Petri net models for algebraic theories of concurrency

extended abstract
  • Rob van Glabbeek
  • Frits Vaandrager
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 259)


In this paper we discuss the issue of interleaving semantics versus True concurrency in an algebraic setting. We present various equivalence notions on Petri nets which can be used in the construction of algebraic models:
  1. (a)

    the occurrence net equivalence of Nielsen, Plotkin & Winskel;

  2. (b)

    bisimulation equivalence, which leads to a model which is isomorphic to the graph model of Baeten, Bergstra & Klop;

  3. (c)

    the concurrent bisimulation equivalence, which is also described by Nielsen & Thiagarajan, and Goltz;

  4. (d)

    partial order equivalences which are inspired by work of Pratt, and Boudol & Castellani.


A central role in the paper will be played by the notion of real-time consistency. We show that, besides occurrence net equivalence, none of the equivalences mentioned above (including the partial order equivalences!) is real-time consistent. Therefore we introduce the notion of ST-bisimulation equivalence, which is real-time consistent. Moreover a complete proof system will be presented for those finite ST-bisimulation processes in which no action can occur concurrently with itself.


Proof System Atomic Action Concurrent System Closed Term Springer LNCS 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [AR]
    Aalbersberg, IJ.J. & G. Rozenberg, Theory of traces, Technical Report No. 86-16, Institute of Applied Mathematics and Computer Science, University of Leiden, 1986.Google Scholar
  2. [BB]
    Baeten, J.C.M. & J.A. Bergstra, Global renaming operators in concrete process algebra, CWI Report CS-R8521, Amsterdam, 1985.Google Scholar
  3. [BBK1]
    Baeten, J.C.M., J.A. Bergstra & J.W. Klop, On the consistency of Koomen's fair abstraction rule, CWI Report CS-R8511, Amsterdam, 1985, to appear in Theor. Comp. Sci.Google Scholar
  4. [BBK2]
    Baeten, J.C.M., J.A. Bergstra & J.W. Klop, An operational semantics for process algebra, CWI Report CS-R8522, Amsterdam, 1985, to appear in: Proc. Banach semester, Warschau 1985, North-Holland.Google Scholar
  5. [BK1]
    Bergstra, J.A. & J.W. Klop, Process algebra for synchronous communication, Information & Control 60 (1/3), 1984, pp. 109–137.Google Scholar
  6. [BK2]
    Bergstra, J.A. & J.W. Klop, Process Algebra: Specification and Verification in Bisimulation Semantics, In: Proc. CWI Symposium Math. & Comp. Sci. (M. Hazewinkel, J.K. Lenstra & L.G.L.T. Meertens, eds.), North Holland, 1986, pp. 61–94.Google Scholar
  7. [BK3]
    Bergstra, J.A. & J.W. Klop, Algebra of Communicating processes with abstraction, Theor. Comp. Sci. 37(1), 1985, pp. 77–121.CrossRefGoogle Scholar
  8. [BC]
    Boudol, G. & I. Castellani, On the semantics of concurrency: partial orders and transition systems, Rapports de Recherche No 550, INRIA, Centre Sophia Antipolis, 1986.Google Scholar
  9. [CCG]
    Carlier, J., Chretienne & C. Girault, Modelling scheduling problems with timed Petri nets, In: Advances in Petri Nets 1984 (G. Rozenberg, ed.), Springer LNCS 188, 1985, pp. 62–82.Google Scholar
  10. [GV]
    van Glabbeek, R.J. & F.W. Vaandrager, Petri net models for algebraic theories of concurrency, to appear as: CWI Report CS-R87.., Amsterdam, 1987.Google Scholar
  11. [G]
    Goltz, U., Building Structured Petri Nets, Arbeitspapiere der GMD 223, Sankt Augustin, 1986.Google Scholar
  12. [GM]
    Goltz, U. & A. Mycroft, On the relationship of CCS and Petri nets, In: Proc. ICALP 84 (J. Paredaens, ed.), Springer LNCS 172, 1984, pp. 196–208.Google Scholar
  13. [Ho]
    Hospers, J., An Introduction to Philosophical Analysis, second edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967.Google Scholar
  14. [M1]
    Mazurkiewicz, A., Concurrent program schemes and their interpretations, Report DAIMI PB-78, Computer Science Department, Aarhus University, Aarhus, 1978.Google Scholar
  15. [M2]
    Mazurkiewicz, A., Semantics of concurrent systems: a modular fixed-point trace approach, In: Advances in Petri Nets 1984 (G. Rozenberg, ed.), Springer LNCS 188, 1985, pp. 353–375.Google Scholar
  16. [Mi]
    Milner, R., A calculus for Communicating Systems, Springer LNCS 92, 1980.Google Scholar
  17. [NPW]
    Nielsen, M., G.D. Plotkin & G. Winskel, Petri nets, event structures and domains, part I. Theor. Comp. Sci., 13(1). 1981, pp. 85–108.CrossRefGoogle Scholar
  18. [NT]
    Nielsen, M. & P.S. Thiagarajan, Degrees of Non-Determinism and Concurrency: A Petri Net View, In: Proc. of the 5th Conf. on Found. of Softw. Techn. and Theor. Comp. Sci. (M. Joseph & R. Shyamasundar, eds.), Springer LNCS 181, 1984, pp. 89–118.Google Scholar
  19. [Pa]
    Park, D.M.R., Concurrency and automata on infinite sequences, Proc. 5th GI Conference (P. Deussen, ed.), Springer LNCS 104, 1981, pp. 167–183.Google Scholar
  20. [Pe]
    Petri, C.A., Kommunikation mit Automaten, Schriften des Institutes für Instrumentelle Mathematik, Bonn, 1962.Google Scholar
  21. [Po]
    Pomello, L., Some equivalence notions for concurrent systems. An overview. In: Advances in Petri Nets 1985 (G. Rozenberg, ed.), Springer LNCS 222, 1986, pp. 381–400.Google Scholar
  22. [Pr1]
    Pratt, V.R., On the Composition of Processes, Proc. of the 9th POPL, 1982, pp. 213–223.Google Scholar
  23. [Pr2]
    Pratt, V.R., Modelling Concurrency with Partial Orders, International Journal of Parallel Programming, Vol. 15, No. 1, 1986, pp. 33–71.CrossRefGoogle Scholar
  24. [RR]
    Reed, G.M. & A.W. Roscoe, A Timed Model for Communicating Sequential Processes, In: Proc. ICALP 86 (L. Kott, ed.), Springer LNCS 226, 1986, pp. 314–323.Google Scholar
  25. [R]
    Reisig, W., Petri Nets, An Introduction, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1985.Google Scholar
  26. [RT]
    Rozenberg, G. & P.S. Thiagarajan, Petri nets: basic notions, structure, behaviour. In: Current Trends in Concurrency, Overviews and Tutorials (J.W. de Bakker, W.P. de Roever, G. Rozenberg, eds.), Springer LNCS 224, 1986, pp. 585–668.Google Scholar
  27. [W1]
    Winskel, G., Event structure semantics for CCS and related languages, In: Proc. 9th ICALP (M. Nielsen & E.M. Schmidt, eds.), Springer LNCS 140, 1982, pp. 561–576.Google Scholar
  28. [W2]
    Winskel, G., A new definition of morphism on Petri net, In: Proc. STACS 84 (M. Fontet, K. Mehlhorn, eds.), Springer LNCS 166, 1984, pp. 140–150.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Rob van Glabbeek
    • 1
  • Frits Vaandrager
    • 1
  1. 1.Centre for Mathematics and Computer ScienceAmsterdamThe Netherlands

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