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

String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure