Reductions of petri nets and unfolding of propositional logic programs

  • Laurent Fribourg
  • Hans Olsén
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1207)


We present in this paper a new method for computing the reachability set associated with a Petri net. The method proceeds, first, by coding the reachability problem into a predicate defined by a logic program with arithmetic constraints, then by transforming the encoding program. Each transition of the Petri net is encoded as a clause. The guards appearing in the clauses correspond to the enabling conditions of the corresponding transitions. In order to characterize arithmetically the least model of the transformed program, we define a system of two reduction rules which apply iteratively to the program. When the process of reduction terminates with success, it generates a formula of Presburger arithmetic, which contains the components of the initial marking as parameters. In the particular case of a BPP-net (i.e., a Petri net where every transition has exactly one input place), the process is guaranteed to succeed, thus providing us with an exact and generic characterization of the reachability set. We show finally how unfolding of propositional logic programs can be viewed as transformation of BPP-nets.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Ajmone Marsan, G. Balbo, G. Conte, S. Donatelli and G. Franceschinis. Modelling with Generalized Stochastic Petri Nets, John Wiley & Sons, Chichester, 1995.Google Scholar
  2. 2.
    G. Berthelot. “Transformations and Decompositions of Nets”. Advances in Petri Nets, LNCS 254, Springer-Verlag, 1986, pp. 359–376.Google Scholar
  3. 3.
    G.W. Brams. Réseaux de Petri: Théorie et Pratique, Masson, Paris, 1983.Google Scholar
  4. 4.
    S. Christensen. Decidability and Decomposition in Process Algebras. Ph.D. Thesis, University of Edinburgh, CST-105-93, 1993.Google Scholar
  5. 5.
    J. Esparza and M. Nielsen. “Decidability Issues for Petri Nets”. Bulletin of the EATCS, Number 52, Feb. 1994.Google Scholar
  6. 6.
    J. Esparza. “Petri Nets, Commutative Context-Free Grammars, and Basic Parallel Processes”. Proc. of Fundamentals of Computer Theory '95, LNCS 965, 1995, pp. 221–232.Google Scholar
  7. 7.
    J. Jaffar and J.L. Lassez. “Constraint Logic Programming”, Proc. 14th ACM Symp. on Principles of Programming Languages, 1987, pp. 111–119.Google Scholar
  8. 8.
    P. Kanellakis, G. Kuper and P. Revesz. “Constraint Query Languages”. Internal Report, November 1990. (Short version in Proc. 9th ACM Symp. on Principles of Database Systems, Nashville, 1990, pp. 299–313).Google Scholar
  9. 9.
    L. Fribourg and H. Olsén. Datalog Programs with Arithmetical Constraints: Hierarchic, Periodic an Spiralling Least Fixpoints. Technical Report LIENS-95-26, Ecole Normale Supérieure, Paris, November 1995.Google Scholar
  10. 10.
    L. Fribourg and H. Olsén. A Decompositional Approach for Computing Least Fixed-Points of Datalog Programs with Z-counters. Technical Report LIENS-96-12, Ecole Normale Supérieure, Paris, July 1996 (available by anonymous ftp on in /pub/reports/liens or on Scholar
  11. 11.
    S. Ginsburg and E.H. Spanier. “Semigroups, Presburger formulas and languages”. Pacific Journal of Mathematics 16, 1966, pp. 285–296.Google Scholar
  12. 12.
    J. C. Shepherdson, “Unfold/fold transformations of logic programs”. Math. Struct. in Comp. Science, 1992, vol. 2, pp. 143–157Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Laurent Fribourg
    • 1
  • Hans Olsén
    • 2
  1. 1.Ecole Normale Supérieure & CNRSParisFrance
  2. 2.IDA, Linköping UniversityLinköpingSweden

Personalised recommendations