Equivalence Hypergraphs: E-Graphs for Monoidal Theories

The technique of equipping graphs with an equivalence relation, called equality saturation, has recently proved both powerful and practical in program optimisation, particularly for satisfiability modulo theory solvers. We give a categorical semantics to these structures, called e-graphs, in terms of Cartesian categories enriched over a semilattice. We show how this semantics can be generalised to monoidal categories, which opens the door to new applications of e-graph techniques, from algebraic to monoidal theories. Finally, we present a sound and complete combinatorial representation of morphisms in such a category, based on a generalisation of hypergraphs which we call e-hypergraphs. They have the usual advantage that many of their structural equations are absorbed into a general notion of isomorphism.