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