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Efficient Synthesis of Weighted Marked Graphs with Circular Reachability Graph, and Beyond

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Transactions on Petri Nets and Other Models of Concurrency XV

Part of the book series: Lecture Notes in Computer Science ((TOPNOC,volume 12530))

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Abstract

In previous studies, several methods have been developed to synthesise Petri nets from labelled transition systems (LTS), often with structural constraints on the net and on the LTS. In this paper, we focus on Weighted Marked Graphs (WMGs) and Choice-Free (CF) Petri nets, two weighted subclasses of nets in which each place has at most one output; WMGs have the additional constraint that each place has at most one input.

We provide new conditions for checking the existence of a WMG whose reachability graph is isomorphic to a given circular LTS, i.e. forming a single cycle; we develop two new polynomial-time synthesis algorithms dedicated to these constraints: the first one is LTS-based (classical synthesis) while the second one is vector-based (weak synthesis) and more efficient in general. We show that our conditions also apply to CF synthesis in the case of three-letter alphabets, and we discuss the difficulties in extending them to CF synthesis over arbitrary alphabets.

E. Erofeev—Supported by DFG through grant Be 1267/16-1 ASYST.

T. Hujsa—Supported by the STAE foundation/project DAEDALUS, Toulouse, France.

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Notes

  1. 1.

    A set A of k arcs in a LTS G defines a cycle of G if the elements of A can be ordered as a sequence \(a_1 \ldots a_k\) such that, for each \(i \in \{1, \ldots , k\}\), \(a_i = (n_i,\ell _i,n_{i+1})\) and \(n_{k+1} = n_1\), i.e. the i-th arc \(a_i\) goes from node \(n_i\) to node \(n_{i+1}\) until the first node \(n_1\) is reached, closing the path.

  2. 2.

    A place p is a side condition if \({}^\bullet p \cap p^\bullet \ne \emptyset \).

  3. 3.

    The projection of a word \(w\in A^*\) on a set \(A' \subseteq A\) of labels, noted , is the word obtained by erasing in w all the occurrences of labels belonging to \(A\setminus A'\). For example, the projection of the word \(w = \ell _1 \, \ell _2 \, \ell _3 \, \ell _2\) on the set \(\{\ell _1 ,\, \ell _2\}\) is the word \(\ell _1 \, \ell _2 \, \ell _2\).

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Acknowledgements

We would like to thank the anonymous referees for their careful proofreading, their relevant comments and their insightful suggestion of addressing the process mining area.

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Authors and Affiliations

  1. Département d’Informatique, Université Libre de Bruxelles, 1050, Brussels, Belgium

    Raymond Devillers

  2. Department of Computing Science, Carl von Ossietzky Universität Oldenburg, 26111, Oldenburg, Germany

    Evgeny Erofeev

  3. LAAS-CNRS, Université de Toulouse, CNRS, Toulouse, France

    Thomas Hujsa

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Correspondence to Thomas Hujsa .

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Editors and Affiliations

  1. Newcastle University, Newcastle upon Tyne, UK

    Maciej Koutny

  2. Sorbonne Université, Paris, France

    Fabrice Kordon

  3. Università degli Studi di Milano-Bicocca, Milan, Italy

    Lucia Pomello

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Devillers, R., Erofeev, E., Hujsa, T. (2021). Efficient Synthesis of Weighted Marked Graphs with Circular Reachability Graph, and Beyond. In: Koutny, M., Kordon, F., Pomello, L. (eds) Transactions on Petri Nets and Other Models of Concurrency XV. Lecture Notes in Computer Science(), vol 12530. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63079-2_4

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