply by knowing (in the code) whether we’ve tried to normalize the type in question or not – the unification routines, in particular, always assume that a projection type implies that normalization wouldn’t succeed.

A note on terminology

I’m not especially happy with the “projection” and “applicative” terminology I’ve been using. Its’s what Chalk uses, but it’s kind of nonsense – for example, both <T as Iterator>::Item and Vec<T> are “applications” of a type function, from a certain perspective. I’m not sure what’s a better choice though. Perhaps just “unnormalized” and “normalized” (with types like Vec<T> always being immediately considered normalized). Suggestions welcome.

Conclusion

I’ve sketched out how associated type normalization works in chalk and how it compares to rustc. I’d like to change rustc over to this strategy, and plan to open up an issue soon describing a strategy. I’ll post a link to it in the internals comment thread once I do.

There are other interesting directions we could go with associated type equality. For example, I was pursuing for some time a strategy based on congruence closure, and even implemented (in ena) an extended version of the algorithm described here. However, I’ve not been able to figure out how to combine congruence closure with things like implication goals – it seems to get quite complicated. I understand that there are papers tackling this topic (e.g, Selsam and de Moura), but haven’t yet had time to read it.

Comments?

I’ll be monitoring [the internals thread] for comments and discussion. =)

Footnotes

  1. Projection is a very common bit of jargon in PL circles, though it typically refers to accessing a field, not a type. As far as I can tell, no mainstream programmer uses it. Ah well, I’m not aware of a good replacement.