Abstract
The reachability problem for Petri nets is a central problem of net theory. The problem is known to be decidable by inductive invariants definable in the Presburger arithmetic. When the reachability set is definable in the Presburger arithmetic, the existence of such an inductive invariant is immediate. However, in this case, the computation of a Presburger formula denoting the reachability set is an open problem. Recently this problem got closed by proving that if the reachability set of a Petri net is definable in the Presburger arithmetic, then the Petri net is flat, i.e. its reachability set can be obtained by runs labeled by words in a bounded language. As a direct consequence, classical algorithms based on acceleration techniques effectively compute a formula in the Presburger arithmetic denoting the reachability set.
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
Esparza, J., Nielsen, M.: Decidability issues for petri nets - a survey. Bulletin of the European Association for Theoretical Computer Science 52, 245–262 (1994)
Figueira, D., Figueira, S., Schmitz, S., Schnoebelen, P.: Ackermannian and primitive-recursive bounds with dickson’s lemma. In: Proc. of LICS 2011, pp. 269–278. IEEE Computer Society (2011)
Fribourg, L.: Petri nets, flat languages and linear arithmetic. In: Alpuente, M. (ed.) Proc. of WFLP 2000, pp. 344–365 (2000)
Ginsburg, S., Spanier, E.H.: Bounded regular sets. Proceedings of the American Mathematical Society 17(5), 1043–1049 (1966)
Ginsburg, S., Spanier, E.H.: Semigroups, Presburger formulas and languages. Pacific Journal of Mathematics 16(2), 285–296 (1966)
Hauschildt, D.: Semilinearity of the Reachability Set is Decidable for Petri Nets. PhD thesis, University of Hamburg (1990)
Hopcroft, J.E., Pansiot, J.-J.: On the reachability problem for 5-dimensional vector addition systems. Theoritical Computer Science 8, 135–159 (1979)
Rao Kosaraju, S.: Decidability of reachability in vector addition systems (preliminary version). In: Proc. of STOC 1982, pp. 267–281. ACM (1982)
Lambert, J.L.: A structure to decide reachability in petri nets. Theoretical Computer Science 99(1), 79–104 (1992)
Leroux, J.: The general vector addition system reachability problem by Presburger inductive invariants. In: Proc. of LICS 2009, pp. 4–13. IEEE Computer Society (2009)
Leroux, J.: Presburger vector addition systems. In: Proc. LICS 2013, IEEE Computer Society (to appear, 2013)
Leroux, J., Sutre, G.: Flat counter automata almost everywhere! In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 489–503. Springer, Heidelberg (2005)
Mayr, E.W.: An algorithm for the general petri net reachability problem. In: Proc. of STOC 1981, pp. 238–246. ACM (1981)
Sacerdote, G.S., Tenney, R.L.: The decidability of the reachability problem for vector addition systems (preliminary version). In: Proc. of STOC 1977, pp. 61–76. ACM (1977)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this paper
Cite this paper
Leroux, J. (2013). Acceleration for Petri Nets. In: Van Hung, D., Ogawa, M. (eds) Automated Technology for Verification and Analysis. Lecture Notes in Computer Science, vol 8172. Springer, Cham. https://doi.org/10.1007/978-3-319-02444-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-02444-8_1
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02443-1
Online ISBN: 978-3-319-02444-8
eBook Packages: Computer ScienceComputer Science (R0)