Demystifying Reachability in Vector Addition Systems

Leroux, Jérôme; Schmitz, Sylvain

doi:10.1109/lics.2015.16

Cited by 86 publications

(77 citation statements)

References 37 publications

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“…5], this is a reasonable assumption in practice. Our complexity analysis uses a fragment of LTL studied by Jančar [10] and shares its complexity: at least as hard as reachability (noted 'PNReach' in Table 1), and at most exponentially harder; recall that the complexity of reachability in general Petri nets is a major open problem [22], with a gigantic gap between a forty years old EXPSPACE lower bound [23] and a cubic Ackermann upper bound obtained recently in [24].…”

Section: Pnreachmentioning

confidence: 99%

The Complexity of Diagnosability and Opacity Verification for Petri Nets

Bérard¹,

Haar

Schmitz

et al. 2018

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Abstract. Diagnosability and opacity are two well-studied problems in discrete-event systems. We revisit these two problems with respect to expressiveness and complexity issues.We first relate different notions of diagnosability and opacity. We consider in particular fairness issues and extend the definition of Germanos et al. [ACM TECS, 2015] of weakly fair diagnosability for safe Petri nets to general Petri nets and to opacity questions.Second, we provide a global picture of complexity results for the verification of diagnosability and opacity. We show that diagnosability is NL-complete for finite state systems, PSPACE-complete for safe convergent Petri nets (even with fairness), and EXPSPACE-complete for general Petri nets without fairness, while non diagnosability is inter-reducible with reachability when fault events are not weakly fair. Opacity is ESPACE-complete for safe Petri nets (even with fairness) and undecidable for general Petri nets already without fairness.

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Section: Pnreachmentioning

confidence: 99%

The Complexity of Diagnosability and Opacity Verification for Petri Nets

Bérard¹,

Haar

Schmitz

et al. 2018

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“…The problem is known to be decidable [30], with a cubic Ackermanian complexity upper bound [27], and EXPSPACE-hard [29]. It is open whether the problem has an algorithm that runs in elementary time, i.e., in k-EXPTIME for some number k independent of the input.…”

Section: Population Protocols As Petri Netsmentioning

confidence: 99%

Verification of population protocols

et al. 2016

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Population protocols (Angluin et al. in PODC, 2004) are a formal model of sensor networks consisting of identical mobile devices. Two devices can interact and thereby change their states. Computations are infinite sequences of interactions satisfying a strong fairness constraint. A population protocol is well specified if for every initial configuration C of devices, and every computation starting at C, all devices eventually agree on a consensus value depending only on C. If a protocol is well specified, then it is said to compute the predicate that assigns to each initial configuration its consensus value. While the computational power of well-specified protocols has been extensively studied, the two basic verification problems remain open: Is a given protocol well specified? Does a given protocol compute a given predicate? We prove that both problems are decidable by reduction to the reachability problem of Petri nets. We also give a new proof of the fact that the predicates computed by well-defined protocols are those definable in Presburger arithmetic (Angluin et al. in PODC, 2006).

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“…Indeed, the flat regular expressions are in fact representations of ideals of words over Σ with respect to the subword ordering [Jullien 1969], whereas the structures employed in Kosaraju's θ condition have recently been shown to correspond to ideals of runs of vector addition systems [Leroux and Schmitz 2015].…”

Section: Pumpable Transductionsmentioning

confidence: 99%

Zeno, Hercules, and the Hydra

Lazić

Ouaknine

Worrell

2016

ACM Trans. Comput. Logic

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A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. Metric temporal logic (MTL) is one of the most prominent specification formalisms for real-time systems. Over infinite timed words, full MTL is undecidable, but satisfiability for a syntactially defined safety fragment, called safety MTL, was proved decidable several years ago. Satisfiability for safety MTL is also known to be equivalent to a fair termination problem for a class of channel machines with insertion errors. However, hitherto its precise computational complexity has remained elusive, with only a non-elementary lower bound.Via another equivalent problem, namely termination for a class of rational relations, we show that satisfiability for safety MTL is Ackermann-complete, i.e., among the easiest non-primitive recursive problems. This is surprising since decidability was originally established using Higman's Lemma, suggesting a much higher non-multiply recursive complexity.

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Demystifying Reachability in Vector Addition Systems

Cited by 86 publications

References 37 publications

The Complexity of Diagnosability and Opacity Verification for Petri Nets

The Complexity of Diagnosability and Opacity Verification for Petri Nets

Verification of population protocols

Zeno, Hercules, and the Hydra

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