Semantic foundations of equality saturation

TL;DR

This paper provides a formal theoretical foundation for equality saturation, connecting it to tree automata and the chase, and analyzes its termination properties and complexities.

Contribution

It introduces a fixpoint semantics for equality saturation, linking it to the chase, and offers criteria for termination based on acyclicity.

Findings

Established a fixpoint semantics using tree automata.

Connected equality saturation to the chase process.

Identified conditions ensuring termination of equality saturation.

Abstract

Equality saturation is an emerging technique for program and query optimization developed in the programming language community. It performs term rewriting over an E-graph, a data structure that compactly represents a program space. Despite its popularity, the theory of equality saturation lags behind the practice. In this paper, we define a fixpoint semantics of equality saturation based on tree automata and uncover deep connections between equality saturation and the chase. We characterize the class of chase sequences that correspond to equality saturation. We study the complexities of terminations of equality saturation in three cases: single-instance, all-term-instance, and all-E-graph-instance. Finally, we define a syntactic criterion based on acyclicity that implies equality saturation termination.