A Categorical Semantics for Guarded Petri Nets

A Categorical Semantics for Hierarchical Petri Nets

We show how a particular variety of hierarchical nets, where the firing of a transition in the parent net must correspond to an execution in some child net, can be modelled utilizing a functorial semantics from a free category -- representing the parent net -- to the category of sets and spans between them. This semantics can be internalized via Grothendieck construction, resulting in the category of executions of a Petri net representing the semantics of the overall hierarchical net. We conclude the paper by giving an engineering-oriented overview of how our model of hierarchical nets can be …

Petri网粘合的本质

The Essence of Petri Net Gluings

本文提出一个在组合Petri网时尊重其语义的范畴框架，并证明其能推广先前通过资源共享或变迁同步进行粘合的方法。

谛亚克塔彼得网

Dialectica Petri Nets

本文提出利用谛亚克塔构造作为彼得网的范畴模型，以统一不同类型的转换（如真值、概率等）在同一框架下进行建模，并探讨其与近期范畴网模型的联系。

Petri网的两种多图表示

Two polygraphic presentations of Petri nets

本文给出了Petri网的一个代数和两个多图翻译，所有三个都提供了一种更简单的方法来描述约简并识别其中的一些。第一种将位置视为交换幺半群的生成元，将变迁视为其上的重写规则：这种设置完全等同于Petri网，但缺乏任何图形直观性。第二种把地点看作是一维的细胞，把过渡看作是二维的：这种翻译恢复了图形的意义，但由于它使用了显式的排列，因此带来了许多困难。最后，第三种翻译将地点视为退化的2维细胞，将变迁视为3维细胞：这是一种等同于Petri网的设置，配备了图形解释。

Categories of Nets

We present a unified framework for Petri nets and various variants, such as pre-nets and Kock's whole-grain Petri nets. Our framework is based on a less well-studied notion that we call $Σ$-nets, which allow finer control over whether tokens are treated using the collective or individual token philosophy. We describe three forms of execution semantics in which pre-nets generate strict monoidal categories, $Σ$-nets (including whole-grain Petri nets) generate symmetric strict monoidal categories, and Petri nets generate commutative monoidal categories, all by left adjoint functors. We also const…

A Categorical Semantics for Bounded Petri Nets

We provide a categorical semantics for bounded Petri nets, both in the collective- and individual-token philosophy. In both cases, we describe the process of bounding a net internally, by just constructing new categories of executions of a net using comonads, and externally, using lax-monoidal-lax functors. Our external semantics is non-local, meaning that tokens are endowed with properties that say something about the global state of the net. We then prove, in both cases, that the internal and external constructions are equivalent, by using machinery built on top of the Grothendieck construct…

Whole-grain Petri nets and processes

We present a formalism for Petri nets based on polynomial-style finite-set configurations and etale maps. The formalism supports both a geometric semantics in the style of Goltz and Reisig (processes are etale maps from graphs) and an algebraic semantics in the style of Meseguer and Montanari, in terms of free coloured props, and allows the following unification: for P a Petri net, the Segal space of P-processes is shown to be the free coloured prop-in-groupoids on P. There is also an unfolding semantics à la Winskel, which bypasses the classical symmetry problems: with the new formalism, ever…