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On the Complexity of Verifying Regular Properties on Flat Counter Systems,

  • Stéphane Demri
  • Amit Kumar Dhar
  • Arnaud Sangnier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7966)

Abstract

Among the approximation methods for the verification of counter systems, one of them consists in model-checking their flat unfoldings. Unfortunately, the complexity characterization of model-checking problems for such operational models is not always well studied except for reachability queries or for Past LTL. In this paper, we characterize the complexity of model-checking problems on flat counter systems for the specification languages including first-order logic, linear mu-calculus, infinite automata, and related formalisms. Our results span different complexity classes (mainly from PTime to PSpace) and they apply to languages in which arithmetical constraints on counter values are systematically allowed. As far as the proof techniques are concerned, we provide a uniform approach that focuses on the main issues.

Keywords

Atomic Formula Counter System Propositional Variable Atomic Proposition Kripke Structure 
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.

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References

  1. 1.
    Boigelot, B.: Symbolic methods for exploring infinite state spaces. PhD thesis, Université de Liège (1998)Google Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    Demri, S., Dhar, A.K., Sangnier, A.: Taming Past LTL and Flat Counter Systems. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 179–193. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  5. 5.
    Demri, S., Dhar, A.K., Sangnier, A.: On the complexity of verifying regular properties on flat counter systems (2013), http://arxiv.org/abs/1304.6301
  6. 6.
    Demri, S., Finkel, A., Goranko, V., van Drimmelen, G.: Model-checking \(\textsf{CTL}^{*}\) over flat Presburger counter systems. JANCL 20(4), 313–344 (2010)zbMATHGoogle Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    Jančar, P., Sawa, Z.: A note on emptiness for alternating finite automata with a one-letter alphabet. IPL 104(5), 164–167 (2007)zbMATHCrossRefGoogle Scholar
  9. 9.
    Jurdziński, M.: Deciding the winner in parity games is in UP ∩ co-UP. IPL 68(3), 119–124 (1998)CrossRefGoogle Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Kupferman, O., Vardi, M.: Weak alternating automata are not that weak. ACM Transactions on Computational Logic 2(3), 408–429 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kučera, A., Strejček, J.: The stuttering principle revisited. Acta Informatica 41(7-8), 415–434 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Laroussinie, F., Markey, N., Schnoebelen, P.: Temporal logic with forgettable past. In: LICS 2002, pp. 383–392. IEEE (2002)Google Scholar
  14. 14.
    Leroux, J., Sutre, G.: Flat counter systems are everywhere! In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 489–503. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Markey, N., Schnoebelen, P.: Model checking a path. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 251–265. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  16. 16.
    Markey, N., Schnoebelen, P.: Mu-calculus path checking. IPL 97(6) (2006)Google Scholar
  17. 17.
    Minsky, M.: Computation, Finite and Infinite Machines. Prentice Hall (1967)Google Scholar
  18. 18.
    Miyano, S., Hayashi, T.: Alternating finite automata on ω-words. Theor. Comput. Sci. 32, 321–330 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Piterman, N.: Extending temporal logic with ω-automata. Master’s thesis, The Weizmann Institute of Science (2000)Google Scholar
  20. 20.
    Pottier, L.: Minimal Solutions of Linear Diophantine Systems: Bounds and Algorithms. In: Book, R.V. (ed.) RTA 1991. LNCS, vol. 488, pp. 162–173. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  21. 21.
    Sistla, A., Clarke, E.: The complexity of propositional linear temporal logic. JACM 32(3), 733–749 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Stockmeyer, L.J.: The complexity of decision problems in automata and logic. PhD thesis, MIT (1974)Google Scholar
  23. 23.
    Vardi, M.: A temporal fixpoint calculus. In: POPL 1988, pp. 250–259. ACM (1988)Google Scholar
  24. 24.
    Vardi, M., Wolper, P.: Reasoning about infinite computations. I&C 115 (1994)Google Scholar
  25. 25.
    Wolper, P.: Temporal logic can be more expressive. I&C 56, 72–99 (1983)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Stéphane Demri
    • 2
    • 3
  • Amit Kumar Dhar
    • 1
  • Arnaud Sangnier
    • 1
  1. 1.LIAFAUniv Paris Diderot, Sorbonne Paris Cité, CNRSFrance
  2. 2.New York UniversityUSA
  3. 3.CNRSLSVFrance

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