Continuous Reachability for Unordered Data Petri Nets is in PTime

Gupta, Utkarsh; Shah, Parikshit; Akshay, S.; Hofman, Piotr

doi:10.1007/978-3-030-17127-8_15

Cited by 10 publications

(9 citation statements)

References 29 publications

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“…These extensions have, for instance, been used for the generation of program loop invariants [54], the validation of business processes [59] and the verification of multi-threaded C and Java program skeletons with communication primitives [16,39]. Linear rational and integer arithmetic over-approximations for such extended Petri nets exist [12,9,34,31] and could smoothly be used inside our framework.…”

Section: Resultsmentioning

confidence: 99%

Directed Reachability for Infinite-State Systems

Blondin

Haase

Offtermatt

2021

Tools and Algorithms for the Construction and Analysis of Systems

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Numerous tasks in program analysis and synthesis reduce to deciding reachability in possibly infinite graphs such as those induced by Petri nets. However, the Petri net reachability problem has recently been shown to require non-elementary time, which raises questions about the practical applicability of Petri nets as target models. In this paper, we introduce a novel approach for efficiently semi-deciding the reachability problem for Petri nets in practice. Our key insight is that computationally lightweight over-approximations of Petri nets can be used as distance oracles in classical graph exploration algorithms such as $$\mathsf {A}^{*}$$ A ∗ and greedy best-first search. We provide and evaluate a prototype implementation of our approach that outperforms existing state-of-the-art tools, sometimes by orders of magnitude, and which is also competitive with domain-specific tools on benchmarks coming from program synthesis and concurrent program analysis.

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Section: Resultsmentioning

confidence: 99%

Directed Reachability for Infinite-State Systems

Blondin

Haase

Offtermatt

2021

Tools and Algorithms for the Construction and Analysis of Systems

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“…Claim 16 below), we need to state and prove two key technical facts: cogs appearing in ( 14) are spanned by vectors from ′ , and so is also the vector Δ ≺ 0 t. Our notation below relies on the implicit embedding of into = × ′ , cf. (10), which allows us to consider every vector w ∈ , in particular every cog, as a vector in .…”

Section: Claim 8 Local Solvability Is Decidablementioning

confidence: 99%

“…Recalling the implicite embedding of and ′ into = × ′ , we present v as the sum v = w + v ′ (recall (10)), where v ′ is the projection to all non-main components. Furthermore, we decompose w into w = w↾ ( ) + w ′ .…”

Section: We Construct a Matrix A Asmentioning

confidence: 99%

“…[6,7,28]). In this paper, motivated by recent and potential future applications to analysis of data-enriched models [3,10,12,14], we augment systems of linear equations with atoms [1,27] (also called data values) thus shifting from finite to orbit-finite systems. The infinite sets that we study are constructed using atoms which can only be accessed in a very limited way, namely can only be tested for equality.…”

Section: Introductionmentioning

confidence: 99%

“…In case when tokens may carry tuples of atoms (arbitrary atom dimension) all the standard problems are undecidable [19]. Decidability may be regained by resorting to relaxations: continuous semantics [10] allowing for fractional executions of transitions, or so-called integer semantics [14] dropping non-negativeness restriction on configurations. Both these results have been obtained by reduction to solving certain systems of linear equations.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Solvability of orbit-finite systems of linear equations

Ghosh¹,

Hofman²,

Lasota³

2022

Preprint

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We study orbit-finite systems of linear equations, in the setting of sets with atoms. Our principal contribution is a decision procedure for solvability of such systems. The procedure works for every field (and even commutative ring) under mild effectiveness assumptions, and reduces a given orbit-finite system to a number of finite ones: exponentially many in general, but polynomially many when atom dimension of input systems is fixed. Towards obtaining the procedure we push further the theory of vector spaces generated by orbit-finite sets, and show that each such vector space admits an orbit-finite basis. This fundamental property is a key tool in our development, but should be also of wider interest.

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Linear combinations of unordered data vectors

Hofman

Leroux

Totzke

2017

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

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Data vectors generalise finite multisets: they are finitely supported functions into a commutative monoid. We study the question if a given data vector can be expressed as a finite sum of others, only assuming that 1) the domain is countable and 2) the given set of base vectors is finite up to permutations of the domain.Based on a succinct representation of the involved permutations as integer linear constraints, we derive that positive instances can be witnessed in a bounded subset of the domain.For data vectors over a group we moreover study when a data vector is reversible, that is, if its inverse is expressible using only nonnegative coefficients. We show that if all base vectors are reversible then the expressibility problem reduces to checking membership in finitely generated subgroups. Moreover, checking reversibility also reduces to such membership tests.These questions naturally appear in the analysis of counter machines extended with unordered data: namely, for data vectors over (Z d , +) expressibility directly corresponds to checking state equations for Coloured Petri nets where tokens can only be tested for equality. We derive that in this case, expressibility is in NP, and in P for reversible instances. These upper bounds are tight: they match the lower bounds for standard integer vectors (over singleton domains). ACM Subject Classification F.1.1 Models of Computation Keywords and phrases Computation with atoms, vector addition systemsDigital Object Identifier 10.4230/LIPIcs.xxx.yyy.p IntroductionFinite collections of named values are basic structures used in many areas of theoretical computer science. They can be used for instance to model databases snapshots or define the operational semantics of programming languages. We can formalize these as functions v : D → X from some countable domain D of names or data, into some value space X, and call such functions (X-valued) data vectors. Often the actual names used are not relevant and instead one is interested in data vectors up to renaming, i.e., one wants to consider vectors v and w equivalent if v = w • θ for some permutation θ : D → D of the domain. We consider the case where the value space X has additional algebraic structure. Namely, we focus on data vectors where the values are from some commutative monoid (M, +, 0) and where all but finitely many names are mapped to the neutral element. A natural question then asks if a given data vector is expressible as a sum of vectors from a given set, where the monoid operation is lifted to data vectors pointwise.

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Continuous Reachability for Unordered Data Petri Nets is in PTime

Cited by 10 publications

References 29 publications

Directed Reachability for Infinite-State Systems

Directed Reachability for Infinite-State Systems

Solvability of orbit-finite systems of linear equations

Linear combinations of unordered data vectors

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