Now that NLL has been shipped, I’ve been doing some work revisiting the Polonius project. Polonius is the project that implements the “alias-based formulation” described in my older blogpost. Polonius has come a long way since that post; it’s now quite fast and also experimentally integrated into rustc, where it passes the full test suite.

However, polonius as described is not complete. It describes the core “borrow check” analysis, but there are a number of other checks that the current implementation checks which polonius ignores:

  • Polonius does not account for moves and initialization.
  • Polonius does not check for relations between named lifetimes.

This blog post is focused on the second of those bullet points. It covers the simple cases; hopefully I will soon post a follow-up that targets some of the more complex cases that can arise (specifically, dealing with higher-ranked things).

Brief Polonius review

If you’ve never read the the original Polonius post, you should probably do so now. But if you have, let me briefly review some of the key details that are relevant to this post:

  • Instead of interpreting the 'a notation as the lifetime of a reference (i.e., a set of points), we interpret 'a as a set of loans. We refer to 'a as a “region”1 in order to emphasize this distinction.
  • We call 'a: 'b a subset relation; it means that the loans in 'a must be a subset of the loans in 'b. We track the required subset relations at each point in the program.
  • A loan comes from some borrow expression like &foo. A loan L0 is “live” if some live variable contains a region 'a whose value includes L0. When a loan is live, the “terms of the loan” must be respected: for a shared borrow like &foo, that means the path that was borrowed (foo) cannot be mutated. For a mutable borrow, it means that the path that was borrowed cannot be accessed at all.
    • If an access occurs that violates the terms of a loan, that is an error.

Running Example 1

Let’s give a quick example of some code that should result in an error, but which would not if we only considered the errors that polonius reports today:

fn foo<'a, 'b>(x: &'a [u32], y: &'b [u32]) -> &'a u32 {

Here, we declared that we are returning a &u32 with lifetime 'a (i.e., borrowed from x) but in fact we are returning data with lifetime 'b (i.e., borrowed from y).

Slightly simplified, the MIR for this function looks something like this.

fn foo(_1: &'a [u32], _2: &'b [u32]) -> &'a [u32] {
  _0 = &'X (*_2)[const 0usize]; // S0
  return;                       // S1

As you can see, there’s only really one interesting statement; it borrows from _2 and stores the result into _0, which is the special “return slot” in MIR.

In the case of the parameters _1 and _2, the regions come directly from the method signature. For regions appearing in the function body, we create fresh region variables – in this case, only one, 'X. 'X represents the region assigned to the borrow.

The relevant polonius facts for this function are as follows:

  • base_subset('b, 'X, mid(S0)) – as described in the NLL RFC, “re-borrowing” the referent of a reference (i.e., *_2) creates a subset relation between the region of the region (here, 'b) and the region of the borrow (here, 'X). Written in the notation of the [NLL RFC], this would be the relation 'X: 'b @ mid(S0).
  • base_subset('X, 'a, mid(S0)) – the borrow expression in S0 produces a result of type &'X u32. This is then assigned to _0, which has the type &'a [u32]. The subtyping rules require that 'X: 'a.

Combining the two base_subset relations allows us to conclude that the full subset relation includes subset('b, 'a, mid(S0)) – that is, for the function to be valid, the region 'b must be a subset of the region 'a. This is an error because the regions 'a and 'b are actually parameters to foo; in other words, foo must be valid for any set of regions 'a and 'b, and hence we cannot know if there is a subset relationship between them. This is a different sort of error than the “illegal access” errors that Polonius reported in the past: there is no access at all, in fact, simply subset relations.

Placeholder regions

There is an important distinction between named regions like 'a and 'b and the region 'X we created for a borrow. The definition of foo has to be true for all regions 'a and 'b, but for a region like 'X there only has to be some valid value. This difference is often called being universally quantified (true for all regions) versus existentially quantified (true for some region).

In this post, I will call universally quantified regions like 'a and 'b “placeholder” regions. This is because they don’t really represent a known quantity of loans, but rather a kind of “placeholder” for some unknown set of loans.

We will include a base fact that helps us to identify placeholder regions:

.decl placeholder_region(R1: region)
.input placeholder_region

This fact is true for any placeholder region. So in our example we might have


Note that the actual polonius impl already includes a relation like this2, because we need to account for the fact that placeholder regions are “live” at all points in the control-flow graph, as we always assume there may be future uses of them that we cannot see.

Representing known relations

Even placeholder regions are not totally unknown though. The function signature will often include where clauses (or implied bounds) that indicate some known relationships between placeholder regions. For example, if foo included a where clause like where 'b: 'a, then it would be perfectly legal.

