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 Open(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 ${\mathbb O}$ pen(Petri). We describe two forms of semantics for open Petri nets using symmetric monoidal double functors out of ${\mathbb O}$ pen(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.

中文翻译：

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开放 Petri 网

可以使用开放 Petri 网研究 Petri 网的可达性语义。对我们来说，一个“开放”的 Petri 网是一个通过一组 cospan 将某些位置指定为输入和输出的网络。我们可以通过将一个的输出粘合到另一个的输入来组成开放的 Petri 网。Open Petri 网可以被视为 Open(Petri) 范畴的态射，它在不相交的联合下变为对称幺半群。然而，由于开放 Petri 网的复合只被定义为同构，因此最好将它们视为对称幺半群的态射双倍的类别${\mathbb O}$笔（培养皿）。我们使用对称单曲面双函子描述了开放 Petri 网的两种语义形式${\mathbb O}$笔（培养皿）。第一个是操作语义，为每个开放 Petri 网提供一个类别，其态射是该网络可以执行的过程。这是以组合方式完成的，因此可以在较小的子网上计算这些类别，然后将它们粘合在一起。第二个，可达性语义，简单地说可以从输入的给定标记到达输出的哪些标记。