{\omega}-Petri nets

TL;DR

This paper introduces {}-Petri nets ({}PN), extending Petri nets with -labeled arcs to analyze parametric concurrent systems with dynamic thread creation, providing new algorithms and complexity bounds for key problems.

Contribution

It presents the first thorough analysis of {}PN, including algorithms for reachability, coverability, and termination, and explores their complexity and extensions with reset and transfer arcs.

Findings

{}PN can be transformed into plain Petri nets for reachability analysis.

A practical algorithm for coverability set computation is proposed.

Complexity bounds for reachability, boundedness, coverability, and termination are established.

Abstract

We introduce {\omega}-Petri nets ({\omega}PN), an extension of plain Petri nets with {\omega}-labeled input and output arcs, that is well-suited to analyse parametric concurrent systems with dynamic thread creation. Most techniques (such as the Karp and Miller tree or the Rackoff technique) that have been proposed in the setting of plain Petri nets do not apply directly to {\omega}PN because {\omega}PN define transition systems that have infinite branching. This motivates a thorough analysis of the computational aspects of {\omega}PN. We show that an {\omega}PN can be turned into an plain Petri net that allows to recover the reachability set of the {\omega}PN, but that does not preserve termination. This yields complexity bounds for the reachability, (place) boundedness and coverability problems on {\omega}PN. We provide a practical algorithm to compute a coverability set of the {\omega}PN and to decide termination by adapting the classical Karp and Miller tree construction. We also adapt the Rackoff technique to {\omega}PN, to obtain the exact complexity of the termination problem. Finally, we consider the extension of {\omega}PN with reset and transfer arcs, and show how this extension impacts the decidability and complexity of the aforementioned problems.