Placeholders and universes

From time to time we have to reason about regions that we can't concretely know. For example, consider this program:

// A function that needs a static reference
fn foo(x: &'static u32) { }

fn bar(f: for<'a> fn(&'a u32)) {
       // ^^^^^^^^^^^^^^^^^^^ a function that can accept **any** reference
    let x = 22;

fn main() {

This program ought not to type-check: foo needs a static reference for its argument, and bar wants to be given a function that accepts any reference (so it can call it with something on its stack, for example). But how do we reject it and why?

Subtyping and Placeholders

When we type-check main, and in particular the call bar(foo), we are going to wind up with a subtyping relationship like this one:

fn(&'static u32) <: for<'a> fn(&'a u32)
----------------    -------------------
the type of `foo`   the type `bar` expects

We handle this sort of subtyping by taking the variables that are bound in the supertype and replacing them with universally quantified representatives, denoted like !1 here. We call these regions "placeholder regions" – they represent, basically, "some unknown region".

Once we've done that replacement, we have the following relation:

fn(&'static u32) <: fn(&'!1 u32)

The key idea here is that this unknown region '!1 is not related to any other regions. So if we can prove that the subtyping relationship is true for '!1, then it ought to be true for any region, which is what we wanted.

So let's work through what happens next. To check if two functions are subtypes, we check if their arguments have the desired relationship (fn arguments are contravariant, so we swap the left and right here):

&'!1 u32 <: &'static u32

According to the basic subtyping rules for a reference, this will be true if '!1: 'static. That is – if "some unknown region !1" outlives 'static. Now, this might be true – after all, '!1 could be 'static – but we don't know that it's true. So this should yield up an error (eventually).

What is a universe?

In the previous section, we introduced the idea of a placeholder region, and we denoted it !1. We call this number 1 the universe index. The idea of a "universe" is that it is a set of names that are in scope within some type or at some point. Universes are formed into a tree, where each child extends its parents with some new names. So the root universe conceptually contains global names, such as the lifetime 'static or the type i32. In the compiler, we also put generic type parameters into this root universe (in this sense, there is not just one root universe, but one per item). So consider this function bar:

struct Foo { }

fn bar<'a, T>(t: &'a T) {

Here, the root universe would consist of the lifetimes 'static and 'a. In fact, although we're focused on lifetimes here, we can apply the same concept to types, in which case the types Foo and T would be in the root universe (along with other global types, like i32). Basically, the root universe contains all the names that appear free in the body of bar.

Now let's extend bar a bit by adding a variable x:

fn bar<'a, T>(t: &'a T) {
    let x: for<'b> fn(&'b u32) = ...;

Here, the name 'b is not part of the root universe. Instead, when we "enter" into this for<'b> (e.g., by replacing it with a placeholder), we will create a child universe of the root, let's call it U1:

U0 (root universe)
└─ U1 (child universe)

The idea is that this child universe U1 extends the root universe U0 with a new name, which we are identifying by its universe number: !1.

Now let's extend bar a bit by adding one more variable, y:

fn bar<'a, T>(t: &'a T) {
    let x: for<'b> fn(&'b u32) = ...;
    let y: for<'c> fn(&'c u32) = ...;

When we enter this type, we will again create a new universe, which we'll call U2. Its parent will be the root universe, and U1 will be its sibling:

U0 (root universe)
├─ U1 (child universe)
└─ U2 (child universe)

This implies that, while in U2, we can name things from U0 or U2, but not U1.

Giving existential variables a universe. Now that we have this notion of universes, we can use it to extend our type-checker and things to prevent illegal names from leaking out. The idea is that we give each inference (existential) variable – whether it be a type or a lifetime – a universe. That variable's value can then only reference names visible from that universe. So for example if a lifetime variable is created in U0, then it cannot be assigned a value of !1 or !2, because those names are not visible from the universe U0.

Representing universes with just a counter. You might be surprised to see that the compiler doesn't keep track of a full tree of universes. Instead, it just keeps a counter – and, to determine if one universe can see another one, it just checks if the index is greater. For example, U2 can see U0 because 2 >= 0. But U0 cannot see U2, because 0 >= 2 is false.

