A Critical Pair Enumeration Algorithm for String Diagram Rewriting

TL;DR

This paper introduces an algorithm for automating critical pair analysis in string diagram rewriting systems, enabling efficient confluence checks through hypergraph manipulation, with proven correctness and exhaustiveness.

Contribution

It develops the first automated algorithm for critical pair enumeration in string diagram rewriting within symmetric monoidal categories, excluding Frobenius structures.

Findings

Algorithm successfully enumerates all critical pairs.

Proven correctness and exhaustiveness of the method.

Applicable to string diagrams in symmetric monoidal categories.

Abstract

Critical pair analysis provides a convenient and computable criterion of confluence, which is a fundamental property in rewriting theory, for a wide variety of rewriting systems. Bonchi et al. showed validity of critical pair analysis for rewriting on string diagrams in symmetric monoidal categories. This work aims at automation of critical pair analysis for string diagram rewriting, and develops an algorithm that implements the core part of critical pair analysis. The algorithm enumerates all critical pairs of a given left-connected string diagram rewriting system, and it can be realised by concrete manipulation of hypergraphs. We prove correctness and exhaustiveness of the algorithm, for string diagrams in symmetric monoidal categories without a Frobenius structure.