Completeness Results for Generalized Communication-Free Petri Nets with Arbitrary Edge Multiplicities

  • Ernst W. Mayr
  • Jeremias Weihmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8169)


We investigate gcf-Petri nets, a generalization of communication-free Petri nets allowing arbitrary edge multiplicities, and characterized by the sole restriction that each transition has at most one incoming edge. We use canonical firing sequences with nice properties for gcf-PNs to show that the RecLFS, (zero-)reachability, covering, and boundedness problems of gcf-PNs are in PSPACE. By showing, how PSPACE-Turing machines can be simulated by gss-PNs, a subclass of gcf-PNs where additionally all transitions have at most one outgoing edge, we ultimately prove the PSPACE-completess of these problems for gss/gcf-PNs. Last, we show PSPACE-hardness as well as a doubly exponential space bound for the containment and equivalence problems of gss/gcf-PNs.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Christensen, S.: Distributed bisimularity is decidable for a class of infinite state-space systems. In: Cleaveland, W.R. (ed.) CONCUR 1992. LNCS, vol. 630, pp. 148–161. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  2. 2.
    Christensen, S., Hirshfeld, Y., Moller, F.: Bisimulation equivalence is decidable for basic parallel processes. In: Best, E. (ed.) CONCUR 1993. LNCS, vol. 715, pp. 143–157. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  3. 3.
    Esparza, J.: Petri nets, commutative context-free grammars, and basic parallel processes. Fundamenta Informaticae 31(1), 13–25 (1997)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Esparza, J., Nielsen, M.: Decibility issues for Petri nets - a survey. Journal of Informatik Processing and Cybernetics 30(3), 143–160 (1994)zbMATHGoogle Scholar
  5. 5.
    Ha, L.M., Trung, P.V., Duong, P.T.H.: A polynomial-time algorithm for reachability problem of a subclass of Petri net and chip firing games. In: 2012 IEEE RIVF International Conference on Computing and Communication Technologies, Research, Innovation, and Vision for the Future (RIVF), pp. 1–6 (2012)Google Scholar
  6. 6.
    Howell, R.R., Jancar, P., Rosier, L.E.: Completeness results for single-path Petri nets. Information and Computation 106(2), 253–265 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Howell, R.R., Rosier, L.E.: Completeness results for conflict-free vector replacement systems. Journal of Computer and System Sciences 37(3), 349–366 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Howell, R.R., Rosier, L.E., Yen, H.-C.: Normal and sinkless Petri nets. In: Csirik, J.A., Demetrovics, J. (eds.) FCT 1989. LNCS, vol. 380, pp. 234–243. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  9. 9.
    Huynh, D.T.: The complexity of semilinear sets. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 324–337. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  10. 10.
    Huynh, D.T.: Commutative grammars: The complexity of uniform word problems. Information and Control 57(1), 21–39 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Huynh, D.T.: A simple proof for the sum upper bound of the inequivalence problem for semilinear sets. Elektronische Informationsverarbeitung und Kybernetik, 147–156 (1986)Google Scholar
  12. 12.
    Mayr, E.W.: An algorithm for the general Petri net reachability problem. In: Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computing, STOC 1981, pp. 238–246. ACM, New York (1981)CrossRefGoogle Scholar
  13. 13.
    Mayr, E.W., Meyer, A.R.: The complexity of the word problems for commutative semigroups and polynomial ideals. Advances in Mathematics 46(3), 305–329 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mayr, E.W., Weihmann, J.: Completeness Results for Generalized Communication-free Petri Nets with Arbitrary Edge Multiplicities. Technical Report TUM-I1335, Institut für Informatik, TU München (Jul 2013)Google Scholar
  15. 15.
    Mayr, E.W., Weihmann, J.: Results on equivalence, boundedness, liveness, and covering problems of BPP-Petri nets. In: Colom, J.-M., Desel, J. (eds.) PETRI NETS 2013. LNCS, vol. 7927, pp. 70–89. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  16. 16.
    Pottier, L.: Minimal solutions of linear diophantine systems: bounds and algorithms. In: Book, R.V. (ed.) RTA 1991. LNCS, vol. 488, pp. 162–173. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  17. 17.
    Taoka, S., Watanabe, T.: Time complexity analysis of the legal firing sequence problem of Petri nets with inhibitor arcs. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E89-A, 3216–3226 (2006)CrossRefGoogle Scholar
  18. 18.
    Yen, H.-C.: On reachability equivalence for BPP-nets. Theoretical Computer Science 179(1-2), 301–317 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ernst W. Mayr
    • 1
  • Jeremias Weihmann
    • 1
  1. 1.Technische Universität MünchenGarchingGermany

Personalised recommendations