Rewriting in Free Hypergraph Categories

TL;DR

This paper characterizes rewriting in free hypergraph categories using hypergraph cospans, enabling decidability results and generalizing previous work beyond single-object categories.

Contribution

It provides a combinatorial hypergraph-based framework for rewriting in free hypergraph categories, extending prior prop-specific approaches.

Findings

Arrows in free hypergraph categories are characterized as cospans of labelled hypergraphs.

Rewriting modulo Frobenius structure corresponds to hypergraph double-pushout rewriting.

Decidability of confluence in these rewriting systems is established.

Abstract

We study rewriting for equational theories in the context of symmetric monoidal categories where there is a separable Frobenius monoid on each object. These categories, also called hypergraph categories, are increasingly relevant: Frobenius structures recently appeared in cross-disciplinary applications, including the study of quantum processes, dynamical systems and natural language processing. In this work we give a combinatorial characterisation of arrows of a free hypergraph category as cospans of labelled hypergraphs and establish a precise correspondence between rewriting modulo Frobenius structure on the one hand and double-pushout rewriting of hypergraphs on the other. This interpretation allows to use results on hypergraphs to ensure decidability of confluence for rewriting in a free hypergraph category. Our results generalise previous approaches where only categories generated by a single object (props) were considered.