String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure

TL;DR

This paper develops a mathematical framework for rewriting string diagrams in symmetric monoidal theories, extending traditional term rewriting to resource-sensitive systems and establishing conditions for soundness and termination.

Contribution

It introduces a convex DPO rewriting approach for arbitrary SMTs, generalizing previous graph rewriting methods and enabling analysis of complex algebraic structures.

Findings

Established a correspondence between string diagram rewriting and convex DPO graph rewriting.

Proved termination for SMTs of Frobenius semi-algebras and bialgebras.

Extended the applicability of diagram rewriting to resource-sensitive systems.

Abstract

Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thoughts as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.