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Synthesis of Petri Nets with Restricted Place-Environments: Classical and Parameterized

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Application and Theory of Petri Nets and Concurrency (PETRI NETS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12734))

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Abstract

Petri net synthesis consists in deciding for a given transition system A whether there exists a Petri net N whose reachability graph is isomorphic to A. In case of a positive decision, N should be constructed. Several works examined the synthesis of Petri net subclasses that restrict, for every place p of the net, the cardinality of its preset or of its postset or both in advance by small natural numbers \(\varrho \) and \(\kappa \), respectively, such as, for example, (weighted) marked graphs and (weighted) T-systems and choice-free nets. In this paper, we study the synthesis aiming at Petri nets, which have such restricted place environments, from the viewpoint of classical and parameterized complexity: We first show that, for any fixed natural numbers \(\varrho \) and \(\kappa \), deciding whether for a given transition system A there is a Petri net N such that (1) its reachability graph is isomorphic to A and (2) for every place p of N the preset of p has at most \(\varrho \) and the postset of p has at most \(\kappa \) elements is doable in polynomial time. Secondly, we introduce a modified version of the problem, namely Environment Restricted Synthesis (ERS, for short), where \(\varrho \) and \(\kappa \) are part of the input and show that ERS is NP-complete. Our methods also imply that ERS parameterized by \(\varrho +\kappa \) is W[2]-hard.

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Acknowledgements

I would like to thank Karsten Wolf, who provided a summary of the data from the Model Checking Contest. Also, I’m very thankful to the anonymous reviewers for their detailed comments and valuable suggestions.

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Authors and Affiliations

  1. Universität Rostock, Institut Für Informatik, Theoretische Informatik, Albert-Einstein-Stra ßE 22, 18059, Rostock, Germany

    Ronny Tredup

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  1. Ronny Tredup
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Correspondence to Ronny Tredup .

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Editors and Affiliations

  1. Université de Genève, Carouge, Switzerland

    Didier Buchs

  2. Universitat Politècnica de Catalunya, Barcelona, Spain

    Josep Carmona

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Tredup, R. (2021). Synthesis of Petri Nets with Restricted Place-Environments: Classical and Parameterized. In: Buchs, D., Carmona, J. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2021. Lecture Notes in Computer Science(), vol 12734. Springer, Cham. https://doi.org/10.1007/978-3-030-76983-3_15

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  • DOI: https://doi.org/10.1007/978-3-030-76983-3_15

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