Model Checking Vector Addition Systems with one zero-test

Bonnet, Rémi; Finkel, Alain; Leroux, Jérôme; Zeitoun, Marc

doi:10.2168/lmcs-8(2:11)2012

Cited by 6 publications

(14 citation statements)

References 35 publications

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“…Here, by Petri net language, we mean sequences of transition labels of runs from an initial to a final marking. Czerwiński and Martens [8] exhibit a simple reduction of the diagonal problem for Petri net languages to the place boundedness problem for Petri nets with one inhibitor arc, which was proven decidable by Bonnet, Finkel, Leroux, and Zeitoun [4]. Since the Petri net languages are well-known to be a full trio [21], Theorem 1 yields an alternative algorithm for downward closures of Petri net languages.…”

Section: Basic Notions and Resultsmentioning

confidence: 99%

An Approach to Computing Downward Closures

Zetzsche

2015

Lecture Notes in Computer Science

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Abstract. The downward closure of a word language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of any language is regular. While the downward closure appears to be a powerful abstraction, algorithms for computing a finite automaton for the downward closure of a given language have been established only for few language classes.This work presents a simple general method for computing downward closures. For language classes that are closed under rational transductions, it is shown that the computation of downward closures can be reduced to checking a certain unboundedness property.This result is used to prove that downward closures are computable for (i) every language class with effectively semilinear Parikh images that are closed under rational transductions, (ii) matrix languages, and (iii) indexed languages (equivalently, languages accepted by higher-order pushdown automata of order 2).

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Section: Basic Notions and Resultsmentioning

confidence: 99%

An Approach to Computing Downward Closures

Zetzsche

2015

Lecture Notes in Computer Science

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“…Here, we present an alternative approach to this problem. We show that the diagonal problem for VAS languages is decidable by reduction to the place-boundedness problem for VASs with one zero test, which has been shown decidable (Bonnet et al, 2010(Bonnet et al, , 2012. A k-dimensional (labeled) vector addition system with one zero-test, or (labeled) VAS z over the alphabet A is a tuple M = (A, T, r z , δ 0 , δ 1 , ℓ, s, t) such that (A, T, δ 0 , δ 1 , ℓ, s, t) is a VAS and additionally r z ∈ T is a distinguished zero-test transition.…”

Section: Decidable Classesmentioning

confidence: 98%

A Note on Decidable Separability by Piecewise Testable Languages

Czerwiński

Martens

Rooijen

et al. 2015

Fundamentals of Computation Theory

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Piecewise testable languages form the first level of the Straubing-Thérien hierarchy. The membership problem for this level is decidable and testing if the language of a DFA is piecewise testable is NL-complete. The question has not yet been addressed for NFAs. We fill in this gap by showing that it is PSpace-complete. The main result is then the lower-bound complexity of separability of regular languages by piecewise testable languages. Two regular languages are separable by a piecewise testable language if the piecewise testable language includes one of them and is disjoint from the other. For languages represented by NFAs, separa-bility by piecewise testable languages is known to be decidable in PTime. We show that it is PTime-hard and that it remains PTime-hard even for minimal DFAs.

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“…In the following we show that the reachability graph of a conservative EOS is a well structured transitions system [1,8,3]. A quasi order is a reflexive and transitive binary relation.…”

Section: Proofmentioning

confidence: 99%

A Survey of Decidability Results for Elementary Object Systems

Köhler-Bußmeier

2014

Fundamenta Informaticae

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This contribution presents recent results on Elementary Object Systems (EOS). Object nets are Petri nets which have Petri nets as tokens -an approach known as the nets-within-nets paradigm.In this work we study the relationship of EOS to existing Petri net formalisms. It turns out that EOS are equivalent to counter programs.But even for the restricted subclass of conservative EOS reachability and liveness are undecidable problems. On the other hand for other properties like boundedness are still decidable for conservative EOS.We also study the sub-class of generalised state machines, which is worth mentioning since it combines decidability of many theoretically interesting properties with a quite rich practical modelling expressiveness.states. The algebraic extension of objects nets -discussed in [20] -even allows operations on the nettokens, like sequential or parallel composition. This is a concise way to express the self-modification of net-tokens at run-time in an algebraic setting.Object Nets can be seen as the Petri net perspective on mobility, in contrast to the Ambient Calculus [4] or the π-calculus [36], which form the process algebra perspective.It is quite natural to use object nets to model mobility and mobile agents (cf. [16]). Each place of the system net describes a location that may hosts agents, which are net-tokens. Mobility can be modelled by moving net-tokens from one place to another. This hierarchy forms a useful abstraction of the system: on a high level the agent system and on a lower level of the hierarchy the agent itself.Example 1.1. Figure 1 shows an object net system of a simple mobile household robot inside a building with several rooms. The mobile robot is modelled as a net-token that moves from room to room in the system net, which models the building. The agent has a very simple plan: Firstly, it synchronises with the environment via the channel ch 1 and then via the channel ch 2 . The synchronisation for ch 1 is provided in the kitchen; the synchronisation for ch 2 is provided in the living room.One possible run is the following: Initially, the agent is in the state q 1 and is located at place p 1 in the bedroom (cf. Figure 1). The agent moves into the bath by t 1 and moves into the kitchen by t 2 . Here, it synchronises its transition u with the environment transition t 3 over the channel ch 1 . Then the agent is in the state q 2 and is located at p 6 . The agent moves into the living room by t 7 and synchronises v with t 8 over the channel ch 2 . Finally, the agent is located at p 5 and has reached its final state q 3 . w − → m ′ holds. The set of reachable markings is RS (m 0 ) := {m | ∃w ∈ T * : m 0 w − → m}. Elementary Object SystemsAn elementary object system (EOS) is composed of a system net, which is a p/t net N = ( P , T , pre, post) and a set of object nets N = {N 1 , . . . , N n }, which are p/t nets given as N = (P N , T N , pre N ,

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Model Checking Vector Addition Systems with one zero-test

Cited by 6 publications

References 35 publications

An Approach to Computing Downward Closures

An Approach to Computing Downward Closures

A Note on Decidable Separability by Piecewise Testable Languages

A Survey of Decidability Results for Elementary Object Systems

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