Sesqui-pushout (SqPO) rewriting along non-linear rules and for monic matches is well-known to permit the modeling of fusing and cloning of vertices and edges, yet to date, no construction of a suitable concurrency theorem was available. The lack of such a theorem, in turn, rendered compositional reasoning for such rewriting systems largely infeasible. We develop in this paper a suitable concurrency theorem for non-linear SqPO-rewriting in categories that are quasi-topoi (subsuming the example of adhesive categories) and with matches required to be regular monomorphisms of the given category. Our construction reveals an interesting "backpropagation effect" in computing rule compositions. We derive in addition a concurrency theorem for non-linear double pushout (DPO) rewriting in rm-adhesive categories. Our results open non-linear SqPO and DPO semantics to the rich static analysis techniques available from concurrency, rule algebra and tracelet theory.

中文翻译：

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非线性重写理论的并发定理

众所周知，沿非线性规则进行Sesqui-pushout（SqPO）重写和单项匹配可以对顶点和边的融合和克隆进行建模，但迄今为止，尚无合适的并发定理的构造。缺乏这样的定理反过来使得这种重写系统的组成推理在很大程度上是不可行的。我们在本文中为非线性SqPO重写开发了一种合适的并发定理，其类别为拟拓扑（包含粘合剂类别的示例），并且必须具有匹配项才能成为给定类别的规则单态性。我们的构造在计算规则组合时揭示了一个有趣的“反向传播效应”。我们还导出了用于rm胶粘剂类别中的非线性双推出（DPO）重写的并发定理。