Reachability in Two-Dimensional Vector Addition Systems with States Is PSPACE-Complete

Blondin, Michael; Finkel, Alain; Göller, Stefan; Haase, Christoph; McKenzie, Pierre

doi:10.1109/lics.2015.14

Cited by 40 publications

(116 citation statements)

References 29 publications

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“…The latter include investigating implications for the reachability problem for fixed-dimension flat vector addition systems with states (cf. [35,3,12]).…”

Section: Discussionmentioning

confidence: 99%

The reachability problem for Petri nets is not elementary

Czerwiński

Lasota

Lazić

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modelling and analysis of hardware, software and database systems, as well as chemical, biological and business processes. The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of verification. Decidability was proved by Mayr in his seminal STOC 1981 work, and the currently best published upper bound is non-primitive recursive Ackermannian of Leroux and Schmitz from LICS 2019. We establish a non-elementary lower bound, i.e. that the reachability problem needs a tower of exponentials of time and space. Until this work, the best lower bound has been exponential space, due to Lipton in 1976. The new lower bound is a major breakthrough for several reasons. Firstly, it shows that the reachability problem is much harder than the coverability (i.e., state reachability) problem, which is also ubiquitous but has been known to be complete for exponential space since the late 1970s. Secondly, it implies that a plethora of problems from formal languages, logic, concurrent systems, process calculi and other areas, that are known to admit reductions from the Petri nets reachability problem, are also not elementary. Thirdly, it makes obsolete the currently best lower bounds for the reachability problems for two key extensions of Petri nets: with branching and with a pushdown stack.At the heart of our proof is a novel gadget so called the factorial amplifier that, assuming availability of counters that are zero testable and bounded by k, guarantees to produce arbitrarily large pairs of values whose ratio is exactly the factorial of k. We also develop a novel construction that uses arbitrarily large pairs of values with ratio R to provide zero testable counters that are bounded by R. Repeatedly composing the factorial amplifier with itself by means of the construction then enables us to compute in linear time Petri nets that simulate Minsky machines whose counters are bounded by a tower of exponentials, which yields the non-elementary lower bound. By refining this scheme further, we in fact establish hardness for h-exponential space already for Petri nets with h + 13 counters. *

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“…The latter include investigating implications for the reachability problem for fixed-dimension flat vector addition systems with states (cf. [35,3,12]).…”

Section: Discussionmentioning

confidence: 99%

The reachability problem for Petri nets is not elementary

Czerwiński

Lasota

Lazić

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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“…Here, not much is known except regarding coverability and reachability in twodimensional VASS: these problems are PSPACE-complete if counter updates are encoded in binary [3] and NL-complete if updates are encoded in unary [5].…”

Section: Discussionmentioning

confidence: 99%

Coverability Is Undecidable in One-Dimensional Pushdown Vector Addition Systems with Resets

Schmitz¹,

Zetzsche²

2019

Lecture Notes in Computer Science

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We consider the model of pushdown vector addition systems with resets. These consist of vector addition systems that have access to a pushdown stack and have instructions to reset counters. For this model, we study the coverability problem. In the absence of resets, this problem is known to be decidable for one-dimensional pushdown vector addition systems, but decidability is open for general pushdown vector addition systems. Moreover, coverability is known to be decidable for reset vector addition systems without a pushdown stack. We show in this note that the problem is undecidable for one-dimensional pushdown vector addition systems with resets.

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“…The set pref p q is nothing but the reachability set of a 2-VASS VpN, Nq in control state pq, pq, from the initial configuration ppq 0 , p 0 q, 0, 0q. We build on a result of [4] which describes the reachability set in terms of sets reachable via a finite set of linear path schemes, a notion that we are going to recall now.…”

Section: Proof Of Proposition 18mentioning

confidence: 99%

Regular separability of one counter automata

Lasota

Czerwiński

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Abstract. The regular separability problem asks, for two given languages, if there exists a regular language including one of them but disjoint from the other. Our main result is decidability, and PSpace-completeness, of the regular separability problem for languages of one counter automata without zero tests (also known as one counter nets). This contrasts with undecidability of the regularity problem for one counter nets, and with undecidability of the regular separability problem for one counter automata, which is our second result.

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Reachability in Two-Dimensional Vector Addition Systems with States Is PSPACE-Complete

Cited by 40 publications

References 29 publications

The reachability problem for Petri nets is not elementary

The reachability problem for Petri nets is not elementary

Coverability Is Undecidable in One-Dimensional Pushdown Vector Addition Systems with Resets

Regular separability of one counter automata

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