ω-Petri Nets

Abstract

We introduce ω-Petri nets (ωPN), an extension of plain Petri nets with ωlabeled 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 an plain Petri net that allows to recover the reachability set of the ωPN, but that does not preserve termination. This yields complexity bounds for the reachability, (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. ReachabilityDecidable and EX- PSPACE-hard (4) Undecidable (6) Undecidable (6) Place-boundedness EXPSPACE-c (4) Boundedness Decidable (6) Coverability Decidable and Ackerman-hard (6) Problem ωPN ωOPN+T, ωOPN+R ωIPN+T, ωIPN+R Termination EXPSPACE-c (5) Undecidable (6) Decidable and Ackerman-hard (6) complexity of (plain) PN problems apply to ωPN too. However, it does not preserve termination. Thus, we study, in Section 5, as a third contribution, an extension of the self-covering path technique due to Rackoff [19]. This technique allows to provide a direct proof of EXPSPACE upper bounds for several classical decision problems, and in particular, this allows to prove EXPSPACE completeness of the termination problem.Finally, in Section 6, as a additional contribution, and to get a complete picture, we consider extensions of ωPN with reset and transfer arcs [7]. For those extensions, the decidability results for reset and transfer nets (without ω arcs) also apply to our extension with the notable exception of the termination problem that becomes, as we show here, undecidable. The summary of our results are given in Table 1.Related works ωPN are well-structured transition systems [10]. The set saturation technique [1] and so symbolic backward analysis can be applied to them while the finite tree unfolding is not applicable because of the infinite branching property of ωPN. For the same reason, ωPN are not well-structured nets [11].In [3], Bradzil et al. extends the Rackoff technique to VASS games with ω output arcs. While this extension of the Rackoff technique is technically close to ours, we cannot directly use their results to solve the termination problem of ωPN.Several works (see for instance [4,5] rely on Petri nets to model parametric systems and perform parametrised verification. However, in all these works, the dynamic creation of thread...

Queue-Dispatch Asynchronous Systems

“…We introduce the, up to our knowledge, first formal model that grasps the core of Gcd, and that allows to derive basic results on the decidability of verification question thereupon. Due to the obvious undecidability issues of the model, we currently focus on several under-and over-approximative approaches (e.g., language bounded verification, graph minor based abstractions, novel Petri net extensions [12]) as well as enhancements for additional Gcd features like task groups, priorities, and timer events.…”

“…Thus by extending the proof of Proposition 5: A terminates, then A is guaranteed to terminate. We have recently shown that the termination problem is decidable for Petri nets with ω-arcs [12]. Hence, also extended termination is decidable on the previous abstraction.…”

On the Verification of Concurrent, Asynchronous Programs with Waiting Queues

“…Formally, in an ω-PN, each transition is a pair t = (I t , O t ), where I t and O t are both functions from P to N ∪ {ω}. The semantics of O t ( p) = ω(I t ( p) = ω, respectively) is that the firing of t produces (consumes) a nondeterministically chosen, yet finite number of tokens in (from) p. The coverability and the termination problems are both decidable for ω-PN [Geeraerts et al 2013b].…”

“…To recover a practical procedure for analyzing EQDAS, we propose using a recently introduced extension of Petri nets [Geeraerts et al 2013b], called ω-Petri net (ω-PN for short). ω-PN transitions with an arbitrary but fix number of tokens, and extends classical Petri nets by permitting arcs to be labeled by ω.…”

Handling Infinitely Branching WSTS

“…However, WSTS such as inserting FIFO automata [7], inserting automata [6] and ω-Petri nets [17], that can arbitrarily increase some values, are intrinsically infinitely branching, and any finitely branching WSTS parameterized with an infinite set of initial states (such as broadcast protocols [10]) also inherits an infinitely branching state. For instance, Geeraerts, Heußner, Praveen and Raskin argue in [17] that parametric concurrent systems with dynamic Supported by the French Agence Nationale de la Recherche, REACHARD (grant ANR-11-BS02-001), by the Fonds québécois de la recherche sur la nature et les technologies, by the Natural Sciences and Engineering Research Council of Canada and by the "Chaire DIGITEO, ENS Cachan -École Polytechnique". thread creation can naturally be modelled by some classes of infinitely branching systems, like ω-Petri nets, i.e.…”