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Towards a Model-Checker for Counter Systems

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Automated Technology for Verification and Analysis (ATVA 2006)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 4218))

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Abstract

This paper deals with model-checking of fragments and extensions of CTL* on infinite-state Presburger counter systems, where the states are vectors of integers and the transitions are determined by means of relations definable within Presburger arithmetic. We have identified a natural class of admissible counter systems (ACS) for which we show that the quantification over paths in CTL* can be simulated by quantification over tuples of natural numbers, eventually allowing translation of the whole Presburger-CTL* into Presburger arithmetic, thereby enabling effective model checking. We have provided evidence that our results are close to optimal with respect to the class of counter systems described above. Finally, we design a complete semi-algorithm to verify first-order LTL properties over trace-flattable counter systems, extending the previous underlying FAST semi-algorithm to verify reachability questions over flattable counter systems.

Supported by CNRS/NRF project No 15469.

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References

  1. Burkart, O., Caucal, D., Moller, F., Steffen, B.: Verification of infinite structures. In: Handbook of Process Algebra, pp. 545–623. Elsevier, Amsterdam (2001)

    Chapter  Google Scholar 

  2. Bruyère, V., Dall’Olio, E., Raskin, J.F.: Durations, parametric model-checking in timed automata with presburger arithmetic. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 687–698. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  3. Bouajjani, A., Echahed, R., Habermehl, P.: On the verification problem of nonregular properties for nonregular processes. In: LICS 1995, pp. 123–133 (1995)

    Google Scholar 

  4. Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of pushdown automata: Application to model checking. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997)

    Google Scholar 

  5. Bardin, S., Finkel, A., Leroux, J.: FASTer acceleration of counter automata in practice. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 576–590. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  6. Bardin, S., Finkel, A., Leroux, J., Petrucci, L.: FAST: Fast Acceleration of Symbolic Transition systems. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 118–121. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  7. Bardin, S., Finkel, A., Leroux, J., Schnoebelen, P.: Flat acceleration in symbolic model checking. In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 474–488. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  8. Bultan, T., Gerber, R., Pugh, W.: Symbolic model checking of infinite state systems using Presburger arithmetic. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 400–411. Springer, Heidelberg (1997)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  11. 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 

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

    Chapter  Google Scholar 

  13. Cortier, V.: About the decision of reachability for register machines. Theoretical Informatics and Applications 36(4), 341–358 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Demri, S.: Temporal logics. Lecture notes for MPRI, 2005/2006 www.lsv.ens-cachan.fr/~demri/

  15. Dang, Z., San Pietro, P., Kemmerer, R.: Presburger liveness verification of discrete timed automata. TCS 299, 413–438 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  17. Finkel, A., Leroux, J.: How to compose Presburger accelerations: Applications to broadcast protocols. In: Stumptner, M., Corbett, D.R., Brooks, M. (eds.) Canadian AI 2001. LNCS (LNAI), vol. 2256, pp. 145–156. Springer, Heidelberg (2001)

    Google Scholar 

  18. Fribourg, L., Olsén, H.: Proving safety properties of infinite state systems by compilation into presburger arithmetic. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 213–227. Springer, Heidelberg (1997)

    Google Scholar 

  19. Finkel, A., Willems, B., Wolper, P.: A direct symbolic approach to model checking pushdown systems (extended abstract). In: INFINITY 1997. ENTCS, vol. 9, Elsevier Science, Amsterdam (1997)

    Google Scholar 

  20. Ibarra, O.: Reversal-bounded multicounter machines and their decision problems. J. ACM 25(1), 116–133 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ibarra, O., Su, J., Dang, Z., Bultan, T., Kemmerer, A.: Counter machines: Decidable properties and applications to verification problems. In: Nielsen, M., Rovan, B. (eds.) MFCS 2000. LNCS, vol. 1893, pp. 426–435. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  22. Lagarias, J.: The 3x + 1 problem and its generalizations. The American Mathematical Monthly 92(1), 3–23 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  23. Leroux, J.: Algorithmique de la vérification des systèmes à compteurs. Approximation et accélération. Implémentation de l’outil FAST. PhD thesis, ENS de Cachan, France (2003)

    Google Scholar 

  24. Leroux, J.: Regular acceleration for number decision diagrams. Technical Report 1385-06, LABRI (January 2006)

    Google Scholar 

  25. 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)

    Chapter  Google Scholar 

  26. Minsky, M.: Computation, Finite and Infinite Machines. Prentice-Hall, Englewood Cliffs (1967)

    MATH  Google Scholar 

  27. Schuele, T., Schneider, K.: Global vs. local model checking: A comparison of verification techniques for infinite state systems. In: SEFM 2004, pp. 67–76. IEEE, Los Alamitos (2004)

    Google Scholar 

  28. Walukiewicz, I.: Pushdown processes: games and model-checking. I & C 164(2), 234–263 (2001)

    MATH  MathSciNet  Google Scholar 

  29. Wolper, P.: Temporal logic can be more expressive. I & C 56, 72–99 (1983)

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. LSV/CNRS UMR 8643 & INRIA Futurs projet SECSI & ENS Cachan,  

    S. Demri & A. Finkel

  2. University of the Witwatersrand, Johannesburg

    V. Goranko & G. van Drimmelen

Authors
  1. S. Demri
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  2. A. Finkel
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  3. V. Goranko
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  4. G. van Drimmelen
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Editor information

Editors and Affiliations

  1. VERIMAG, 2 Avenue de Vignate, Centre Equitation, 38610, Grenoble-Gières, France

    Susanne Graf

  2. Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China

    Wenhui Zhang

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Demri, S., Finkel, A., Goranko, V., van Drimmelen, G. (2006). Towards a Model-Checker for Counter Systems. In: Graf, S., Zhang, W. (eds) Automated Technology for Verification and Analysis. ATVA 2006. Lecture Notes in Computer Science, vol 4218. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11901914_36

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  • DOI: https://doi.org/10.1007/11901914_36

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-47237-7

  • Online ISBN: 978-3-540-47238-4

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