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// Copyright 2012 The Rust Project Developers. See the COPYRIGHT
// file at the top-level directory of this distribution and at
// http://rust-lang.org/COPYRIGHT.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.
/*!
Region inference module.
# Introduction
Region inference uses a somewhat more involved algorithm than type
inference. It is not the most efficient thing ever written though it
seems to work well enough in practice (famous last words). The reason
that we use a different algorithm is because, unlike with types, it is
impractical to hand-annotate with regions (in some cases, there aren't
even the requisite syntactic forms). So we have to get it right, and
it's worth spending more time on a more involved analysis. Moreover,
regions are a simpler case than types: they don't have aggregate
structure, for example.
Unlike normal type inference, which is similar in spirit to H-M and thus
works progressively, the region type inference works by accumulating
constraints over the course of a function. Finally, at the end of
processing a function, we process and solve the constraints all at
once.
The constraints are always of one of three possible forms:
- ConstrainVarSubVar(R_i, R_j) states that region variable R_i
must be a subregion of R_j
- ConstrainRegSubVar(R, R_i) states that the concrete region R
(which must not be a variable) must be a subregion of the varibale R_i
- ConstrainVarSubReg(R_i, R) is the inverse
# Building up the constraints
Variables and constraints are created using the following methods:
- `new_region_var()` creates a new, unconstrained region variable;
- `make_subregion(R_i, R_j)` states that R_i is a subregion of R_j
- `lub_regions(R_i, R_j) -> R_k` returns a region R_k which is
the smallest region that is greater than both R_i and R_j
- `glb_regions(R_i, R_j) -> R_k` returns a region R_k which is
the greatest region that is smaller than both R_i and R_j
The actual region resolution algorithm is not entirely
obvious, though it is also not overly complex. I'll explain
the algorithm as it currently works, then explain a somewhat
more complex variant that would probably scale better for
large graphs (and possibly all graphs).
## Snapshotting
It is also permitted to try (and rollback) changes to the graph. This
is done by invoking `start_snapshot()`, which returns a value. Then
later you can call `rollback_to()` which undoes the work.
Alternatively, you can call `commit()` which ends all snapshots.
Snapshots can be recursive---so you can start a snapshot when another
is in progress, but only the root snapshot can "commit".
# Resolving constraints
The constraint resolution algorithm is not super complex but also not
entirely obvious. Here I describe the problem somewhat abstractly,
then describe how the current code works, and finally describe a
better solution that is as of yet unimplemented. There may be other,
smarter ways of doing this with which I am unfamiliar and can't be
bothered to research at the moment. - NDM
## The problem
Basically our input is a directed graph where nodes can be divided
into two categories: region variables and concrete regions. Each edge
`R -> S` in the graph represents a constraint that the region `R` is a
subregion of the region `S`.
Region variable nodes can have arbitrary degree. There is one region
variable node per region variable.
Each concrete region node is associated with some, well, concrete
region: e.g., a free lifetime, or the region for a particular scope.
Note that there may be more than one concrete region node for a
particular region value. Moreover, because of how the graph is built,
we know that all concrete region nodes have either in-degree 1 or
out-degree 1.
Before resolution begins, we build up the constraints in a hashmap
that maps `Constraint` keys to spans. During resolution, we construct
the actual `Graph` structure that we describe here.
## Our current algorithm
We divide region variables into two groups: Expanding and Contracting.
Expanding region variables are those that have a concrete region
predecessor (direct or indirect). Contracting region variables are
all others.
We first resolve the values of Expanding region variables and then
process Contracting ones. We currently use an iterative, fixed-point
procedure (but read on, I believe this could be replaced with a linear
walk). Basically we iterate over the edges in the graph, ensuring
that, if the source of the edge has a value, then this value is a
subregion of the target value. If the target does not yet have a
value, it takes the value from the source. If the target already had
a value, then the resulting value is Least Upper Bound of the old and
new values. When we are done, each Expanding node will have the
smallest region that it could possibly have and still satisfy the
constraints.
We next process the Contracting nodes. Here we again iterate over the
edges, only this time we move values from target to source (if the
source is a Contracting node). For each contracting node, we compute
its value as the GLB of all its successors. Basically contracting
nodes ensure that there is overlap between their successors; we will
ultimately infer the largest overlap possible.
### A better algorithm
Fixed-point iteration is not necessary. What we ought to do is first
identify and remove strongly connected components (SCC) in the graph.
Note that such components must consist solely of region variables; all
of these variables can effectively be unified into a single variable.
Once SCCs are removed, we are left with a DAG. At this point, we can
walk the DAG in toplogical order once to compute the expanding nodes,
and again in reverse topological order to compute the contracting
nodes. The main reason I did not write it this way is that I did not
feel like implementing the SCC and toplogical sort algorithms at the
moment.
# Skolemization and functions
One of the trickiest and most subtle aspects of regions is dealing
with the fact that region variables are bound in function types. I
strongly suggest that if you want to understand the situation, you
read this paper (which is, admittedly, very long, but you don't have
to read the whole thing):
http://research.microsoft.com/en-us/um/people/simonpj/papers/higher-rank/
Although my explanation will never compete with SPJ's (for one thing,
his is approximately 100 pages), I will attempt to explain the basic
problem and also how we solve it. Note that the paper only discusses
subtyping, not the computation of LUB/GLB.
