String diagram rewrite theory III: Confluence with and without Frobenius

Abstract In this paper, we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorially as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For standard DPO without interfaces, terminating systems is, in general, undecidable. Nevertheless, show here that DPOI, and hence is decidable. We apply result give effective procedures deciding local symmetric monoidal theories Frobenius structure by critical pair analysis. latter, introduce new notion path joinability pairs, enables finitely many joins a lifted an arbitrary context spite strong non-local constraints placed generic theory.