Towards Term-based Verification of Diagrammatic Equivalence

2026-02-11

AI Summary

This work addresses the challenge of verifying equivalence between string diagrams under different syntactic representations by proposing a normalization method based on term rewriting systems, with a focus on two key classes of string diagrams arising in quantum circuit equivalence verification. By introducing two-dimensional diagrammatic terms generated through sequential and parallel composition, and designing rewrite rules within a framework of deformation equivalence, the paper establishes—for the first time—a provably terminating and confluent normalization system for these diagram classes. The termination and confluence properties are rigorously formalized and verified using the Isabelle/HOL proof assistant, thereby providing a solid theoretical foundation and a reliable implementation pathway for automated reasoning about string diagram equivalence.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Quantum Machine Learning

Application Category

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

Abstract

A string diagram is a two-dimensional graphical representation that can be described as a one-dimensional term generated from a set of primitives using sequential and parallel compositions. Since different syntactic terms may represent the same diagram, this syntax is quotiented by a collection of coherence equations expressing equivalence up to deformation. This work lays foundations for automated reasoning about diagrammatic equivalence, motivated primarily by the verification of quantum circuit equivalences. We consider two classes of diagrams, for which we introduce normalizing term rewriting systems that equate diagrammatically equivalent terms. In both cases, we prove termination and confluence with the help of the proof assistant Isabelle/HOL.

Problem

• string diagram
• diagrammatic equivalence
• term rewriting
• quantum circuit verification
• coherence equations

Innovation

• string diagrams
• term rewriting
• diagrammatic equivalence
• Isabelle/HOL
• quantum circuit verification