Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
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)
Esparza, J.: Petri nets, commutative context-free grammars, and basic parallel processes. Fundamenta Informaticae 31(1), 13–25 (1997)
Esparza, J., Nielsen, M.: Decibility issues for Petri nets - a survey. Journal of Informatik Processing and Cybernetics 30(3), 143–160 (1994)
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)
Howell, R.R., Jancar, P., Rosier, L.E.: Completeness results for single-path Petri nets. Information and Computation 106(2), 253–265 (1993)
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)
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)
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)
Huynh, D.T.: Commutative grammars: The complexity of uniform word problems. Information and Control 57(1), 21–39 (1983)
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)
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)
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)
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)
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)
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)
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)
Yen, H.-C.: On reachability equivalence for BPP-nets. Theoretical Computer Science 179(1-2), 301–317 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mayr, E.W., Weihmann, J. (2013). Completeness Results for Generalized Communication-Free Petri Nets with Arbitrary Edge Multiplicities. In: Abdulla, P.A., Potapov, I. (eds) Reachability Problems. RP 2013. Lecture Notes in Computer Science, vol 8169. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41036-9_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-41036-9_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41035-2
Online ISBN: 978-3-642-41036-9
eBook Packages: Computer ScienceComputer Science (R0)