A Critical Pair Enumeration Algorithm for String Diagram Rewriting

2026-03-10

AI Summary

This work addresses the problem of confluence verification in string diagram rewriting systems by proposing an algorithm that automatically enumerates critical pairs. The approach leverages hypergraph operations to systematically generate critical pairs within the framework of symmetric monoidal categories, and it is the first to support string diagram rewriting systems that do not rely on Frobenius structures. The algorithm is formally proven to be both sound and complete, thereby providing the first computable and automatable tool for confluence analysis in this setting. This advancement significantly extends the applicability and formal expressiveness of existing string diagram rewriting techniques, enabling broader use in categorical and compositional reasoning contexts.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesKnowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingComputer Vision: Visual Reasoning & Symbolic Representations

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• critical pair
• string diagram rewriting
• confluence
• symmetric monoidal categories
• rewriting systems

Innovation

• critical pair enumeration
• string diagram rewriting
• symmetric monoidal categories
• hypergraph manipulation
• confluence