🤖 AI Summary
This work addresses the tension between *completeness* and *structural naturality* in string diagram rewriting—where conventional approaches impose restrictive Frobenius algebraic structure, limiting applicability in practical domains such as quantum circuits and graphical machine learning models. We propose a lightweight rewriting theory relying solely on commutative (co)monoid structure, eschewing the Frobenius condition entirely. Grounded in symmetric monoidal categories, our framework integrates double-pushout (DPO) hypergraph rewriting with string diagram syntax to yield the first sound and complete rewriting system for this setting. By significantly weakening algebraic prerequisites while preserving semantic fidelity and logical completeness, our approach ensures rigorous, structure-preserving reasoning. The resulting theory provides a more general and conceptually streamlined algebraic foundation for graphical, interpretable, and compositional reasoning across diverse computational models.
📝 Abstract
String diagrams are pictorial representations for morphisms of symmetric monoidal categories. They constitute an intuitive and expressive graphical syntax, which has found application in a very diverse range of fields including concurrency theory, quantum computing, control theory, machine learning, linguistics, and digital circuits. Rewriting theory for string diagrams relies on a combinatorial interpretation as double-pushout rewriting of certain hypergraphs. As previously studied, there is a `tension' in this interpretation: in order to make it sound and complete, we either need to add structure on string diagrams (in particular, Frobenius algebra structure) or pose restrictions on double-pushout rewriting (resulting in 'convex' rewriting). From the string diagram viewpoint, imposing a full Frobenius structure may not always be natural or desirable in applications, which motivates our study of a weaker requirement: commutative monoid structure. In this work we characterise string diagram rewriting modulo commutative monoid equations, via a sound and complete interpretation in a suitable notion of double-pushout rewriting of hypergraphs.
