Advertisement

Synthesis of Weighted Marked Graphs from Constrained Labelled Transition Systems: A Geometric Approach

  • Raymond Devillers
  • Evgeny Erofeev
  • Thomas HujsaEmail author
Chapter
  • 177 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11790)

Abstract

Recent studies investigated the problems of analysing Petri nets and synthesising them from labelled transition systems (LTS) with two labels (transitions) only. In this paper, we extend these works by providing new conditions for the synthesis of Weighted Marked Graphs (WMGs), a well-known and useful class of weighted Petri nets in which each place has at most one input and one output.

Some of these new conditions do not restrict the number of labels; the other ones consider up to 3 labels. Additional constraints are investigated: when the LTS is either finite or infinite, and either cyclic or acyclic. We show that one of these conditions, developed for 3 labels, does not extend to 4 nor to 5 labels. Also, we tackle geometrically the WMG-solvability of finite, acyclic LTS with any number of labels.

Keywords

Weighted Petri net Marked graph Synthesis Labelled transition system Cycles Cyclic words Circular solvability Theory of regions Geometric interpretation 

Notes

Acknowledgements

We would like to thank the anonymous referees for their involvement and useful suggestions.

References

  1. 1.
    Murata, T.: Petri nets: properties, analysis and applications. Proc. IEEE 77(4), 541–580 (1989)CrossRefGoogle Scholar
  2. 2.
    Teruel, E., Silva, M.: Structure theory of equal conflict systems. Theoret. Comput. Sci. 153(1&2), 271–300 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hujsa, T., Devillers, R.: On liveness and deadlockability in subclasses of weighted Petri nets. In: van der Aalst, W., Best, E. (eds.) PETRI NETS 2017. LNCS, vol. 10258, pp. 267–287. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-57861-3_16CrossRefzbMATHGoogle Scholar
  4. 4.
    Desel, J., Esparza, J.: Free Choice Petri Nets. Cambridge Tracts in Theoretical Computer Science, vol. 40. Cambridge University Press, New York (1995)CrossRefGoogle Scholar
  5. 5.
    Teruel, E., Colom, J.M., Silva, M.: Choice-free Petri nets: a model for deterministic concurrent systems with bulk services and arrivals. IEEE Trans. Syst. Man Cybern. Part A 27(1), 73–83 (1997)CrossRefGoogle Scholar
  6. 6.
    Hujsa, T., Delosme, J.-M., Munier-Kordon, A.: On the reversibility of well-behaved weighted choice-free systems. In: Ciardo, G., Kindler, E. (eds.) PETRI NETS 2014. LNCS, vol. 8489, pp. 334–353. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-07734-5_18CrossRefzbMATHGoogle Scholar
  7. 7.
    Commoner, F., Holt, A., Even, S., Pnueli, A.: Marked directed graphs. J. Comput. Syst. Sci. 5(5), 511–523 (1971)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Teruel, E., Chrzastowski-Wachtel, P., Colom, J.M., Silva, M.: On weighted T-systems. In: Jensen, K. (ed.) ICATPN 1992. LNCS, vol. 616, pp. 348–367. Springer, Heidelberg (1992).  https://doi.org/10.1007/3-540-55676-1_20CrossRefGoogle Scholar
  9. 9.
    Best, E., Hujsa, T., Wimmel, H.: Sufficient conditions for the marked graph realisability of labelled transition systems. Theoret. Comput. Sci. 750, 101–116 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Devillers, R., Hujsa, T.: Analysis and synthesis of weighted marked graph Petri nets. In: Khomenko, V., Roux, O.H. (eds.) PETRI NETS 2018. LNCS, vol. 10877, pp. 19–39. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-91268-4_2CrossRefzbMATHGoogle Scholar
  11. 11.
    Delosme, J.M., Hujsa, T., Munier-Kordon, A.: Polynomial sufficient conditions of well-behavedness for weighted join-free and choice-free systems. In: 13th International Conference on Application of Concurrency to System Design, pp. 90–99, July 2013Google Scholar
  12. 12.
