E-Graphs With Bindings

TL;DR

This paper extends equality saturation and e-graphs to handle variable bindings in lambda calculus by using categorical semantics and hierarchical hypergraph representations, enabling effective rewriting with bindings.

Contribution

It introduces a categorical extension for e-graphs to support variable bindings, along with a combinatorial hypergraph model and DPO rewriting, bridging term rewriting and graph rewriting.

Findings

Categorical framework for e-graphs with bindings

Hierarchical hypergraph representation of terms

Equivalence of term rewriting and DPO rewriting

Abstract

Equality saturation, a technique for program optimisation and reasoning, has gained attention due to the resurgence of equality graphs (e-graphs). E-graphs represent equivalence classes of terms under rewrite rules, enabling simultaneous rewriting across a family of terms. However, they struggle in domains like $\lambda$-calculus that involve variable binding, due to a lack of native support for bindings. Building on recent work interpreting e-graphs categorically as morphisms in semilattice-enriched symmetric monoidal categories, we extend this framework to closed symmetric monoidal categories to handle bindings. We provide a concrete combinatorial representation using hierarchical hypergraphs and introduce a corresponding double-pushout (DPO) rewriting mechanism. Finally, we establish the equivalence of term rewriting and DPO rewriting, with the key property that the combinatorial representation absorbs the equations of the symmetric monoidal category.