Logics for continuous reachability in Petri nets and vector addition systems with states

Blondin, Michael; Haase, Christoph

doi:10.1109/lics.2017.8005068

Cited by 17 publications

(19 citation statements)

References 22 publications

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“…This can be seen on the same lines as the proof of undecidability of continuous reachability for Petri nets with zero tests[12].…”

mentioning

confidence: 68%

“…Finally, we consider continuous, i.e., Q + -reachability for UDPN. We adapt the techniques used for Q + -reachability of Petri nets without data from [11,12] to the setting with data, and obtain a characterization of Q + -reachability for UDPN in Section 7.1. Finally, in Section 7.3, we show how the characterization can be combined with the above bound and compression techniques from [22] to obtain a polynomial sized system of linear equations with implications over Q + .…”

Section: Udpn Reachability and Its Variants: Our Main Resultsmentioning

confidence: 99%

“…The above characterization and its proof are obtained by adapting to the data setting, the techniques developed for continuous reachability in Petri nets (without data) in [11] and [12]. Details are in Appendix 9.3.…”

Section: Characterizing Q + -Reachabilitymentioning

confidence: 99%

“…Linear equations with implications are defined exactly as we use it in [23] but they were introduced in [12]. We also call a system of linear equations with implications a =⇒ system.…”

Section: Encoding Q + -Reachability As Linear Equations With Implicatmentioning

confidence: 99%

“…Finally, with the above bound, we solve continuous reachability in UDPN by adapting the techniques from the non-data setting of [12,25]. We adapt the characterization of continuous reachability to the data setting and next encode it as system of linear equations with implications.…”

Section: Introductionmentioning

confidence: 99%

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Continuous Reachability for Unordered Data Petri Nets is in PTime

Gupta

Shah

Akshay

et al. 2019

Lecture Notes in Computer Science

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Unordered data Petri nets (UDPN) are an extension of classical Petri nets with tokens that carry data from an infinite domain and where transitions may check equality and disequality of tokens. UDPN are well-structured, so the coverability and termination problems are decidable, but with higher complexity than for Petri nets. On the other hand, the problem of reachability for UDPN is surprisingly complex, and its decidability status remains open. In this paper, we consider the continuous reachability problem for UDPN, which can be seen as an over-approximation of the reachability problem. Our main result is a characterization of continuous reachability for UDPN and polynomial time algorithm for solving it. This is a consequence of a combinatorial argument, which shows that if continuous reachability holds then there exists a run using only polynomially many data values.

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“…This can be seen on the same lines as the proof of undecidability of continuous reachability for Petri nets with zero tests[12].…”

mentioning

confidence: 68%

Section: Udpn Reachability and Its Variants: Our Main Resultsmentioning

confidence: 99%

Section: Characterizing Q + -Reachabilitymentioning

confidence: 99%

“…Linear equations with implications are defined exactly as we use it in [23] but they were introduced in [12]. We also call a system of linear equations with implications a =⇒ system.…”

Section: Encoding Q + -Reachability As Linear Equations With Implicatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Continuous Reachability for Unordered Data Petri Nets is in PTime

Gupta

Shah

Akshay

et al. 2019

Lecture Notes in Computer Science

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Separators in Continuous Petri Nets

Blondin

Esparza

2022

Lecture Notes in Computer Science

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Leroux has proved that unreachability in Petri nets can be witnessed by a Presburger separator, i.e. if a marking $$\boldsymbol{m}_\text {src}$$ m src cannot reach a marking $$\boldsymbol{m}_\text {tgt}$$ m tgt , then there is a formula $$\varphi $$ φ of Presburger arithmetic such that: $$\varphi (\boldsymbol{m}_\text {src})$$ φ ( m src ) holds; $$\varphi $$ φ is forward invariant, i.e., $$\varphi (\boldsymbol{m})$$ φ ( m ) and $$\boldsymbol{m} \rightarrow \boldsymbol{m}'$$ m → m ′ imply $$\varphi (\boldsymbol{m}'$$ φ ( m ′ ); and $$\lnot \varphi (\boldsymbol{m}_\text {tgt})$$ ¬ φ ( m tgt ) holds. While these separators could be used as explanations and as formal certificates of unreachability, this has not yet been the case due to their (super-)Ackermannian worst-case size and the (super-)exponential complexity of checking that a formula is a separator. We show that, in continuous Petri nets, these two problems can be overcome. We introduce locally closed separators, and prove that: (a) unreachability can be witnessed by a locally closed separator computable in polynomial time; (b) checking whether a formula is a locally closed separator is in NC (so, simpler than unreachablity, which is P-complete).

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The Tractability Border of Reachability in Simple Vector Addition Systems with States

Chistikov,

Czerwiński,

Mazowiecki

et al. 2024

2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)

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Vector Addition Systems with States (VASS), equivalent to Petri nets, are a well-established model of concurrency. A d-VASS can be seen as directed graph whose edges are labelled by d-dimensional integer vectors. While following a path, the values of d nonnegative integer counters are updated according to the integer labels. The central algorithmic challenge in VASS is the reachability problem: is there a run from a given starting node and counter values to a given target node and counter values? When the input is encoded in binary, reachability is computationally intractable: even in dimension one, it is NPhard.In this paper, we comprehensively characterise the tractability border of the problem when the input is encoded in unary. For our main result, we prove that reachability is NP-hard in unary encoded 3-VASS, even when structure is heavily restricted to be a simple linear-path scheme. This improves upon a recent result of Czerwi ński and Orlikowski [LICS 2022], in both the number of counters and expressiveness of the considered model, as well as answers open questions of Englert, Lazić, and Totzke [LICS 2016] and Leroux [PETRI NETS 2021].The underlying graph structure of a simple linear path scheme (SLPS) is just a path with self-loops at each node. We also study the exceedingly weak model of computation that is SPLS with counter updates in {−1, 0, +1}. Here, we show that reachability is NP-hard when the dimension is bounded by O(α(k)), where α is the inverse Ackermann function and k bounds the size of the SLPS.We complement our result by presenting a polynomial-time algorithm that decides reachability in 2-SLPS when the initial and target configurations are specified in binary. To achieve this, we show that reachability in such instances is well-structured: all loops, except perhaps for a constant number, are taken either

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Logics for continuous reachability in Petri nets and vector addition systems with states

Cited by 17 publications

References 22 publications

Continuous Reachability for Unordered Data Petri Nets is in PTime

Continuous Reachability for Unordered Data Petri Nets is in PTime

Separators in Continuous Petri Nets

The Tractability Border of Reachability in Simple Vector Addition Systems with States

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