How can we get away with this? Doesn't this mean that we would allow U2 to also see U1? The answer is that, yes, we would, if that question ever arose. But because of the structure of our type checker etc, there is no way for that to happen. In order for something happening in the universe U1 to "communicate" with something happening in U2, they would have to have a shared inference variable X in common. And because everything in U1 is scoped to just U1 and its children, that inference variable X would have to be in U0. And since X is in U0, it cannot name anything from U1 (or U2). This is perhaps easiest to see by using a kind of generic "logic" example:

exists<X> {
   forall<Y> { ... /* Y is in U1 ... */ }
   forall<Z> { ... /* Z is in U2 ... */ }

Here, the only way for the two foralls to interact would be through X, but neither Y nor Z are in scope when X is declared, so its value cannot reference either of them.

Universes and placeholder region elements

But where does that error come from? The way it happens is like this. When we are constructing the region inference context, we can tell from the type inference context how many placeholder variables exist (the InferCtxt has an internal counter). For each of those, we create a corresponding universal region variable !n and a "region element" placeholder(n). This corresponds to "some unknown set of other elements". The value of !n is {placeholder(n)}.

At the same time, we also give each existential variable a universe (also taken from the InferCtxt). This universe determines which placeholder elements may appear in its value: For example, a variable in universe U3 may name placeholder(1), placeholder(2), and placeholder(3), but not placeholder(4). Note that the universe of an inference variable controls what region elements can appear in its value; it does not say region elements will appear.

Placeholders and outlives constraints

In the region inference engine, outlives constraints have the form:

V1: V2 @ P

where V1 and V2 are region indices, and hence map to some region variable (which may be universally or existentially quantified). The P here is a "point" in the control-flow graph; it's not important for this section. This variable will have a universe, so let's call those universes U(V1) and U(V2) respectively. (Actually, the only one we are going to care about is U(V1).)

When we encounter this constraint, the ordinary procedure is to start a DFS from P. We keep walking so long as the nodes we are walking are present in value(V2) and we add those nodes to value(V1). If we reach a return point, we add in any end(X) elements. That part remains unchanged.

But then after that we want to iterate over the placeholder placeholder(x) elements in V2 (each of those must be visible to U(V2), but we should be able to just assume that is true, we don't have to check it). We have to ensure that value(V1) outlives each of those placeholder elements.

Now there are two ways that could happen. First, if U(V1) can see the universe x (i.e., x <= U(V1)), then we can just add placeholder(x) to value(V1) and be done. But if not, then we have to approximate: we may not know what set of elements placeholder(x) represents, but we should be able to compute some sort of upper bound B for it – some region B that outlives placeholder(x). For now, we'll just use 'static for that (since it outlives everything) – in the future, we can sometimes be smarter here (and in fact we have code for doing this already in other contexts). Moreover, since 'static is in the root universe U0, we know that all variables can see it – so basically if we find that value(V2) contains placeholder(x) for some universe x that V1 can't see, then we force V1 to 'static.

Extending the "universal regions" check

After all constraints have been propagated, the NLL region inference has one final check, where it goes over the values that wound up being computed for each universal region and checks that they did not get 'too large'. In our case, we will go through each placeholder region and check that it contains only the placeholder(u) element it is known to outlive. (Later, we might be able to know that there are relationships between two placeholder regions and take those into account, as we do for universal regions from the fn signature.)

Put another way, the "universal regions" check can be considered to be checking constraints like:

{placeholder(1)}: V1

where {placeholder(1)} is like a constant set, and V1 is the variable we made to represent the !1 region.