    Hujsa, T., Delosme, J.M., Munier-Kordon, A.: Polynomial sufficient conditions of well-behavedness and home markings in subclasses of weighted Petri nets. ACM Trans. Embed. Comput. Syst. 13(4s), 141:1–141:25 (2014)CrossRefGoogle Scholar
  13. 13.
    Barylska, K., Best, E., Erofeev, E., Mikulski, L., Piatkowski, M.: On binary words being Petri net solvable. In: Proceedings of the International Workshop on Algorithms & Theories for the Analysis of Event Data, ATAED 2015, Brussels, Belgium, pp. 1–15 (2015)Google Scholar
  14. 14.
    Barylska, K., Best, E., Erofeev, E., Mikulski, L., Piatkowski, M.: Conditions for Petri net solvable binary words. Trans. Petri Nets Other Models Concurr. 11, 137–159 (2016)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Erofeev, E., Barylska, K., Mikulski, L., Piatkowski, M.: Generating all minimal Petri net unsolvable binary words. In: Proceedings of the Prague Stringology Conference 2016, Prague, Czech Republic, pp. 33–46 (2016)Google Scholar
  16. 16.
    Erofeev, E., Wimmel, H.: Reachability graphs of two-transition Petri nets. In: Proceedings of the International Workshop on Algorithms & Theories for the Analysis of Event Data 2017, Zaragoza, Spain, pp. 39–54 (2017)Google Scholar
  17. 17.
    Best, E., Devillers, R.: Synthesis and reengineering of persistent systems. Acta Inf. 52(1), 35–60 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hujsa, T., Delosme, J.M., Munier-Kordon, A.: On liveness and reversibility of equal-conflict Petri nets. Fundamenta Informaticae 146(1), 83–119 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hujsa, T.: Contribution to the study of weighted Petri nets. Ph.D. thesis, Pierre and Marie Curie University, Paris, France (2014)Google Scholar
  20. 20.
    Devillers, R., Erofeev, E., Hujsa, T.: Synthesis of weighted marked graphs from constrained labelled transition systems. In: Proceedings of the International Workshop on Algorithms & Theories for the Analysis of Event Data, Bratislava, Slovakia, pp. 75–90 (2018)Google Scholar
  21. 21.
    Crespi-Reghizzi, S., Mandrioli, D.: A decidability theorem for a class of vector-addition systems. Inf. Process. Lett. 3(3), 78–80 (1975)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Devillers, R.: Products of transition systems and additions of Petri nets. In: Desel, J., Yakovlev, A. (eds.) Proceedings of 16th International Conference on Application of Concurrency to System Design (ACSD 2016), pp. 65–73 (2016)Google Scholar
  23. 23.
    Devillers, R.: Factorisation of transition systems. Acta Informatica 55, 339–362 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Best, E., Erofeev, E., Schlachter, U., Wimmel, H.: Characterising Petri net solvable binary words. In: Kordon, F., Moldt, D. (eds.) PETRI NETS 2016. LNCS, vol. 9698, pp. 39–58. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-39086-4_4CrossRefzbMATHGoogle Scholar
  25. 25.
    Doignon, J.P.: Convexity in cristallographical lattices. J. Geom. 3(1), 71–85 (1973)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Keller, R.M.: A fundamental theorem of asynchronous parallel computation. In: Feng, T. (ed.) Parallel Processing. LNCS, vol. 24, pp. 102–112. Springer, Heidelberg (1975).  https://doi.org/10.1007/3-540-07135-0_113CrossRefGoogle Scholar
  27. 27.
    Badouel, E., Bernardinello, L., Darondeau, P.: Petri Net Synthesis. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-47967-4CrossRefzbMATHGoogle Scholar
  28. 28.
    David, R., Alla, H.: Discrete, Continuous, and Hybrid Petri Nets, 2nd edn. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-10669-9CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Raymond Devillers
    • 1
  • Evgeny Erofeev
    • 2
  • Thomas Hujsa
    • 3
    Email author
  1. 1.Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  2. 2.Department of Computing ScienceCarl von Ossietzky Universität OldenburgOldenburgGermany
  3. 3.LAAS-CNRS, Université de Toulouse, CNRSToulouseFrance

Personalised recommendations