The problem we are addressing is that there is a kind of subtyping
between functions with bound region parameters. Consider, for
example, whether the following relation holds:
fn(&'a int) <: &fn(&'b int)? (Yes, a => b)
The answer is that of course it does. These two types are basically
the same, except that in one we used the name `a` and one we used
the name `b`.
In the examples that follow, it becomes very important to know whether
a lifetime is bound in a function type (that is, is a lifetime
parameter) or appears free (is defined in some outer scope).
Therefore, from now on I will write the bindings explicitly, using a
notation like `fn<a>(&'a int)` to indicate that `a` is a lifetime
parameter.
Now let's consider two more function types. Here, we assume that the
`self` lifetime is defined somewhere outside and hence is not a
lifetime parameter bound by the function type (it "appears free"):
fn<a>(&'a int) <: &fn(&'self int)? (Yes, a => self)
This subtyping relation does in fact hold. To see why, you have to
consider what subtyping means. One way to look at `T1 <: T2` is to
say that it means that it is always ok to treat an instance of `T1` as
if it had the type `T2`. So, with our functions, it is always ok to
treat a function that can take pointers with any lifetime as if it
were a function that can only take a pointer with the specific
lifetime `&self`. After all, `&self` is a lifetime, after all, and
the function can take values of any lifetime.
You can also look at subtyping as the *is a* relationship. This amounts
to the same thing: a function that accepts pointers with any lifetime
*is a* function that accepts pointers with some specific lifetime.
So, what if we reverse the order of the two function types, like this:
fn(&'self int) <: &fn<a>(&'a int)? (No)
Does the subtyping relationship still hold? The answer of course is
no. In this case, the function accepts *only the lifetime `&self`*,
so it is not reasonable to treat it as if it were a function that
accepted any lifetime.
What about these two examples:
fn<a,b>(&'a int, &'b int) <: &fn<a>(&'a int, &'a int)? (Yes)
fn<a>(&'a int, &'a int) <: &fn<a,b>(&'a int, &'b int)? (No)
Here, it is true that functions which take two pointers with any two
lifetimes can be treated as if they only accepted two pointers with
the same lifetime, but not the reverse.
## The algorithm
Here is the algorithm we use to perform the subtyping check:
1. Replace all bound regions in the subtype with new variables
2. Replace all bound regions in the supertype with skolemized
equivalents. A "skolemized" region is just a new fresh region
name.
3. Check that the parameter and return types match as normal
4. Ensure that no skolemized regions 'leak' into region variables
visible from "the outside"
Let's walk through some examples and see how this algorithm plays out.
#### First example
We'll start with the first example, which was:
1. fn<a>(&'a T) <: &fn<b>(&'b T)? Yes: a -> b
After steps 1 and 2 of the algorithm we will have replaced the types
like so:
1. fn(&'A T) <: &fn(&'x T)?
Here the upper case `&A` indicates a *region variable*, that is, a
region whose value is being inferred by the system. I also replaced
`&b` with `&x`---I'll use letters late in the alphabet (`x`, `y`, `z`)
to indicate skolemized region names. We can assume they don't appear
elsewhere. Note that neither the sub- nor the supertype bind any
region names anymore (as indicated by the absence of `<` and `>`).
The next step is to check that the parameter types match. Because
parameters are contravariant, this means that we check whether:
&'x T <: &'A T
Region pointers are contravariant so this implies that
&A <= &x
must hold, where `<=` is the subregion relationship. Processing
*this* constrain simply adds a constraint into our graph that `&A <=
&x` and is considered successful (it can, for example, be satisfied by
choosing the value `&x` for `&A`).
So far we have encountered no error, so the subtype check succeeds.
#### The third example
Now let's look first at the third example, which was:
3. fn(&'self T) <: &fn<b>(&'b T)? No!
After steps 1 and 2 of the algorithm we will have replaced the types
like so:
3. fn(&'self T) <: &fn(&'x T)?
This looks pretty much the same as before, except that on the LHS
`&self` was not bound, and hence was left as-is and not replaced with
a variable. The next step is again to check that the parameter types
match. This will ultimately require (as before) that `&self` <= `&x`
must hold: but this does not hold. `self` and `x` are both distinct
free regions. So the subtype check fails.
#### Checking for skolemization leaks
You may be wondering about that mysterious last step in the algorithm.
So far it has not been relevant. The purpose of that last step is to
catch something like *this*:
fn<a>() -> fn(&'a T) <: &fn() -> fn<b>(&'b T)? No.
Here the function types are the same but for where the binding occurs.
The subtype returns a function that expects a value in precisely one
region. The supertype returns a function that expects a value in any
region. If we allow an instance of the subtype to be used where the
supertype is expected, then, someone could call the fn and think that
the return value has type `fn<b>(&'b T)` when it really has type
`fn(&'a T)` (this is case #3, above). Bad.
So let's step through what happens when we perform this subtype check.
We first replace the bound regions in the subtype (the supertype has
no bound regions). This gives us:
fn() -> fn(&'A T) <: &fn() -> fn<b>(&'b T)?
