The reachability semantics for Petri nets can be studied using open Petri nets. For us an "open" Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category $\mathsf{Open}(\mathsf{Petri})$, which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category $\mathbb{O}\mathbf{pen}(\mathsf{Petri})$. We describe two forms of semantics for open Petri nets using symmetric monoidal double functors out of $\mathbb{O}\mathbf{pen}(\mathsf{Petri})$. The first, an operational semantics, gives for each open Petri net a category whose morphisms are the processes that this net can carry out. This is done in a compositional way, so that these categories can be computed on smaller subnets and then glued together. The second, a reachability semantics, simply says which markings of the outputs can be reached from a given marking of the inputs.
翻译:使用开放的 Petrii 网可以使用开放式 Petrii 网的可达性语义 。 对于我们来说, “ 开放的” Petri 网是一个由某些地方指定为输入和输出的集合。 我们可以通过将一个网的输出插入到另一个网的输入中来拼写打开 Petri 网。 打开 Petri 网可以被当作一个类别$\ mathsf{ Open} (\mathsf{Petri}) 的形态学, 在一个类别中, 使用 $\ mathb* (mathsf{Petri}) 的对称性单项性单项性。 但是, 对我们来说, 开放 Petri 网的组合只能被定义为非正态化的组合。 将每个对称性单项的单项的单项组合作为正态化单项的双类。 运行式网状的分类可以形成一个固定的缩略性, 并且可以形成一个对等式的 。