The Emptiness Problem for Valence Automata or: Another Decidable Extension of Petri Nets

  • Georg ZetzscheEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9328)


This work studies which storage mechanisms in automata permit decidability of the reachability problem. The question is formalized using valence automata, an abstract model that generalizes automata with storage. For each of a variety of storage mechanisms, one can choose a (typically infinite) monoid M such that valence automata over M are equivalent to (one-way) automata with this type of storage.

In fact, many interesting storage mechanisms can be realized by monoids defined by finite graphs, called graph monoids. Hence, we study for which graph monoids the emptiness problem for valence automata is decidable. A particular model realized by graph monoids is that of Petri nets with a pushdown stack. For these, decidability is a long-standing open question and we do not answer it here.

However, if one excludes subgraphs corresponding to this model, a characterization can be achieved. This characterization yields a new extension of Petri nets with a decidable reachability problem. Moreover, we provide a description of those storage mechanisms for which decidability remains open. This leads to a natural model that generalizes both pushdown Petri nets and priority multicounter machines.


Free Product Storage Mechanism Reachability Problem Counter Machine Looped Vertex 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berstel, J.: Transductions and Context-Free Languages. Teubner, Stuttgart (1979)CrossRefzbMATHGoogle Scholar
  2. 2.
    Diekert, V., Rozenberg, G. (eds.): The Book of Traces. World Scientific, Singapore (1995)Google Scholar
  3. 3.
    Kambites, M.: Formal Languages and Groups as Memory. Commun. Algebra 37, 193–208 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kambites, M., Silva, P.V., Steinberg, B.: On the rational subset problem for groups. J. Algebra 309, 622–639 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Leroux, J., Sutre, G., Totzke, P.: On the coverability problem for pushdown vector addition systems in one dimension. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 324–336. Springer, Heidelberg (2015) CrossRefGoogle Scholar
  6. 6.
    Lohrey, M.: The rational subset membership problem for groups: A survey (to appear)Google Scholar
  7. 7.
    Lohrey, M., Steinberg, B.: The submonoid and rational subset membership problems for graph groups. J. Algebra 320(2), 728–755 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Reinhardt, K.: Reachability in petri nets with inhibitor arcs. In: Proc. of RP 2008 (2008)Google Scholar
  9. 9.
    van Leeuwen, J.: A generalisation of Parikh’s theorem in formal language theory. In: Loeckx, J. (ed.) Automata, Languages and Programming. LNCS, vol. 14, pp. 17–26. Springer, Heidelberg (1974) CrossRefGoogle Scholar
  10. 10.
    Wolk, E.S.: A Note on ‘The Comparability Graph of a Tree’. P. Am. Math. Soc. 16(1), 17–20 (1965)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Zetzsche, G.: Computing downward closures for stacked counter automata. In: Proc. of STACS 2015 (2015)Google Scholar
  12. 12.
    Zetzsche, G.: Silent transitions in automata with storage. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 434–445. Springer, Heidelberg (2013) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Concurrency Theory Group, Fachbereich InformatikTechnische Universität KaiserslauternKaiserslauternGermany

Personalised recommendations