On Causal Semantics of Petri Nets

  • Rob J. van Glabbeek
  • Ursula Goltz
  • Jens-Wolfhard Schicke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6901)


We consider approaches for causal semantics of Petri nets, explicitly representing dependencies between transition occurrences. For one-safe nets or condition/event-systems, the notion of process as defined by Carl Adam Petri provides a notion of a run of a system where causal dependencies are reflected in terms of a partial order. A wellknown problem is how to generalise this notion for nets where places may carry several tokens. Goltz and Reisig have defined such a generalisation by distinguishing tokens according to their causal history. However, this so-called individual token interpretation is often considered too detailed. A number of approaches have tackled the problem of defining a more abstract notion of process, thereby obtaining a so-called collective token interpretation. Here we give a short overview on these attempts and then identify a subclass of Petri nets, called structural conflict nets, where the interplay between conflict and concurrency due to token multiplicity does not occur. For this subclass, we define abstract processes as equivalence classes of Goltz-Reisig processes. We justify this approach by showing that we obtain exactly one maximal abstract process if and only if the underlying net is conflict-free with respect to a canonical notion of conflict.


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  1. [BD87]
    Best, E., Devillers, R.R.: Sequential and concurrent behaviour in Petri net theory. Theoretical Computer Science 55(1), 87–136 (1987), doi:10.1016/0304-3975(87)90090-9; See also: Best, E., Devillers, R.R.: Interleaving and Partial Orders in Concurrency: A Formal Comparison. In: Wirsing, M. (ed.) Formal Description of Programming Concepts III, pp. 299–321. North-Holland, Amsterdam (1987)MathSciNetCrossRefMATHGoogle Scholar
  2. [BMO09]
    Barylska, K., Mikulski, Ł., Ochmański, E.: Nonviolence Petri Nets. In: Proceedings Workshop on Concurrency, Specification and Programming, CS&P, pp. 50–59 (2009)Google Scholar
  3. [DMM89]
    Degano, P., Meseguer, J., Montanari, U.: Axiomatizing Net Computations and Processes. In: Proceedings Fourth Annual Symposium on Logic in Computer Science, LICS 1989, pp. 175–185. IEEE, Pacific Grove (1989); see also Degano, P., Meseguer, J., Montanari, U.: Axiomatizing Net Computations and Processes. Acta Informatica 33(5), 641–667 (1996), doi:10.1007/BF03036469Google Scholar
  4. [Eng91]
    Engelfriet, J.: Branching Processes of Petri Nets. Acta Informatica 28(6), 575–591 (1991)MathSciNetCrossRefMATHGoogle Scholar
  5. [GG01]
    van Glabbeek, R.J., Goltz, U.: Refinement of actions and equivalence notions for concurrent systems. Acta Informatica 37(4/5), 229–327 (2001), doi:10.1007/s002360000041MathSciNetCrossRefMATHGoogle Scholar
  6. [GGS11a]
    van Glabbeek, R.J., Goltz, U., Schicke, J.-W.: Abstract Processes of Place/Transition Systems. Information Processing Letters 111(13), 626–633 (2011), doi:10.1016/j.ipl.2011.03.013MathSciNetCrossRefMATHGoogle Scholar
  7. [GGS11b]
    van Glabbeek, R.J., Goltz, U., Schicke, J.-W.: On Causal Semantics of Petri Nets. Technical Report 2011-06, TU Braunschweig (2011)Google Scholar
  8. [Gla05]
    van Glabbeek, R.J.: The Individual and Collective Token Interpretations of Petri Nets. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 323–337. Springer, Heidelberg (2005), doi:10.1007/11539452_26CrossRefGoogle Scholar
  9. [Gol86]
    Goltz, U.: How Many Transitions may be in Conflict? Petri Net Newsletter 25, 4–9 (1986)Google Scholar
  10. [Gol87]
    Goltz, U.: On condition/event representations of place/transition nets. In: Concurrency and Nets: Advances in Petri Nets. LNCS, pp. 217–231. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  11. [GP95]
    van Glabbeek, R.J., Plotkin, G.D.: Configuration Structures (extended abstract). In: Kozen, D. (ed.) Proceedings LICS 1995, pp. 199–209 (1995); See also van Glabbeek, R.J., Plotkin, G.D.: Configuration Structures, Event Structures and Petri Nets. Theoretical Computer Science 410(41), 4111–4159, doi:10.1016/j.tcs.2009.06.014 (2009)Google Scholar
  12. [GR83]
    Goltz, U., Reisig, W.: The Non-Sequential Behaviour of Petri Nets. Information and Control 57(2-3), 125–147 (1983)MathSciNetCrossRefMATHGoogle Scholar
  13. [GSW80]
    Genrich, H.J., Stankiewicz-Wiechno, E.: A Dictionary of Some Basic Notions of Net Theory. In: Brauer, W. (ed.) Advanced Course: Net Theory and Applications. LNCS, vol. 84, pp. 519–531. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  14. [HKT95]
    Hoogers, P.W., Kleijn, H.C.M., Thiagarajan, P.S.: A Trace Semantics for Petri Nets. Information and Computation 117, 98–114 (1995)MathSciNetCrossRefMATHGoogle Scholar
  15. [Maz89]
    Mazurkiewicz, A.W.: Concurrency, Modularity, and Synchronization. In: Kreczmar, A., Mirkowska, G. (eds.) MFCS 1989. LNCS, vol. 379, pp. 577–598. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  16. [Maz95]
    Mazurkiewicz, A.W.: Introduction to Trace Theory. In: Diekert, V., Rozenberg, G. (eds.) The Book of Traces, pp. 3–41. World Scientific, Singapore (1995)CrossRefGoogle Scholar
  17. [MM88]
    Meseguer, J., Montanari, U.: Petri Nets Are Monoids: A New Algebraic Foundation for Net theory. In: Proceedings Third Annual Symposium on Logic in Computer Science, LICS 1988, Edinburgh, Scotland, pp. 155–164. IEEE, Los Alamitos (1988)Google Scholar
  18. [MMS97]
    Meseguer, J., Montanari, U., Sassone, V.: On the Semantics of Place/Transition Petri Nets. Mathematical Structures in Computer Science 7(4), 359–397 (1997)MathSciNetCrossRefMATHGoogle Scholar
  19. [NPW81]
    Nielsen, M., Plotkin, G.D., Winskel, G.: Petri nets, event structures and domains, part I. Theoretical Computer Science 13(1), 85–108 (1981), doi:10.1016/0304-3975(81)90112-2MathSciNetCrossRefMATHGoogle Scholar
  20. [Och89]
    Ochmański, E.: Personal communication (1989)Google Scholar
  21. [Pet77]
    Petri, C.A.: Non-sequential Processes. GMD-ISF Report 77.05, GMD (1977)Google Scholar
  22. [Vog91]
    Vogler, W.: Executions: a new partial-order semantics of Petri nets. Theoretical Computer Science 91(2), 205–238 (1991), doi:10.1016/0304-3975(91)90084-FMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Rob J. van Glabbeek
    • 1
    • 2
  • Ursula Goltz
    • 3
  • Jens-Wolfhard Schicke
    • 3
  1. 1.NICTASydneyAustralia
  2. 2.School of Comp. Sc. and EngineeringUniv. of New South WalesSydneyAustralia
  3. 3.Institute for Programming and Reactive SystemsTU BraunschweigGermany

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