String diagram rewrite theory II: Rewriting with symmetric monoidal structure

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