Canonicalization

Canonicalization is the process of isolating an inference value from its context. It is a key part of implementing canonical queries, and you may wish to read the parent chapter to get more context.

Canonicalization is really based on a very simple concept: every inference variable is always in one of two states: either it is unbound, in which case we don't know yet what type it is, or it is bound, in which case we do. So to isolate some data-structure T that contains types/regions from its environment, we just walk down and find the unbound variables that appear in T; those variables get replaced with "canonical variables", starting from zero and numbered in a fixed order (left to right, for the most part, but really it doesn't matter as long as it is consistent).

So, for example, if we have the type X = (?T, ?U), where ?T and ?U are distinct, unbound inference variables, then the canonical form of X would be (?0, ?1), where ?0 and ?1 represent these canonical placeholders. Note that the type Y = (?U, ?T) also canonicalizes to (?0, ?1). But the type Z = (?T, ?T) would canonicalize to (?0, ?0) (as would (?U, ?U)). In other words, the exact identity of the inference variables is not important – unless they are repeated.

We use this to improve caching as well as to detect cycles and other things during trait resolution. Roughly speaking, the idea is that if two trait queries have the same canonical form, then they will get the same answer. That answer will be expressed in terms of the canonical variables (?0, ?1), which we can then map back to the original variables (?T, ?U).

Canonicalizing the query

To see how it works, imagine that we are asking to solve the following trait query: ?A: Foo<'static, ?B>, where ?A and ?B are unbound. This query contains two unbound variables, but it also contains the lifetime 'static. The trait system generally ignores all lifetimes and treats them equally, so when canonicalizing, we will also replace any free lifetime with a canonical variable (Note that 'static is actually a free lifetime variable here. We are not considering it in the typing context of the whole program but only in the context of this trait reference. Mathematically, we are not quantifying over the whole program, but only this obligation). Therefore, we get the following result:

?0: Foo<'?1, ?2>

Sometimes we write this differently, like so:

for<T,L,T> { ?0: Foo<'?1, ?2> }

This for<> gives some information about each of the canonical variables within. In this case, each T indicates a type variable, so ?0 and ?2 are types; the L indicates a lifetime variable, so ?1 is a lifetime. The canonicalize method also gives back a CanonicalVarValues array OV with the "original values" for each canonicalized variable:

[?A, 'static, ?B]

We'll need this vector OV later, when we process the query response.

Executing the query

Once we've constructed the canonical query, we can try to solve it. To do so, we will wind up creating a fresh inference context and instantiating the canonical query in that context. The idea is that we create a substitution S from the canonical form containing a fresh inference variable (of suitable kind) for each canonical variable. So, for our example query:

for<T,L,T> { ?0: Foo<'?1, ?2> }

the substitution S might be:

S = [?A, '?B, ?C]

We can then replace the bound canonical variables (?0, etc) with these inference variables, yielding the following fully instantiated query:

?A: Foo<'?B, ?C>

Remember that substitution S though! We're going to need it later.

OK, now that we have a fresh inference context and an instantiated query, we can go ahead and try to solve it. The trait solver itself is explained in more detail in another section, but suffice to say that it will compute a certainty value (Proven or Ambiguous) and have side-effects on the inference variables we've created. For example, if there were only one impl of Foo, like so:

impl<'a, X> Foo<'a, X> for Vec<X>
where X: 'a
{ ... }

then we might wind up with a certainty value of Proven, as well as creating fresh inference variables '?D and ?E (to represent the parameters on the impl) and unifying as follows:

  • '?B = '?D
  • ?A = Vec<?E>
  • ?C = ?E

We would also accumulate the region constraint ?E: '?D, due to the where clause.

In order to create our final query result, we have to "lift" these values out of the query's inference context and into something that can be reapplied in our original inference context. We do that by re-applying canonicalization, but to the query result.

Canonicalizing the query result

As discussed in the parent section, most trait queries wind up with a result that brings together a "certainty value" certainty, a result substitution var_values, and some region constraints. To create this, we wind up re-using the substitution S that we created when first instantiating our query. To refresh your memory, we had a query

for<T,L,T> { ?0: Foo<'?1, ?2> }

for which we made a substutition S:

S = [?A, '?B, ?C]

We then did some work which unified some of those variables with other things. If we "refresh" S with the latest results, we get:

S = [Vec<?E>, '?D, ?E]

These are precisely the new values for the three input variables from our original query. Note though that they include some new variables (like ?E). We can make those go away by canonicalizing again! We don't just canonicalize S, though, we canonicalize the whole query response QR:

QR = {
    certainty: Proven,             // or whatever
    var_values: [Vec<?E>, '?D, ?E] // this is S
    region_constraints: [?E: '?D], // from the impl
    value: (),                     // for our purposes, just (), but
                                   // in some cases this might have
                                   // a type or other info
}

The result would be as follows:

Canonical(QR) = for<T, L> {
    certainty: Proven,
    var_values: [Vec<?0>, '?1, ?2]
    region_constraints: [?2: '?1],
    value: (),
}

(One subtle point: when we canonicalize the query result, we do not use any special treatment for free lifetimes. Note that both references to '?D, for example, were converted into the same canonical variable (?1). This is in contrast to the original query, where we canonicalized every free lifetime into a fresh canonical variable.)

Now, this result must be reapplied in each context where needed.

Processing the canonicalized query result

In the previous section we produced a canonical query result. We now have to apply that result in our original context. If you recall, way back in the beginning, we were trying to prove this query:

?A: Foo<'static, ?B>

We canonicalized that into this:

for<T,L,T> { ?0: Foo<'?1, ?2> }

and now we got back a canonical response:

for<T, L> {
    certainty: Proven,
    var_values: [Vec<?0>, '?1, ?2]
    region_constraints: [?2: '?1],
    value: (),
}

We now want to apply that response to our context. Conceptually, how we do that is to (a) instantiate each of the canonical variables in the result with a fresh inference variable, (b) unify the values in the result with the original values, and then (c) record the region constraints for later. Doing step (a) would yield a result of

{
      certainty: Proven,
      var_values: [Vec<?C>, '?D, ?C]
                       ^^   ^^^ fresh inference variables
      region_constraints: [?C: '?D],
      value: (),
}

Step (b) would then unify:

?A with Vec<?C>
'static with '?D
?B with ?C

And finally the region constraint of ?C: 'static would be recorded for later verification.

(What we actually do is a mildly optimized variant of that: Rather than eagerly instantiating all of the canonical values in the result with variables, we instead walk the vector of values, looking for cases where the value is just a canonical variable. In our example, values[2] is ?C, so that means we can deduce that ?C := ?B and'?D := 'static`. This gives us a partial set of values. Anything for which we do not find a value, we create an inference variable.)