We can represent the known relationships using an input:

.decl known_base_subset(R1: region, R2: region)
.input known_base_subset

Naturally these known relations are transitive, so we can define a known_subset rule to encode that:

.decl known_subset(R1: region, R2: region)

known_subset(R1, R2) :- known_base_subset(R1, R2).
known_subset(R1, R3) :- known_base_subset(R1, R2), known_subset(R2, R3).

In our example of foo, there are no where clauses nor implied bounds, so these relations are empty. If there were a where clause like where 'b: 'a, however, then we would have a known_base_subset('b, 'a) fact. Similarly, per out implied bounds rules, such an input fact might be derived from an argument with a type like &'a &'b u32, where there are ‘nested’ regions.

Detecting illegal subset relations

We can now extend the polonius rules to report errors for cases like our running example. The basic idea is this: if the function requires a subset relationship 'r1: 'r2 between two placeholder regions 'r1 and 'r2, then it must be a “known subset”, or else we have an error. We can encode this like so:

.decl subset_error(R1: region, R2: region, P:point)

subset_error(R1, R2, P) :-
  subset(R1, R2, P),      // `R1: R2` required at `P`
  placeholder_region(R1), // `R1` is a placeholder
  placeholder_region(R2), // `R2` is also a placeholder
  !known_subset(R1, R2).  // `R1: R2` is not a "known subset" relation.

In our example program, we can clearly derive subset_error('b, 'a, mid(S0)), and hence we have an error:

  • we saw earlier that subset('a, 'b, mid(S0)) holds
  • as 'a is a placeholder region, placeholder_region('a) will appear in the input (same for 'b)
  • finally, the known_base_subset (and hence known_subset) relation in our example is empty

Sidenote on negative reasoning and stratification. This rule makes use of negative reasoning in the form of the !known_subset(R1, R2) predicate. Negative reasoning is fine in datalog so long as the program is “stratified” – in particular, we must be able to compute the entire known_subset relation without having to compute subset_error. In this case, the program is trivialy stratified – known_subset depends only on the input relation known_base_subset.)

Observation about borrowing local data

It is interesting to walk through a different example. This is another case where we expect an error, but in this case the error arises because we are returning a reference to the stack:

fn bar<'a>(x: &'a [u32]) -> &'a u32 {
    let stack_slot = x[0];

Polonius will report an error for this case, but not because of the mechanisms in this blog post. What happens instead is that we create a loan for the borrow expression &stack_slot, we’ll call it L0. When the borrow is returned, this loan L0 winds up being a member of the 'a region. It is therefore “live” when the storage for stack_slot is popped from the stack, which is an error: you can’t pop the storage for a stack slot where there are live loans that have reference it.


This post describes a simple extension to the polonius rules that covers errors arising from subset relations. Unlike the prior rules, these errors are not triggered by any “access”, but rather simply the creation of a (transitive) subset relation between two placeholder regions.

Unfortunately, this is not the complete story around region checking errors. In particular, this post ignored subset relations that can arise from “higher-ranked” types like for<'a> fn(&'a u32). Handling these properly requires us to introduce a bit more logic and will be covered in a follow-up.

Comments, if any, should be posted in the internals thread dedicated to my previous polonius post

Appendix: A (potentially) more efficient formulation

The subset_error formulation above relied on the transitive subset relation to work, because we wanted to report errors any time that one placeholder wound up being forced to be a subset of another. In the more optimized polonius implementations, we don’t compute the full transitive relation, so it might be useful to create a new relation subset_placeholder that is specific to placeholder regions:

.decl subset_placeholder(R1: region, R2: region, P:point)

The idea is that subset_placeholder(R1, R2, P) means that, at the point P, we know that R1: R2 must hold, where R1 is a placeholder. You can express this via a “base” rule:

subset_placeholder(R1, R2, P) :-
  subset(R1, R2, P),      // `R1: R2` required at `P`
  placeholder_region(R1). // `R1` is a placeholder

and a transitive rule:

subset_placeholder(R1, R3, P) :-
  subset_placeholder(R1, R2, P), // `R1: R2` at P where `R1` is a placeholder
  subset(R2, R3, P).      // `R2: R3` required at `P`

Then we reformulate the subset_error rule to be based on subset_placeholder:

.decl subset_error(R1: region, R2: region, P:point)

subset_error(R1, R2, P) :-
  subset_placeholder(R1, R2, P), // `R1: R2` required at `P`
  placeholder_region(R2), // `R2` is also a placeholder
  !known_subset(R1, R2).  // `R1: R2` is not a "known subset" relation.


  1. The term “region” is not an especially good fit, but it’s common in academia.

  2. Currently called universal_region, though I plan to rename it.