Skip to main content

The Context-Freeness Problem Is coNP-Complete for Flat Counter Systems

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNPSE,volume 8837)

Abstract

Bounded languages have recently proved to be an important class of languages for the analysis of Turing-powerful models. For instance, bounded context-free languages are used to under-approximate the behaviors of recursive programs. Ginsburg and Spanier have shown in 1966 that a bounded language \(L \subseteq a_1^* \cdots a_d^*\) is context-free if, and only if, its Parikh image is a stratifiable semilinear set. However, the question whether a semilinear set is stratifiable, hereafter called the stratifiability problem, was left open, and remains so. In this paper, we give a partial answer to this problem. We focus on semilinear sets that are given as finite systems of linear inequalities, and we show that stratifiability is coNP-complete in this case. Then, we apply our techniques to the context-freeness problem for flat counter systems, that asks whether the trace language of a counter system intersected with a bounded regular language is context-free. As main result of the paper, we show that this problem is coNP-complete.

Keywords

  • Regular Language
  • Counter System
  • Integral Cone
  • Presburger Arithmetic
  • Emptiness Problem

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.

This work was supported by the ANR project ReacHard (ANR-11-BS02-001).

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-11936-6_19
  • Chapter length: 16 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-11936-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   79.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Annichini, A., Bouajjani, A., Sighireanu, M.: TReX: A tool for reachability analysis of complex systems. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 368–372. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  2. Bardin, S., Finkel, A., Leroux, J., Petrucci, L.: Fast: acceleration from theory to practice. Int. J. Software Tools Technology Transfer (STTT) 10(5), 401–424 (2008)

    CrossRef  Google Scholar 

  3. Boigelot, B., Wolper, P.: Symbolic verification with periodic sets. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 55–67. Springer, Heidelberg (1994)

    CrossRef  Google Scholar 

  4. Bouajjani, A., Habermehl, P.: Symbolic reachability analysis of FIFO-channel systems with nonregular sets of configurations. Theor. Comput. Sci. 221(1-2), 211–250 (1999)

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. 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)

    CrossRef  Google Scholar 

  6. Cadilhac, M., Finkel, A., McKenzie, P.: Bounded parikh automata. Int. J. Found. Comput. Sci. 23(8), 1691–1710 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. Chambart, P., Finkel, A., Schmitz, S.: Forward analysis and model checking for trace bounded WSTS. In: Kristensen, L.M., Petrucci, L. (eds.) PETRI NETS 2011. LNCS, vol. 6709, pp. 49–68. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  8. Comon, H., Jurski, Y.: Multiple counters automata, safety analysis and Presburger arithmetic. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998)

    CrossRef  Google Scholar 

  9. Esparza, J., Ganty, P., Majumdar, R.: A perfect model for bounded verification. In: Proc. LICS 2012, pp. 285–294. IEEE Computer Society (2012)

    Google Scholar 

  10. Finkel, A., Iyer, S.P., Sutre, G.: Well-abstracted transition systems: Application to FIFO automata. Information and Computation 181(1), 1–31 (2003)

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Ginsburg, S., Spanier, E.H.: Bounded ALGOL-like languages. Trans. Amer. Math. Soc. 113, 333–368 (1964)

    MathSciNet  MATH  Google Scholar 

  12. Ginsburg, S., Spanier, E.H.: Bounded regular sets. Proc. Amer. Math. Soc. 17(5), 1043–1049 (1966)

    MathSciNet  CrossRef  MATH  Google Scholar 

  13. Ginsburg, S., Spanier, E.H.: Semigroups, Presburger formulas and languages. Pacific J. Math. 16(2), 285–296 (1966)

    MathSciNet  CrossRef  MATH  Google Scholar 

  14. Ginsburg, S.: The Mathematical Theory of Context-Free Languages. McGraw-Hill (1966)

    Google Scholar 

  15. Hopcroft, J., Pansiot, J.J.: On the reachability problem for 5-dimensional vector addition systems. Theor. Comput. Sci. 8(2), 135–159 (1979)

    MathSciNet  CrossRef  MATH  Google Scholar 

  16. Ibarra, O.H., Seki, S.: Characterizations of bounded semilinear languages by one-way and two-way deterministic machines. Int. J. Found. Comput. Sci. 23(6), 1291–1306 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

  17. Ibarra, O.H., Seki, S.: On the open problem of Ginsburg concerning semilinear sets and related problems. Theor. Comput. Sci. 501, 11–19 (2013)

    MathSciNet  CrossRef  MATH  Google Scholar 

  18. Leroux, J., Penelle, V., Sutre, G.: On the context-freeness problem for vector addition systems. In: Proc. LICS 2013, pp. 43–52. IEEE Computer Society (2013)

    Google Scholar 

  19. Leroux, J., Praveen, M., Sutre, G.: A relational trace logic for vector addition systems with application to context-freeness. In: D’Argenio, P.R., Melgratti, H. (eds.) CONCUR 2013. LNCS, vol. 8052, pp. 137–151. Springer, Heidelberg (2013)

    Google Scholar 

  20. Liu, L., Weiner, P.: A characterization of semilinear sets. J. Comput. Syst. Sci. 4(4), 299–307 (1970)

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. Schrijver, A.: Theory of Linear and Integer Programming. John Wiley and Sons, New York (1987)

    Google Scholar 

  22. Schwer, S.R.: The context-freeness of the languages associated with vector addition systems is decidable. Theor. Comput. Sci. 98(2), 199–247 (1992)

    MathSciNet  CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Leroux, J., Penelle, V., Sutre, G. (2014). The Context-Freeness Problem Is coNP-Complete for Flat Counter Systems. In: Cassez, F., Raskin, JF. (eds) Automated Technology for Verification and Analysis. ATVA 2014. Lecture Notes in Computer Science, vol 8837. Springer, Cham. https://doi.org/10.1007/978-3-319-11936-6_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-11936-6_19

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-11935-9

  • Online ISBN: 978-3-319-11936-6

  • eBook Packages: Computer ScienceComputer Science (R0)