Now we compare the return types, which are covariant, and hence we have:
fn(&'A T) <: &fn<b>(&'b T)?
Here we skolemize the bound region in the supertype to yield:
fn(&'A T) <: &fn(&'x T)?
And then proceed to compare the argument types:
&'x T <: &'A T
&A <= &x
Finally, this is where it gets interesting! This is where an error
*should* be reported. But in fact this will not happen. The reason why
is that `A` is a variable: we will infer that its value is the fresh
region `x` and think that everything is happy. In fact, this behavior
is *necessary*, it was key to the first example we walked through.
The difference between this example and the first one is that the variable
`A` already existed at the point where the skolemization occurred. In
the first example, you had two functions:
fn<a>(&'a T) <: &fn<b>(&'b T)
and hence `&A` and `&x` were created "together". In general, the
intention of the skolemized names is that they are supposed to be
fresh names that could never be equal to anything from the outside.
But when inference comes into play, we might not be respecting this
rule.
So the way we solve this is to add a fourth step that examines the
constraints that refer to skolemized names. Basically, consider a
non-directed verison of the constraint graph. Let `Tainted(x)` be the
set of all things reachable from a skolemized variable `x`.
`Tainted(x)` should not contain any regions that existed before the
step at which the skolemization was performed. So this case here
would fail because `&x` was created alone, but is relatable to `&A`.
## Computing the LUB and GLB
The paper I pointed you at is written for Haskell. It does not
therefore considering subtyping and in particular does not consider
LUB or GLB computation. We have to consider this. Here is the
algorithm I implemented.
First though, let's discuss what we are trying to compute in more
detail. The LUB is basically the "common supertype" and the GLB is
"common subtype"; one catch is that the LUB should be the
*most-specific* common supertype and the GLB should be *most general*
common subtype (as opposed to any common supertype or any common
subtype).
Anyway, to help clarify, here is a table containing some
function pairs and their LUB/GLB:
```
Type 1 Type 2 LUB GLB
fn<a>(&a) fn(&X) fn(&X) fn<a>(&a)
fn(&A) fn(&X) -- fn<a>(&a)
fn<a,b>(&a, &b) fn<x>(&x, &x) fn<a>(&a, &a) fn<a,b>(&a, &b)
fn<a,b>(&a, &b, &a) fn<x,y>(&x, &y, &y) fn<a>(&a, &a, &a) fn<a,b,c>(&a,&b,&c)
```
### Conventions
I use lower-case letters (e.g., `&a`) for bound regions and upper-case
letters for free regions (`&A`). Region variables written with a
dollar-sign (e.g., `$a`). I will try to remember to enumerate the
bound-regions on the fn type as well (e.g., `fn<a>(&a)`).
### High-level summary
Both the LUB and the GLB algorithms work in a similar fashion. They
begin by replacing all bound regions (on both sides) with fresh region
inference variables. Therefore, both functions are converted to types
that contain only free regions. We can then compute the LUB/GLB in a
straightforward way, as described in `combine.rs`. This results in an
interim type T. The algorithms then examine the regions that appear
in T and try to, in some cases, replace them with bound regions to
yield the final result.
To decide whether to replace a region `R` that appears in `T` with a
bound region, the algorithms make use of two bits of information.
First is a set `V` that contains all region variables created as part
of the LUB/GLB computation. `V` will contain the region variables
created to replace the bound regions in the input types, but it also
contains 'intermediate' variables created to represent the LUB/GLB of
individual regions. Basically, when asked to compute the LUB/GLB of a
region variable with another region, the inferencer cannot oblige
immediately since the valuese of that variables are not known.
Therefore, it creates a new variable that is related to the two
regions. For example, the LUB of two variables `$x` and `$y` is a
fresh variable `$z` that is constrained such that `$x <= $z` and `$y
<= $z`. So `V` will contain these intermediate variables as well.
The other important factor in deciding how to replace a region in T is
the function `Tainted($r)` which, for a region variable, identifies
all regions that the region variable is related to in some way
(`Tainted()` made an appearance in the subtype computation as well).
### LUB
The LUB algorithm proceeds in three steps:
1. Replace all bound regions (on both sides) with fresh region
inference variables.
2. Compute the LUB "as normal", meaning compute the GLB of each
pair of argument types and the LUB of the return types and
so forth. Combine those to a new function type `F`.
3. Replace each region `R` that appears in `F` as follows:
- Let `V` be the set of variables created during the LUB
computational steps 1 and 2, as described in the previous section.
- If `R` is not in `V`, replace `R` with itself.
- If `Tainted(R)` contains a region that is not in `V`,
replace `R` with itself.
- Otherwise, select the earliest variable in `Tainted(R)` that originates
from the left-hand side and replace `R` with the bound region that
this variable was a replacement for.
So, let's work through the simplest example: `fn(&A)` and `fn<a>(&a)`.
In this case, `&a` will be replaced with `$a` and the interim LUB type
`fn($b)` will be computed, where `$b=GLB(&A,$a)`. Therefore, `V =
{$a, $b}` and `Tainted($b) = { $b, $a, &A }`. When we go to replace
`$b`, we find that since `&A \in Tainted($b)` is not a member of `V`,
we leave `$b` as is. When region inference happens, `$b` will be
resolved to `&A`, as we wanted.
Let's look at a more complex one: `fn(&a, &b)` and `fn(&x, &x)`. In
this case, we'll end up with a (pre-replacement) LUB type of `fn(&g,
&h)` and a graph that looks like:
```
$a $b *--$x
\ \ / /
\ $h-* /
$g-----------*
```
Here `$g` and `$h` are fresh variables that are created to represent
the LUB/GLB of things requiring inference. This means that `V` and
`Tainted` will look like:
```
V = {$a, $b, $g, $h, $x}
Tainted($g) = Tainted($h) = { $a, $b, $h, $g, $x }
```
Therefore we replace both `$g` and `$h` with `$a`, and end up
with the type `fn(&a, &a)`.
### GLB
The procedure for computing the GLB is similar. The difference lies
in computing the replacements for the various variables. For each
region `R` that appears in the type `F`, we again compute `Tainted(R)`
and examine the results:
1. If `R` is not in `V`, it is not replaced.
2. Else, if `Tainted(R)` contains only variables in `V`, and it
contains exactly one variable from the LHS and one variable from
the RHS, then `R` can be mapped to the bound version of the
variable from the LHS.
3. Else, if `Tainted(R)` contains no variable from the LHS and no
variable from the RHS, then `R` can be mapped to itself.
4. Else, `R` is mapped to a fresh bound variable.
These rules are pretty complex. Let's look at some examples to see
how they play out.
Out first example was `fn(&a)` and `fn(&X)`. In this case, `&a` will
be replaced with `$a` and we will ultimately compute a
(pre-replacement) GLB type of `fn($g)` where `$g=LUB($a,&X)`.
Therefore, `V={$a,$g}` and `Tainted($g)={$g,$a,&X}. To find the
replacement for `$g` we consult the rules above:
- Rule (1) does not apply because `$g \in V`
- Rule (2) does not apply because `&X \in Tainted($g)`
- Rule (3) does not apply because `$a \in Tainted($g)`
- Hence, by rule (4), we replace `$g` with a fresh bound variable `&z`.
So our final result is `fn(&z)`, which is correct.
The next example is `fn(&A)` and `fn(&Z)`. In this case, we will again
have a (pre-replacement) GLB of `fn(&g)`, where `$g = LUB(&A,&Z)`.
Therefore, `V={$g}` and `Tainted($g) = {$g, &A, &Z}`. In this case,
by rule (3), `$g` is mapped to itself, and hence the result is
`fn($g)`. This result is correct (in this case, at least), but it is
indicative of a case that *can* lead us into concluding that there is
no GLB when in fact a GLB does exist. See the section "Questionable
Results" below for more details.
The next example is `fn(&a, &b)` and `fn(&c, &c)`. In this case, as
before, we'll end up with `F=fn($g, $h)` where `Tainted($g) =
Tainted($h) = {$g, $h, $a, $b, $c}`. Only rule (4) applies and hence
we'll select fresh bound variables `y` and `z` and wind up with
`fn(&y, &z)`.
For the last example, let's consider what may seem trivial, but is
not: `fn(&a, &a)` and `fn(&b, &b)`. In this case, we'll get `F=fn($g,
$h)` where `Tainted($g) = {$g, $a, $x}` and `Tainted($h) = {$h, $a,
$x}`. Both of these sets contain exactly one bound variable from each
side, so we'll map them both to `&a`, resulting in `fn(&a, &a)`, which
is the desired result.
### Shortcomings and correctness
You may be wondering whether this algorithm is correct. The answer is
"sort of". There are definitely cases where they fail to compute a
result even though a correct result exists. I believe, though, that
if they succeed, then the result is valid, and I will attempt to
convince you. The basic argument is that the "pre-replacement" step
computes a set of constraints. The replacements, then, attempt to
satisfy those constraints, using bound identifiers where needed.
For now I will briefly go over the cases for LUB/GLB and identify
their intent:
- LUB:
- The region variables that are substituted in place of bound regions
are intended to collect constraints on those bound regions.
- If Tainted(R) contains only values in V, then this region is unconstrained
and can therefore be generalized, otherwise it cannot.
- GLB:
- The region variables that are substituted in place of bound regions
are intended to collect constraints on those bound regions.
- If Tainted(R) contains exactly one variable from each side, and
only variables in V, that indicates that those two bound regions
must be equated.
- Otherwise, if Tainted(R) references any variables from left or right
side, then it is trying to combine a bound region with a free one or
multiple bound regions, so we need to select fresh bound regions.
Sorry this is more of a shorthand to myself. I will try to write up something
more convincing in the future.
#### Where are the algorithms wrong?
- The pre-replacement computation can fail even though using a
bound-region would have succeeded.
- We will compute GLB(fn(fn($a)), fn(fn($b))) as fn($c) where $c is the
GLB of $a and $b. But if inference finds that $a and $b must be mapped
to regions without a GLB, then this is effectively a failure to compute
the GLB. However, the result `fn<$c>(fn($c))` is a valid GLB.
*/

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src/librustc/middle/typeck/infer/region_inference/mod.rs
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// option. This file may not be copied, modified, or distributed
// except according to those terms.
/*!
Region inference module.
