ω-Petri Nets: Algorithms and Complexity

Abstract

We introduce ω-Petri nets (ωPN), an extension of plain Petri nets with ω-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 ωPN because ωPN define transition systems that have infinite branching. This motivates a thorough analysis of the computational aspects of ωPN. We show that an ωPN can be turned into a plain Petri net that allows us to recover the reachability set of the ωPN, but that does not preserve termination (an ωPN terminates iff it admits no infinitely long execution). This yields complexity bounds for the reachability, boundedness, place boundedness and coverability problems on ωPN. We provide a practical algorithm to compute a coverability set of the ωPN and to decide termination by adapting the classical Karp and Miller tree construction. We also adapt the Rackoff technique to ωPN, to obtain the exact complexity of the termination problem. Finally, we consider the extension of ωPN with reset and transfer arcs, and show how this extension impacts the decidability and complexity of the aforementioned problems.

Parametric Verification: An Introduction

“…preT-PPN) can only add behaviours. Second we can reduce universal coverability in preT-PPNs to simultaneous unboundedness, and existential coverability in postT-PPNs to coverability in the ω Petri nets of [Gee+15]. Both of these reductions can be found in [Dav+17].…”

Forward Analysis for WSTS, Part III: Karp-Miller Trees

“…We claim that the class of very-WSTS includes vector addition systems, vector addition systems with states, Petri nets, ω-Petri nets [GHPR15], post-self-modifying nets [Val78] and strongly increasing ω-recursive nets [FMP04] for which Karp-Miller algorithms were known.…”

“…In particular, in the completion of an ω-Petri net, an arc from t to p labeled by ω increases the contents of p to ω whenever t is fired. See Figure 1 for an example of an ω-Petri net, and [GHPR15] for precise definitions. firing t in x, provided that t is enabled in x.…”

“…Since ω-Petri nets encompass all three models, we only give a proof for that model. For a definition of ω-Petri nets, see either Section 4 or [GHPR15]. Let S be an ω-Petri net with places P and transitions T .…”

“…Nonetheless, some recent Petri nets extensions, e.g. ω-Petri nets [GHPR15] and unordered data Petri nets [HLL + 16], benefit from algorithms in the style of Karp and Miller. Hence, there is hope of finding a general framework of WSTS with Karp-Miller-like algorithms.…”