Skip to main content
SpringerLink
Log in

Taming Past LTL and Flat Counter Systems

  • Conference paper
Automated Reasoning (IJCAR 2012)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 7364))

Included in the following conference series:

Abstract

Reachability and LTL model-checking problems for flat counter systems are known to be decidable but whereas the reachability problem can be shown in NP, the best known complexity upper bound for the latter problem is made of a tower of several exponentials. Herein, we show that the problem is only NP-complete even if LTL admits pasttime operators and arithmetical constraints on counters. Actually, the NP upper bound is shown by adequately combining a new stuttering theorem for Past LTL and the property of small integer solutions for quantifier-free Presburger formulae. Other complexity results are proved, for instance for restricted classes of flat counter systems.

Supported by ANR project REACHARD ANR-11-BS02-001.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Boigelot, B.: Symbolic methods for exploring infinite state spaces. PhD thesis, Université de Liège (1998)

    Google Scholar 

  2. Borosh, I., Treybig, L.: Bounds on positive integral solutions of linear Diophantine equations. American Mathematical Society 55, 299–304 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bozga, M., Iosif, R., Konečný, F.: Fast Acceleration of Ultimately Periodic Relations. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 227–242. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  4. Comon, H., Cortier, V.: Flatness Is Not a Weakness. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 262–276. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  5. Comon, H., Jurski, Y.: Multiple Counter Automata, Safety Analysis and PA. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  6. Demri, S., Dhar, A., Sangnier, A.: Taming past LTL and flat counter systems. Technical report (2012), ArXiv

    Google Scholar 

  7. Demri, S., Finkel, A., Goranko, V., van Drimmelen, G.: Model-checking \(\textsf{CTL}^{*}\) over flat Presburger counter systems. JANCL 20(4), 313–344 (2010)

    Google Scholar 

  8. Dixon, C., Fisher, M., Konev, B.: Temporal Logic with Capacity Constraints. In: Konev, B., Wolter, F. (eds.) FroCos 2007. LNCS (LNAI), vol. 4720, pp. 163–177. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  9. Esparza, J., Finkel, A., Mayr, R.: On the verification of broadcast protocols. In: LICS 1999, pp. 352–359 (1999)

    Google Scholar 

  10. Etessami, K., Wilke, T.: An until hierarchy and other applications of an Ehrenfeucht-Fraïssé game for temporal logic. I&C 160(1–2), 88–108 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Finkel, A., Leroux, J.: How to Compose Presburger-Accelerations: Applications to Broadcast Protocols. In: Agrawal, M., Seth, A.K. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 145–156. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  12. Finkel, A., Lozes, É., Sangnier, A.: Towards Model-Checking Programs with Lists. In: Archibald, M., Brattka, V., Goranko, V., Löwe, B. (eds.) ILC 2007. LNCS (LNAI), vol. 5489, pp. 56–86. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  13. Gabbay, D.: The Declarative Past and Imperative Future. In: Banieqbal, B., Pnueli, A., Barringer, H. (eds.) Temporal Logic in Specification. LNCS, vol. 398, pp. 409–448. Springer, Heidelberg (1989)

    Chapter  Google Scholar 

  14. Haase, C., Kreutzer, S., Ouaknine, J., Worrell, J.: Reachability in Succinct and Parametric One-Counter Automata. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 369–383. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  15. Kuhtz, L.: Model Checking Finite Paths and Trees. PhD thesis, Universität des Saarlandes (2010)

    Google Scholar 

  16. Kuhtz, L., Finkbeiner, B.: Weak Kripke Structures and LTL. In: Katoen, J.-P., König, B. (eds.) CONCUR 2011. LNCS, vol. 6901, pp. 419–433. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  17. Kučera, A., Strejček, J.: The stuttering principle revisited. Acta Informatica 41(7–8), 415–434 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Laroussinie, F., Markey, N., Schnoebelen, P.: Temporal logic with forgettable past. In: LICS 2002, pp. 383–392. IEEE (2002)

    Google Scholar 

  19. Laroussinie, F., Schnoebelen, P.: Specification in CTL + past for verification in CTL. I&C 156, 236–263 (2000)

    MathSciNet  MATH  Google Scholar 

  20. 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)

    Chapter  Google Scholar 

  21. Lutz, C.: NE\({\textsc{xp}}\)T\({\textsc{ime}}\)-Complete Description Logics with Concrete Domains. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 45–60. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  22. Minsky, M.: Computation, Finite and Infinite Machines. Prentice Hall (1967)

    Google Scholar 

  23. Peled, D., Wilke, T.: Stutter-invariant temporal properties are expressible without the next-time operator. IPL 63, 243–246 (1997)

    Article  MathSciNet  Google Scholar 

  24. Rackoff, C.: The covering and boundedness problems for vector addition systems. TCS 6(2), 223–231 (1978)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LSV, CNRS, ENS Cachan, INRIA, France

    Stéphane Demri

  2. LIAFA, Univ Paris Diderot, Sorbonne Paris Cité, CNRS, France

    Amit Kumar Dhar & Arnaud Sangnier

Authors
  1. Stéphane Demri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amit Kumar Dhar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arnaud Sangnier
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Fakultät für Informatik, Technische Universität Wien, Favoritenstr. 9, E185/2, 1040, Wien, Austria

    Bernhard Gramlich

  2. École Polytechnique, INRIA Saclay and Laboratoire d’Informatique, Route de Saclay, 91128, Palaiseau Cedex, France

    Dale Miller

  3. School of Computer Science, University of Manchester, Oxford Road, M13 9PL, Manchester, UK

    Uli Sattler

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Demri, S., Dhar, A.K., Sangnier, A. (2012). Taming Past LTL and Flat Counter Systems. In: Gramlich, B., Miller, D., Sattler, U. (eds) Automated Reasoning. IJCAR 2012. Lecture Notes in Computer Science(), vol 7364. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31365-3_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-31365-3_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31364-6

  • Online ISBN: 978-3-642-31365-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation