In my previous post I mentioned that local reasoning in EGraphs is likely crucial to Cornelius’s success, and suggested equality refinement as one possible approach. Here I’ll outline what equality refinement is as well as some of the problems it has and possible solutions.

## Equality Refinement

In equality refinement I use the fact that a and b are identical in the then branch of(phi (== a b) a b) to transform the term into (phi (== a b) b b). To accomplish this I would use rules like:

refine: (phi (== ?a ?b) ?t ?e) => (phi (== ?a ?b) (swap ?a ?b ?t) ?e)

;; Swapping
swap-match: (swap ?a ?b ?a)          => ?b
swap-plus:  (swap ?a ?b (+ ?l ?r))   => (+  (swap ?a ?b ?l) (swap ?a ?b ?r))
;; etc



The refine rule rewrites the phi node to a new phi node by updating then branch ?t to a new term (swap ?a ?b ?t). A swap node has the following semantics:

(swap ?a ?b ?t) is the term ?t with each instance of term ?a replaced with term ?b.

This is similar to substitution in the lambda calculus, only there are no free and bound variables to worry about.

Rules like swap-plus handle the recursive transformation on the AST, and swap-match handles the cases where ?a should be replaced by ?b.

### Pass-through

I’d also like to handle base cases where no match happened. For instance, if a swap encounters a var, an int, or a bool that doesn’t satisfy swap-match, I’d like to pass the term through; (swap (var a) 7 (var b)) should simplify to (var b).

The obvious way to implement this is with:

swap-var-pass-through:   (swap ?a ?b ?c) => ?c if is_var("?c")
swap-const-pass-through: (swap ?a ?b ?c) => ?c if is_const("?c")


### Naive Equality Refinement + Pass-through Fails

The above technique fails: it over identifies terms! Consider the following case:

(phi arbitrary-condition
(phi (== (var a) (var b))
(var a)
(var b))
(var a))


With the above implementation of equality refinement, the above program gets rewritten to (var b)! How?

Well, EGraphs reason globally, and they take the transitive symmetric closure of rewrites. So if term a rewrites to both term b and term c, then b and c will be identified with each other in the EGraph. This might not seem surprising when you read it, but it can introduce some subtle bugs.

In our example, applying equality refinement yields something like

(phi arbitrary-condition
(phi (== (var a) (var b))
(swap (var a) (var b) (var a))
(var b))
(var a))


Either of the following two rules can fire on term (swap (var a) (var b) (var a)).

1. swap-match: this rewrites the swap term to (var b); this is the desired effect
2. swap-var-pass-through: this rewrites the swap term to (var a).

In a normal rewrite system, this firing would replace the swap node. An EGraph, however, will retain the swap node so that the other rule can be applied as well. Thus, the swap node will be rewritten to both (var a) and (var b), and all three nodes have been identified as equivalent globally.

This is a common theme in EGraphs: either of the above rules are correct. Rewrite rules that are sound in a normal rewrite system can be unsound in an EGraph. This can be highly counter-intuitive and I’ve spent many long hours trying to track down subtle bugs that are consequences of this.

### Fixing Equality Refinement + Pass-through?

Suppose I add something like the following to our rules:

swap-var-pass-through: (swap ?a ?b ?c) => ?c
if is_var("?c") && are_not_same_nodes("?a", "?c")
swap-const-pass-through: (swap ?a ?b ?c) => ?c
if is_const("?c") && are_not_same_nodes("?a", "?c")


Does this save us? I’m not sure. Sticking with just vars and consts for the moment, this might work. One thing that I am trying to avoid is ever (globally) identifying a ground term, either a const or a var, with another ground term.

I’ll mention now, since it’s probably not obvious: a var is an unknown term. It is passed in by the user, and it can be anything. These should probably be called params, since these are parameterizing our programs. So, if I ever prove that (var a) is equal to (var b) globally, I’ve lost, because I’ve proven that our user always passes in the same values for parameters a and b; this is clearly nonsense!

Likewise, if I prove that (var a) is equal to 7, I’ve proven that our user always passes in the number 7 for parameter a. Again, balderdash!

Finally, if I prove that 7 and 8 are equal, things are dire indeed.

Ground terms must be separate.

I actually need to be more careful than just not rewriting ground terms: what if I prove that (and (var a) (var b)) is equal to (var a)? Well then I’ve proven that our dear user always passes in parameters that satisfy the condition (== (var a) (var b)).

There are a number of ways I can put my foot in it. But let’s begin by keeping ground terms separate.

I’m going to adopt the following terminology: a value term is a term without any swap nodes, while a swapping term is a term that is not a value term. This is a little complicated since I am dealing with equivalence classes of terms, but I’ll worry about that later.

The errant global identifications are coming from swapping terms being rewritten to multiple distinct value terms (here, I use distinct to mean that the different terms are not already in the same eclass). I don’t need to worry about value terms being rewritten to swapping terms: the only place this happens is in the refine rule, and here the rewrite is rooted at a phi node. It’s only the roots of the rewritten nodes (and possibly their parents, via congruence) that get identified: rewriting (foo a b) to (foo b c) only identifies two foo nodes, not a with b or b with c. But identifying a phi node with another phi node with appropriate values swapped causes no problems: this is precisely what I want.

TODO: worry about effects of congruence (I don’t think this will be a problem but it might be, and may be sticky to prove).

So I want to prove that a swapping term w/ a swap root only has a single valid rewrite (I’m not considering rewrites of subterms). Let’s restate the rules rewriting swap nodes:

swap-match:              (swap ?a ?b ?a)        => ?b
swap-plus:               (swap ?a ?b (+ ?l ?r)) => (+  (swap ?a ?b ?l) (swap ?a ?b ?r))
swap-var-pass-through:   (swap ?a ?b ?c)        => ?c
if is_var("?c") && are_not_same_nodes("?a", "?c")
swap-const-pass-through: (swap ?a ?b ?c)        => ?c
if is_const("?c") && are_not_same_nodes("?a", "?c")


The first thing that pops out at me is that swap-plus is dangerous: I should guard it against ?a being (+ ?l ?r).

swap-match:              (swap ?a ?b ?a)        => ?b
swap-plus:               (swap ?a ?b (+ ?l ?r)) => (+  (swap ?a ?b ?l) (swap ?a ?b ?r))
if are_not_same_nodes("?a", "(+ ?l ?r)")
swap-var-pass-through:   (swap ?a ?b ?c)        => ?c
if is_var("?c") && are_not_same_nodes("?a", "?c")
swap-const-pass-through: (swap ?a ?b ?c)        => ?c
if is_const("?c") && are_not_same_nodes("?a", "?c")


This almost looks good, but there is still a problem. Consider the following term:

(swap 0 (var a) (+ 0 0))


At some point (+ 0 0) will be identified as 0. Applying the swap to (+ 0 0) produces (+ (var a) (var a)), while applying it to 0 produces (var a); we’ve proved that (== (var a) (+ (var a) (var a)), which is bogus.

I’m not sure we can avoid this just with our rewrite rules. We might need to use some other mechanism, perhaps via an analysis.