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The Power of Prime Cycles

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Application and Theory of Petri Nets and Concurrency (PETRI NETS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9698))

Abstract

In this paper, we shall examine properties of labelled transition systems which are motivated by system synthesis. Most of them are necessary conditions for synthesis by Petri nets to be successful. They can be checked in a pre-synthesis phase, allowing the immediate rejection of transition systems not satisfying them as non-synthesisable. The order of checking such conditions plays an important role in pre-synthesis optimisation. It is particularly desirable to know which conditions are implied by others, especially if the latter can be machine-verified more simply than the former. The purpose of this paper is to describe some mathematical results exhibiting a number of such implications.

Two properties called strong cycle-consistency and full backward determinism, respectively, are particularly hard to check. They are generalised counterparts of the marking equation of Petri net theory. We show that under some circumstances, they may be deduced from other properties which are easier to check. Amongst these other properties, the prime cycle property plays a particularly important role, not just because it is strong enough to imply others, but also because it is interesting to be checked on its own, if synthesis is targetted towards choice-free Petri nets.

E. Best—Supported by DFG (German Research Foundation) through grants Be 1267/15-1 ARS (Algorithms for Reengineering and Synthesis) and Be 1267/14-1 CAVER (Comparative Analysis and Verification for Correctness-Critical Systems).

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Notes

  1. 1.

    This algorithm has a complexity of \(O(n{\cdot }e{\cdot }\gamma )\) where n is the number of nodes of a graph, e the number of edges, and \(\gamma \) the number of small cycles. Thus, in our context, a bad upper bound is \(O(n^3{\cdot }m)\) where \(n=|S|\) and \(m=|T|\), which, however, still compares favourably with \(O(n^6)\) for the full synthesis algorithm.

  2. 2.

    Since, in brute force form, they involve, for all states \(s\in S\), Parikh-comparisons of pairs of – possibly long – paths emanating from s.

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Acknowledgments

The authors would like to thank Harro Wimmel and Valentin Spreckels for carefully commenting on a draft version of this paper and for checking and confirming that the assumption of weak cycle-consistency can indeed be circumvented in the proof of Theorem 2. The authors are also indebted to the anonymous reviewers for useful detailed comments.

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Authors and Affiliations

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

    Eike Best

  2. Département d’Informatique, Université Libre de Bruxelles, Boulevard du Triomphe - C.P. 212, B-1050, Bruxelles, Belgium

    Raymond Devillers

Authors
  1. Eike Best
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Correspondence to Eike Best .

Editor information

Editors and Affiliations

  1. Université Pierre and Marie Curie, Paris, France

    Fabrice Kordon

  2. Department Informatik, Universität Hamburg Department Informatik, Hamburg, Germany

    Daniel Moldt

A Error messages of synet and APT (and pre-synthesis)

A Error messages of synet and APT (and pre-synthesis)

When asked to synthesise \( TS _2\) (Fig. 1), synet [12] outputs

figure a

while APT’s [24] output is:

figure b

Instead, our suggested pre-synthesis algorithm would issue an error message such as:

figure c

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Best, E., Devillers, R. (2016). The Power of Prime Cycles. In: Kordon, F., Moldt, D. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2016. Lecture Notes in Computer Science(), vol 9698. Springer, Cham. https://doi.org/10.1007/978-3-319-39086-4_5

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  • DOI: https://doi.org/10.1007/978-3-319-39086-4_5

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  • Online ISBN: 978-3-319-39086-4

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