We develop a novel method to analyze the dynamics of stochastic rewriting systems evolving over finitary adhesive, extensive categories. Our formalism is based on the so-called rule algebra framework and exhibits an intimate relationship between the combinatorics of the rewriting rules (as encoded in the rule algebra) and the dynamics which these rules generate on observables (as encoded in the stochastic mechanics formalism). We introduce the concept of combinatorial conversion, whereby under certain technical conditions the evolution equation for (the exponential generating function of) the statistical moments of observables can be expressed as the action of certain differential operators on formal power series. This permits us to formulate the novel concept of moment-bisimulation, whereby two dynamical systems are compared in terms of their evolution of sets of observables that are in bijection. In particular, we exhibit non-trivial examples of graphical rewriting systems that are moment-bisimilar to certain discrete rewriting systems (such as branching processes or the larger class of stochastic chemical reaction systems). Our results point towards applications of a vast number of existing well-established exact and approximate analysis techniques developed for chemical reaction systems to the far richer class of general stochastic rewriting systems.

中文翻译：

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随机重写系统的组合转换和矩互模拟

我们开发了一种新方法来分析随机重写系统在有限的粘合剂、广泛的类别上演变的动态。我们的形式主义基于所谓的规则代数框架，并展示了重写规则的组合（如规则代数中编码）与这些规则在可观察量上产生的动力学（如随机力学形式主义中编码）之间的密切关系. 我们引入组合转换的概念，即在一定的技术条件下，可观测的统计矩（的指数生成函数）的演化方程可以表示为某些微分算子对形式幂级数的作用。这使我们能够制定矩双模拟的新概念，从而比较两个动力系统在双射中的可观测集合的演化。特别是，我们展示了与某些离散重写系统（例如分支过程或更大类别的随机化学反应系统）具有矩相似性的图形重写系统的非平凡示例。我们的结果表明，为化学反应系统开发的大量现有的、完善的精确和近似分析技术应用于更丰富的一般随机重写系统类别。我们展示了与某些离散重写系统（例如分支过程或更大类别的随机化学反应系统）具有矩相似性的图形重写系统的非平凡示例。我们的结果表明，为化学反应系统开发的大量现有的、完善的精确和近似分析技术应用于更丰富的一般随机重写系统类别。我们展示了与某些离散重写系统（例如分支过程或更大类别的随机化学反应系统）具有矩相似性的图形重写系统的非平凡示例。我们的结果表明，为化学反应系统开发的大量现有的、完善的精确和近似分析技术应用于更丰富的一般随机重写系统类别。