# A Categorical Semantics for Hierarchical Petri Nets

@inproceedings{Genovese2021ACS, title={A Categorical Semantics for Hierarchical Petri Nets}, author={Fabrizio Genovese and Jelle Herold}, booktitle={GCM@STAF}, year={2021} }

We show how a particular flavor 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…

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## 46 References

### A Categorical Semantics for Bounded Petri Nets

- Computer ScienceElectronic Proceedings in Theoretical Computer Science
- 2022

This work provides a categorical semantics for bounded Petri nets, both in the collective- and individual-token philosophy, and proves that the internal and external constructions are equivalent, by using machinery built on top of the Grothendieck construction.

### Categories of Nets

- Computer Science2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2021

This work presents a unified framework for Petri nets and various variants, based on a less well-studied notion that it calls Σ-nets, which allow fine-grained control over whether each transition behaves according to the collective or individual token philosophy.

### Executions in (Semi-)Integer Petri Nets are Compact Closed Categories

- Computer Science, MathematicsQPL
- 2018

This work uses compact closeness to infer that allowing negative tokens in a Petri net makes the causal relations between transition firings non-trivial, and this is used to model interesting phenomena in economics and computer science.

### Open Petri Nets: Non-deterministic Processes and Compositionality

- Computer ScienceICGT
- 2008

It is shown that the non-deterministic process semantics is compositional with respect to the composition operation over ranked open nets, a result which did not hold for basic open nets.

### Compositional Modeling of Reactive Systems Using Open Nets

- Computer ScienceCONCUR
- 2001

This work defines a truly concurrent semantics for open nets by extending the Goltz-Reisig process semantics of Petri nets, and introduces a composition operation over open nets, characterized as a pushout in the corresponding category, suitable to model both interaction through open places and synchronization of transitions.

### A Categorical Semantics for Guarded Petri Nets

- Computer ScienceICGT
- 2020

This work builds on the correspondence between Petri nets and free symmetric strict monoidal categories and shows how the guard semantics can be internalized in the net itself, using the Grothendieck construction.

### Encoding Synchronous Interactions Using Labelled Petri Nets

- Computer ScienceCOORDINATION
- 2014

Through the encoding, the firing semantics of Petri nets models the standard operational semantics of CSP processes, which is both preserved and reflected, and allows for describing by net semantics the standard CSP observational equivalences.

### Petri nets based on Lawvere theories

- Computer ScienceMathematical Structures in Computer Science
- 2020

This definition of Q-net is functorial with respect to change in Lawvere theory, and it is shown how this can be used to construct the semantics for Petri nets, pre-nets, integer nets, and elementary net systems.

### On the Category of Petri Net Computations

- MathematicsTAPSOFT
- 1995

This work identifies a coreflection right adjoint to \({\cal Q}\)[_] and characterize its replete image, thus yielding an axiomatization of the category of net computations.

### A Concept of Hierarchical Petri Nets with Building Blocks

- Computer ScienceApplications and Theory of Petri Nets
- 1991

The main purpose of the concept is to facilitate the modeling of large real-world systems, rather than using them for theoretical considerations, and form the foundation of an editor-simulator-tool with a direct manipulation interface for this type of nets.