Back to our example

OK, so far so good. Now let's walk through what would happen with our first example:

fn(&'static u32) <: fn(&'!1 u32) @ P  // this point P is not imp't here

The region inference engine will create a region element domain like this:

{ CFG; end('static); placeholder(1) }
  ---  ------------  ------- from the universe `!1`
  |    'static is always in scope
  all points in the CFG; not especially relevant here

It will always create two universal variables, one representing 'static and one representing '!1. Let's call them Vs and V1. They will have initial values like so:

Vs = { CFG; end('static) } // it is in U0, so can't name anything else
V1 = { placeholder(1) }

From the subtyping constraint above, we would have an outlives constraint like

'!1: 'static @ P

To process this, we would grow the value of V1 to include all of Vs:

Vs = { CFG; end('static) }
V1 = { CFG; end('static), placeholder(1) }

At that point, constraint propagation is complete, because all the outlives relationships are satisfied. Then we would go to the "check universal regions" portion of the code, which would test that no universal region grew too large.

In this case, V1 did grow too large – it is not known to outlive end('static), nor any of the CFG – so we would report an error.

Another example

What about this subtyping relationship?

for<'a> fn(&'a u32, &'a u32)
for<'b, 'c> fn(&'b u32, &'c u32)

Here we would replace the bound region in the supertype with a placeholder, as before, yielding:

for<'a> fn(&'a u32, &'a u32)
fn(&'!1 u32, &'!2 u32)

then we instantiate the variable on the left-hand side with an existential in universe U2, yielding the following (?n is a notation for an existential variable):

fn(&'?3 u32, &'?3 u32)
fn(&'!1 u32, &'!2 u32)

Then we break this down further:

&'!1 u32 <: &'?3 u32
&'!2 u32 <: &'?3 u32

and even further, yield up our region constraints:

'!1: '?3
'!2: '?3

Note that, in this case, both '!1 and '!2 have to outlive the variable '?3, but the variable '?3 is not forced to outlive anything else. Therefore, it simply starts and ends as the empty set of elements, and hence the type-check succeeds here.

(This should surprise you a little. It surprised me when I first realized it. We are saying that if we are a fn that needs both of its arguments to have the same region, we can accept being called with arguments with two distinct regions. That seems intuitively unsound. But in fact, it's fine, as I tried to explain in this issue on the Rust issue tracker long ago. The reason is that even if we get called with arguments of two distinct lifetimes, those two lifetimes have some intersection (the call itself), and that intersection can be our value of 'a that we use as the common lifetime of our arguments. -nmatsakis)

Final example

Let's look at one last example. We'll extend the previous one to have a return type:

for<'a> fn(&'a u32, &'a u32) -> &'a u32
for<'b, 'c> fn(&'b u32, &'c u32) -> &'b u32

Despite seeming very similar to the previous example, this case is going to get an error. That's good: the problem is that we've gone from a fn that promises to return one of its two arguments, to a fn that is promising to return the first one. That is unsound. Let's see how it plays out.

First, we replace the bound region in the supertype with a placeholder:

for<'a> fn(&'a u32, &'a u32) -> &'a u32
fn(&'!1 u32, &'!2 u32) -> &'!1 u32

Then we instantiate the subtype with existentials (in U2):

fn(&'?3 u32, &'?3 u32) -> &'?3 u32
fn(&'!1 u32, &'!2 u32) -> &'!1 u32

And now we create the subtyping relationships:

&'!1 u32 <: &'?3 u32 // arg 1
&'!2 u32 <: &'?3 u32 // arg 2
&'?3 u32 <: &'!1 u32 // return type

And finally the outlives relationships. Here, let V1, V2, and V3 be the variables we assign to !1, !2, and ?3 respectively:

V1: V3
V2: V3
V3: V1

Those variables will have these initial values:

V1 in U1 = {placeholder(1)}
V2 in U2 = {placeholder(2)}
V3 in U2 = {}

Now because of the V3: V1 constraint, we have to add placeholder(1) into V3 (and indeed it is visible from V3), so we get:

V3 in U2 = {placeholder(1)}

then we have this constraint V2: V3, so we wind up having to enlarge V2 to include placeholder(1) (which it can also see):

V2 in U2 = {placeholder(1), placeholder(2)}

Now constraint propagation is done, but when we check the outlives relationships, we find that V2 includes this new element placeholder(1), so we report an error.