Rewriting for Symmetric Monoidal Categories with Commutative (Co)Monoid Structure

TL;DR

This paper develops a sound and complete rewriting framework for string diagrams in symmetric monoidal categories with commutative monoid structures, addressing limitations of previous approaches that required Frobenius algebra structures.

Contribution

It introduces a novel interpretation of string diagram rewriting modulo commutative monoid equations, expanding the applicability of graphical rewriting techniques.

Findings

Provides a sound and complete hypergraph rewriting interpretation

Addresses limitations of Frobenius algebra structures in string diagrams

Enables more natural applications in various fields

Abstract

String diagrams are pictorial representations for morphisms of symmetric monoidal categories. They constitute an intuitive and expressive graphical syntax, which has found application in a very diverse range of fields including concurrency theory, quantum computing, control theory, machine learning, linguistics, and digital circuits. Rewriting theory for string diagrams relies on a combinatorial interpretation as double-pushout rewriting of certain hypergraphs. As previously studied, there is a `tension' in this interpretation: in order to make it sound and complete, we either need to add structure on string diagrams (in particular, Frobenius algebra structure) or pose restrictions on double-pushout rewriting (resulting in 'convex' rewriting). From the string diagram viewpoint, imposing a full Frobenius structure may not always be natural or desirable in applications, which motivates our study of a weaker requirement: commutative monoid structure. In this work we characterise string diagram rewriting modulo commutative monoid equations, via a sound and complete interpretation in a suitable notion of double-pushout rewriting of hypergraphs.