Abstract
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.
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
Berstel, J.: Transductions and Context-Free Languages. Teubner, Stuttgart (1979)
Diekert, V., Rozenberg, G. (eds.): The Book of Traces. World Scientific, Singapore (1995)
Kambites, M.: Formal Languages and Groups as Memory. Commun. Algebra 37, 193–208 (2009)
Kambites, M., Silva, P.V., Steinberg, B.: On the rational subset problem for groups. J. Algebra 309, 622–639 (2007)
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)
Lohrey, M.: The rational subset membership problem for groups: A survey (to appear)
Lohrey, M., Steinberg, B.: The submonoid and rational subset membership problems for graph groups. J. Algebra 320(2), 728–755 (2008)
Reinhardt, K.: Reachability in petri nets with inhibitor arcs. In: Proc. of RP 2008 (2008)
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)
Wolk, E.S.: A Note on ‘The Comparability Graph of a Tree’. P. Am. Math. Soc. 16(1), 17–20 (1965)
Zetzsche, G.: Computing downward closures for stacked counter automata. In: Proc. of STACS 2015 (2015)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Zetzsche, G. (2015). The Emptiness Problem for Valence Automata or: Another Decidable Extension of Petri Nets. In: Bojanczyk, M., Lasota, S., Potapov, I. (eds) Reachability Problems. RP 2015. Lecture Notes in Computer Science(), vol 9328. Springer, Cham. https://doi.org/10.1007/978-3-319-24537-9_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-24537-9_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24536-2
Online ISBN: 978-3-319-24537-9
eBook Packages: Computer ScienceComputer Science (R0)