# Introduction
Region inference uses a somewhat more involved algorithm than type
inference. It is not the most efficient thing ever written though it
seems to work well enough in practice (famous last words). The reason
that we use a different algorithm is because, unlike with types, it is
impractical to hand-annotate with regions (in some cases, there aren't
even the requisite syntactic forms). So we have to get it right, and
it's worth spending more time on a more involved analysis. Moreover,
regions are a simpler case than types: they don't have aggregate
structure, for example.
Unlike normal type inference, which is similar in spirit to H-M and thus
works progressively, the region type inference works by accumulating
constraints over the course of a function. Finally, at the end of
processing a function, we process and solve the constraints all at
once.
The constraints are always of one of three possible forms:
- ConstrainVarSubVar(R_i, R_j) states that region variable R_i
must be a subregion of R_j
- ConstrainRegSubVar(R, R_i) states that the concrete region R
(which must not be a variable) must be a subregion of the varibale R_i
- ConstrainVarSubReg(R_i, R) is the inverse
# Building up the constraints
Variables and constraints are created using the following methods:
- `new_region_var()` creates a new, unconstrained region variable;
- `make_subregion(R_i, R_j)` states that R_i is a subregion of R_j
- `lub_regions(R_i, R_j) -> R_k` returns a region R_k which is
the smallest region that is greater than both R_i and R_j
- `glb_regions(R_i, R_j) -> R_k` returns a region R_k which is
the greatest region that is smaller than both R_i and R_j
The actual region resolution algorithm is not entirely
obvious, though it is also not overly complex. I'll explain
the algorithm as it currently works, then explain a somewhat
more complex variant that would probably scale better for
large graphs (and possibly all graphs).
## Snapshotting
It is also permitted to try (and rollback) changes to the graph. This
is done by invoking `start_snapshot()`, which returns a value. Then
later you can call `rollback_to()` which undoes the work.
Alternatively, you can call `commit()` which ends all snapshots.
Snapshots can be recursive---so you can start a snapshot when another
is in progress, but only the root snapshot can "commit".
# Resolving constraints
The constraint resolution algorithm is not super complex but also not
entirely obvious. Here I describe the problem somewhat abstractly,
then describe how the current code works, and finally describe a
better solution that is as of yet unimplemented. There may be other,
smarter ways of doing this with which I am unfamiliar and can't be
bothered to research at the moment. - NDM
## The problem
Basically our input is a directed graph where nodes can be divided
into two categories: region variables and concrete regions. Each edge
`R -> S` in the graph represents a constraint that the region `R` is a
subregion of the region `S`.
Region variable nodes can have arbitrary degree. There is one region
variable node per region variable.
Each concrete region node is associated with some, well, concrete
region: e.g., a free lifetime, or the region for a particular scope.
Note that there may be more than one concrete region node for a
particular region value. Moreover, because of how the graph is built,
we know that all concrete region nodes have either in-degree 1 or
out-degree 1.
Before resolution begins, we build up the constraints in a hashmap
that maps `Constraint` keys to spans. During resolution, we construct
the actual `Graph` structure that we describe here.
## Our current algorithm
We divide region variables into two groups: Expanding and Contracting.
Expanding region variables are those that have a concrete region
predecessor (direct or indirect). Contracting region variables are
all others.
We first resolve the values of Expanding region variables and then
process Contracting ones. We currently use an iterative, fixed-point
procedure (but read on, I believe this could be replaced with a linear
walk). Basically we iterate over the edges in the graph, ensuring
that, if the source of the edge has a value, then this value is a
subregion of the target value. If the target does not yet have a
value, it takes the value from the source. If the target already had
a value, then the resulting value is Least Upper Bound of the old and
new values. When we are done, each Expanding node will have the
smallest region that it could possibly have and still satisfy the
constraints.
We next process the Contracting nodes. Here we again iterate over the
edges, only this time we move values from target to source (if the
source is a Contracting node). For each contracting node, we compute
its value as the GLB of all its successors. Basically contracting
nodes ensure that there is overlap between their successors; we will
ultimately infer the largest overlap possible.
### A better algorithm
Fixed-point iteration is not necessary. What we ought to do is first
identify and remove strongly connected components (SCC) in the graph.
Note that such components must consist solely of region variables; all
of these variables can effectively be unified into a single variable.
Once SCCs are removed, we are left with a DAG. At this point, we can
walk the DAG in toplogical order once to compute the expanding nodes,
and again in reverse topological order to compute the contracting
nodes. The main reason I did not write it this way is that I did not
feel like implementing the SCC and toplogical sort algorithms at the
moment.
# Skolemization and functions
One of the trickiest and most subtle aspects of regions is dealing
with the fact that region variables are bound in function types. I
strongly suggest that if you want to understand the situation, you
read this paper (which is, admittedly, very long, but you don't have
to read the whole thing):
http://research.microsoft.com/en-us/um/people/simonpj/papers/higher-rank/
Although my explanation will never compete with SPJ's (for one thing,
his is approximately 100 pages), I will attempt to explain the basic
problem and also how we solve it. Note that the paper only discusses
subtyping, not the computation of LUB/GLB.
