Rewriting modulo symmetric monoidal structure

TL;DR

This paper develops a formal foundation for rewriting string diagrams in symmetric monoidal categories by interpreting them as hypergraph transformations, enabling rigorous analysis and applications across computer science, physics, and control theory.

Contribution

It establishes a precise correspondence between diagram rewriting modulo SMC laws and hypergraph DPO rewriting, generalizing the theoretical framework for diagrammatic reasoning.

Findings

Hypergraph DPO rewriting corresponds to diagram rewriting modulo SMC laws.

The approach is applied to prove termination of non-commutative bimonoid theories.

Provides a soundness condition called convexity for hypergraph rewriting.

Abstract

String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and control theory. An important role in many such approaches is played by equational theories of diagrams, typically oriented and applied as rewrite rules. This paper lays a comprehensive foundation of this form of rewriting. We interpret diagrams combinatorially as typed hypergraphs and establish the precise correspondence between diagram rewriting modulo the laws of SMCs on the one hand and double pushout (DPO) rewriting of hypergraphs, subject to a soundness condition called convexity, on the other. This result rests on a more general characterisation theorem in which we show that typed hypergraph DPO rewriting amounts to diagram rewriting modulo the laws of SMCs with a chosen special Frobenius structure. We illustrate our approach with a proof of termination for the theory of non-commutative bimonoids.