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$$\omega$$-Petri nets: algorithms and complexity. (English) Zbl 1335.68172
Summary: 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 thecomputational aspects of $$\omega$$PN. We show that an $$\omega$$PN can be turned into a plain Petri net that allows us to recover the reachability set of the $$\omega$$PN, but that does not preserve termination (an $$\omega$$PN terminates iff it admits no infinitely long execution). This yields complexity bounds for the reachability, boundedness, 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.

##### MSC:
 68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
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