The problem we are addressing is that there is a kind of subtyping
between functions with bound region parameters. Consider, for
example, whether the following relation holds:
fn(&'a int) <: &fn(&'b int)? (Yes, a => b)
The answer is that of course it does. These two types are basically
the same, except that in one we used the name `a` and one we used
the name `b`.
In the examples that follow, it becomes very important to know whether
a lifetime is bound in a function type (that is, is a lifetime
parameter) or appears free (is defined in some outer scope).
Therefore, from now on I will write the bindings explicitly, using a
notation like `fn<a>(&'a int)` to indicate that `a` is a lifetime
parameter.
Now let's consider two more function types. Here, we assume that the
`self` lifetime is defined somewhere outside and hence is not a
lifetime parameter bound by the function type (it "appears free"):
fn<a>(&'a int) <: &fn(&'self int)? (Yes, a => self)
This subtyping relation does in fact hold. To see why, you have to
consider what subtyping means. One way to look at `T1 <: T2` is to
say that it means that it is always ok to treat an instance of `T1` as
if it had the type `T2`. So, with our functions, it is always ok to
treat a function that can take pointers with any lifetime as if it
were a function that can only take a pointer with the specific
lifetime `&self`. After all, `&self` is a lifetime, after all, and
the function can take values of any lifetime.
You can also look at subtyping as the *is a* relationship. This amounts
to the same thing: a function that accepts pointers with any lifetime
*is a* function that accepts pointers with some specific lifetime.
So, what if we reverse the order of the two function types, like this:
fn(&'self int) <: &fn<a>(&'a int)? (No)
Does the subtyping relationship still hold? The answer of course is
no. In this case, the function accepts *only the lifetime `&self`*,
so it is not reasonable to treat it as if it were a function that
accepted any lifetime.
What about these two examples:
fn<a,b>(&'a int, &'b int) <: &fn<a>(&'a int, &'a int)? (Yes)
fn<a>(&'a int, &'a int) <: &fn<a,b>(&'a int, &'b int)? (No)
Here, it is true that functions which take two pointers with any two
lifetimes can be treated as if they only accepted two pointers with
the same lifetime, but not the reverse.
## The algorithm
Here is the algorithm we use to perform the subtyping check:
1. Replace all bound regions in the subtype with new variables
2. Replace all bound regions in the supertype with skolemized
equivalents. A "skolemized" region is just a new fresh region
name.
3. Check that the parameter and return types match as normal
4. Ensure that no skolemized regions 'leak' into region variables
visible from "the outside"
Let's walk through some examples and see how this algorithm plays out.
#### First example
We'll start with the first example, which was:
1. fn<a>(&'a T) <: &fn<b>(&'b T)? Yes: a -> b
After steps 1 and 2 of the algorithm we will have replaced the types
like so:
1. fn(&'A T) <: &fn(&'x T)?
Here the upper case `&A` indicates a *region variable*, that is, a
region whose value is being inferred by the system. I also replaced
`&b` with `&x`---I'll use letters late in the alphabet (`x`, `y`, `z`)
to indicate skolemized region names. We can assume they don't appear
elsewhere. Note that neither the sub- nor the supertype bind any
region names anymore (as indicated by the absence of `<` and `>`).
The next step is to check that the parameter types match. Because
parameters are contravariant, this means that we check whether:
&'x T <: &'A T
Region pointers are contravariant so this implies that
&A <= &x
must hold, where `<=` is the subregion relationship. Processing
*this* constrain simply adds a constraint into our graph that `&A <=
&x` and is considered successful (it can, for example, be satisfied by
choosing the value `&x` for `&A`).
So far we have encountered no error, so the subtype check succeeds.
#### The third example
Now let's look first at the third example, which was:
3. fn(&'self T) <: &fn<b>(&'b T)? No!
After steps 1 and 2 of the algorithm we will have replaced the types
like so:
3. fn(&'self T) <: &fn(&'x T)?
This looks pretty much the same as before, except that on the LHS
`&self` was not bound, and hence was left as-is and not replaced with
a variable. The next step is again to check that the parameter types
match. This will ultimately require (as before) that `&self` <= `&x`
must hold: but this does not hold. `self` and `x` are both distinct
free regions. So the subtype check fails.
#### Checking for skolemization leaks
You may be wondering about that mysterious last step in the algorithm.
So far it has not been relevant. The purpose of that last step is to
catch something like *this*:
fn<a>() -> fn(&'a T) <: &fn() -> fn<b>(&'b T)? No.
Here the function types are the same but for where the binding occurs.
The subtype returns a function that expects a value in precisely one
region. The supertype returns a function that expects a value in any
region. If we allow an instance of the subtype to be used where the
supertype is expected, then, someone could call the fn and think that
the return value has type `fn<b>(&'b T)` when it really has type
`fn(&'a T)` (this is case #3, above). Bad.
So let's step through what happens when we perform this subtype check.
We first replace the bound regions in the subtype (the supertype has
no bound regions). This gives us:
fn() -> fn(&'A T) <: &fn() -> fn<b>(&'b T)?
Now we compare the return types, which are covariant, and hence we have:
fn(&'A T) <: &fn<b>(&'b T)?
Here we skolemize the bound region in the supertype to yield:
fn(&'A T) <: &fn(&'x T)?
