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.

中文翻译：

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弦图重写理论 III：有和没有 Frobenius 的汇合

在本文中，我们解决了证明字符串图重写的汇合问题，该问题之前被证明在（标记的）超图上具有接口（DPOI）的双推出重写的组合特征。对于没有接口的标准 DPO 重写，用于终止重写系统的汇合通常是不可判定的。尽管如此，我们在这里展示了 DPOI 的汇合，以及因此的字符串图重写，是可判定的。我们应用这一结果，通过临界对分析，给出确定有和没有 Frobenius 结构的对称幺半群理论的局部汇合的有效程序。对于后者，我们引入了关键对的路径可连接性的新概念，