Presheaf automata

TL;DR

This paper introduces presheaf automata, a versatile formalism that generalizes various automata models, develops their theoretical foundations, and demonstrates their relation to Petri nets and language theory.

Contribution

It presents the foundational theory of presheaf automata, extending simulation notions, and shows their capacity to encompass Petri nets and other automata variants.

Findings

Presheaf automata generalize Petri nets and higher-dimensional automata.

Certain finite-type presheaf automata include all Petri nets.

No Kleene theorem exists for some classes of presheaf automata.

Abstract

We introduce presheaf automata as a generalisation of different variants of higher-dimensional automata and other automata-like formalisms, including Petri nets and vector addition systems. We develop the foundations of a language theory for them based on notions of paths and track objects. We also define open maps for presheaf automata, extending the standard notions of simulation and bisimulation for transition systems. Apart from these conceptual contributions, we show that certain finite-type presheaf automata subsume all Petri nets, generalising a previous result by van Glabbeek, which applies to higher-dimensional automata and safe Petri nets. We also present a class of presheaf automata for which there is no Kleene theorem with respect to the notions of rational and regular languages introduced.