And then proceed to compare the argument types:
&'x T <: &'A T
&A <= &x
Finally, this is where it gets interesting! This is where an error
*should* be reported. But in fact this will not happen. The reason why
is that `A` is a variable: we will infer that its value is the fresh
region `x` and think that everything is happy. In fact, this behavior
is *necessary*, it was key to the first example we walked through.
The difference between this example and the first one is that the variable
`A` already existed at the point where the skolemization occurred. In
the first example, you had two functions:
fn<a>(&'a T) <: &fn<b>(&'b T)
and hence `&A` and `&x` were created "together". In general, the
intention of the skolemized names is that they are supposed to be
fresh names that could never be equal to anything from the outside.
But when inference comes into play, we might not be respecting this
rule.
So the way we solve this is to add a fourth step that examines the
constraints that refer to skolemized names. Basically, consider a
non-directed verison of the constraint graph. Let `Tainted(x)` be the
set of all things reachable from a skolemized variable `x`.
`Tainted(x)` should not contain any regions that existed before the
step at which the skolemization was performed. So this case here
would fail because `&x` was created alone, but is relatable to `&A`.
## Computing the LUB and GLB
The paper I pointed you at is written for Haskell. It does not
therefore considering subtyping and in particular does not consider
LUB or GLB computation. We have to consider this. Here is the
algorithm I implemented.
First though, let's discuss what we are trying to compute in more
detail. The LUB is basically the "common supertype" and the GLB is
"common subtype"; one catch is that the LUB should be the
*most-specific* common supertype and the GLB should be *most general*
common subtype (as opposed to any common supertype or any common
subtype).
Anyway, to help clarify, here is a table containing some
function pairs and their LUB/GLB:
```
Type 1 Type 2 LUB GLB
fn<a>(&a) fn(&X) fn(&X) fn<a>(&a)
fn(&A) fn(&X) -- fn<a>(&a)
fn<a,b>(&a, &b) fn<x>(&x, &x) fn<a>(&a, &a) fn<a,b>(&a, &b)
fn<a,b>(&a, &b, &a) fn<x,y>(&x, &y, &y) fn<a>(&a, &a, &a) fn<a,b,c>(&a,&b,&c)
```
### Conventions
I use lower-case letters (e.g., `&a`) for bound regions and upper-case
letters for free regions (`&A`). Region variables written with a
dollar-sign (e.g., `$a`). I will try to remember to enumerate the
bound-regions on the fn type as well (e.g., `fn<a>(&a)`).
### High-level summary
Both the LUB and the GLB algorithms work in a similar fashion. They
begin by replacing all bound regions (on both sides) with fresh region
inference variables. Therefore, both functions are converted to types
that contain only free regions. We can then compute the LUB/GLB in a
straightforward way, as described in `combine.rs`. This results in an
interim type T. The algorithms then examine the regions that appear
in T and try to, in some cases, replace them with bound regions to
yield the final result.
To decide whether to replace a region `R` that appears in `T` with a
bound region, the algorithms make use of two bits of information.
First is a set `V` that contains all region variables created as part
of the LUB/GLB computation. `V` will contain the region variables
created to replace the bound regions in the input types, but it also
contains 'intermediate' variables created to represent the LUB/GLB of
individual regions. Basically, when asked to compute the LUB/GLB of a
region variable with another region, the inferencer cannot oblige
immediately since the valuese of that variables are not known.
Therefore, it creates a new variable that is related to the two
regions. For example, the LUB of two variables `$x` and `$y` is a
fresh variable `$z` that is constrained such that `$x <= $z` and `$y
<= $z`. So `V` will contain these intermediate variables as well.
The other important factor in deciding how to replace a region in T is
the function `Tainted($r)` which, for a region variable, identifies
all regions that the region variable is related to in some way
(`Tainted()` made an appearance in the subtype computation as well).
### LUB
The LUB algorithm proceeds in three steps:
1. Replace all bound regions (on both sides) with fresh region
inference variables.
2. Compute the LUB "as normal", meaning compute the GLB of each
pair of argument types and the LUB of the return types and
so forth. Combine those to a new function type `F`.
3. Replace each region `R` that appears in `F` as follows:
- Let `V` be the set of variables created during the LUB
computational steps 1 and 2, as described in the previous section.
- If `R` is not in `V`, replace `R` with itself.
- If `Tainted(R)` contains a region that is not in `V`,
replace `R` with itself.
- Otherwise, select the earliest variable in `Tainted(R)` that originates
from the left-hand side and replace `R` with the bound region that
this variable was a replacement for.
So, let's work through the simplest example: `fn(&A)` and `fn<a>(&a)`.
In this case, `&a` will be replaced with `$a` and the interim LUB type
`fn($b)` will be computed, where `$b=GLB(&A,$a)`. Therefore, `V =
{$a, $b}` and `Tainted($b) = { $b, $a, &A }`. When we go to replace
`$b`, we find that since `&A \in Tainted($b)` is not a member of `V`,
we leave `$b` as is. When region inference happens, `$b` will be
resolved to `&A`, as we wanted.
