Small Proofs from Congruence Closure

TL;DR

This paper introduces efficient algorithms for generating minimal and small proof certificates in congruence closure, significantly reducing proof size and overhead in SMT solvers and equality saturation engines.

Contribution

It presents a novel O(n^5) optimal proof generation algorithm and a practical O(n log n) greedy algorithm for smaller proofs in congruence closure, implemented in the egg toolkit.

Findings

Greedy algorithm produces substantially smaller proofs than Z3.

First certifying equality saturation engine implemented.

Algorithms operate efficiently with no asymptotic overhead.

Abstract

Satisfiability Modulo Theory (SMT) solvers and equality saturation engines must generate proof certificates from e-graph-based congruence closure procedures to enable verification and conflict clause generation. Smaller proof certificates speed up these activities. Though the problem of generating proofs of minimal size is known to be NP-complete, existing proof minimization algorithms for congruence closure generate unnecessarily large proofs and introduce asymptotic overhead over the core congruence closure procedure. In this paper, we introduce an O(n^5) time algorithm which generates optimal proofs under a new relaxed "proof tree size" metric that directly bounds proof size. We then relax this approach further to a practical O(n \log(n)) greedy algorithm which generates small proofs with no asymptotic overhead. We implemented our techniques in the egg equality saturation toolkit, yielding the first certifying equality saturation engine. We show that our greedy approach in egg quickly generates substantially smaller proofs than the state-of-the-art Z3 SMT solver on a corpus of 3760 benchmarks.