Let's look at a more complex one: `fn(&a, &b)` and `fn(&x, &x)`. In
this case, we'll end up with a (pre-replacement) LUB type of `fn(&g,
&h)` and a graph that looks like:
```
$a $b *--$x
\ \ / /
\ $h-* /
$g-----------*
```
Here `$g` and `$h` are fresh variables that are created to represent
the LUB/GLB of things requiring inference. This means that `V` and
`Tainted` will look like:
```
V = {$a, $b, $g, $h, $x}
Tainted($g) = Tainted($h) = { $a, $b, $h, $g, $x }
```
Therefore we replace both `$g` and `$h` with `$a`, and end up
with the type `fn(&a, &a)`.
### GLB
The procedure for computing the GLB is similar. The difference lies
in computing the replacements for the various variables. For each
region `R` that appears in the type `F`, we again compute `Tainted(R)`
and examine the results:
1. If `R` is not in `V`, it is not replaced.
2. Else, if `Tainted(R)` contains only variables in `V`, and it
contains exactly one variable from the LHS and one variable from
the RHS, then `R` can be mapped to the bound version of the
variable from the LHS.
3. Else, if `Tainted(R)` contains no variable from the LHS and no
variable from the RHS, then `R` can be mapped to itself.
4. Else, `R` is mapped to a fresh bound variable.
These rules are pretty complex. Let's look at some examples to see
how they play out.
Out first example was `fn(&a)` and `fn(&X)`. In this case, `&a` will
be replaced with `$a` and we will ultimately compute a
(pre-replacement) GLB type of `fn($g)` where `$g=LUB($a,&X)`.
Therefore, `V={$a,$g}` and `Tainted($g)={$g,$a,&X}. To find the
replacement for `$g` we consult the rules above:
- Rule (1) does not apply because `$g \in V`
- Rule (2) does not apply because `&X \in Tainted($g)`
- Rule (3) does not apply because `$a \in Tainted($g)`
- Hence, by rule (4), we replace `$g` with a fresh bound variable `&z`.
So our final result is `fn(&z)`, which is correct.
The next example is `fn(&A)` and `fn(&Z)`. In this case, we will again
have a (pre-replacement) GLB of `fn(&g)`, where `$g = LUB(&A,&Z)`.
Therefore, `V={$g}` and `Tainted($g) = {$g, &A, &Z}`. In this case,
by rule (3), `$g` is mapped to itself, and hence the result is
`fn($g)`. This result is correct (in this case, at least), but it is
indicative of a case that *can* lead us into concluding that there is
no GLB when in fact a GLB does exist. See the section "Questionable
Results" below for more details.
The next example is `fn(&a, &b)` and `fn(&c, &c)`. In this case, as
before, we'll end up with `F=fn($g, $h)` where `Tainted($g) =
Tainted($h) = {$g, $h, $a, $b, $c}`. Only rule (4) applies and hence
we'll select fresh bound variables `y` and `z` and wind up with
`fn(&y, &z)`.
For the last example, let's consider what may seem trivial, but is
not: `fn(&a, &a)` and `fn(&b, &b)`. In this case, we'll get `F=fn($g,
$h)` where `Tainted($g) = {$g, $a, $x}` and `Tainted($h) = {$h, $a,
$x}`. Both of these sets contain exactly one bound variable from each
side, so we'll map them both to `&a`, resulting in `fn(&a, &a)`, which
is the desired result.
### Shortcomings and correctness
You may be wondering whether this algorithm is correct. The answer is
"sort of". There are definitely cases where they fail to compute a
result even though a correct result exists. I believe, though, that
if they succeed, then the result is valid, and I will attempt to
convince you. The basic argument is that the "pre-replacement" step
computes a set of constraints. The replacements, then, attempt to
satisfy those constraints, using bound identifiers where needed.
For now I will briefly go over the cases for LUB/GLB and identify
their intent:
- LUB:
- The region variables that are substituted in place of bound regions
are intended to collect constraints on those bound regions.
- If Tainted(R) contains only values in V, then this region is unconstrained
and can therefore be generalized, otherwise it cannot.
- GLB:
- The region variables that are substituted in place of bound regions
are intended to collect constraints on those bound regions.
- If Tainted(R) contains exactly one variable from each side, and
only variables in V, that indicates that those two bound regions
must be equated.
- Otherwise, if Tainted(R) references any variables from left or right
side, then it is trying to combine a bound region with a free one or
multiple bound regions, so we need to select fresh bound regions.
Sorry this is more of a shorthand to myself. I will try to write up something
more convincing in the future.
#### Where are the algorithms wrong?
- The pre-replacement computation can fail even though using a
bound-region would have succeeded.
- We will compute GLB(fn(fn($a)), fn(fn($b))) as fn($c) where $c is the
GLB of $a and $b. But if inference finds that $a and $b must be mapped
to regions without a GLB, then this is effectively a failure to compute
the GLB. However, the result `fn<$c>(fn($c))` is a valid GLB.
*/
/*! See doc.rs */
use middle::ty;
@ -556,7 +30,9 @@ use syntax::ast;
use syntax::opt_vec;
use syntax::opt_vec::OptVec;
#[deriving(Eq,IterBytes)]
mod doc;
#[deriving(Eq, IterBytes)]
enum Constraint {
ConstrainVarSubVar(RegionVid, RegionVid),
ConstrainRegSubVar(Region, RegionVid),