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A Decidability Result for the Model Checking of Infinite-State Systems

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Abstract

We present a decidability result for the model checking of a certain class of properties that can be conveniently expressed as ground formulae of a first-order temporal fragment. The decidability result is obtained by importing into the context of model-checking problems some techniques developed for the combination of decision procedures for the satisfiability of constraints. The general decidability result is then specialized for checking properties of particular interest, such as liveness and safety, and, for the latter case, a more optimized algorithm has been proposed.

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References

  1. Abadi, M.: The power of temporal proofs. Theor. Comp. Sci. 65(1), 35–83 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abdulla, P.A., Cerans, K., Jonsson, B., Tsay, Y.-K.: General decidability theorems for infinite-state systems. In: Proceedings of the 11th IEEE Symposium on Logic in Computer Science (LICS 1996), pp. 313–321. IEEE Computer Society, New Brunswick, NJ, USA (1996)

    Google Scholar 

  3. Barrett, C., Sebastiani, R., Seshia, S., Tinelli, C.: Satisfiability modulo theories. In: Van Maaren, H., Biere, A., Heule, M., Walsh, T. (eds.) The Handbook of Satisfiability, vol. II, chap. 26, pp. 887–925. IOS Press, Amsterdam, The Netherlands (2009)

    Google Scholar 

  4. Bonacina, M.P., Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Decidability and undecidability results for Nelson-Oppen and rewrite-based decision procedures. In: Furbach, U., Shankar, N. (eds.) Proceedings of the 3rd International Joint Conference on Automated Reasoning (IJCAR 2006). Lecture Notes in Computer Science, vol. 4130, pp. 513–527. Springer, Seattle, WA, USA (2006)

    Google Scholar 

  5. Bouajjani, A., Jonsson, B., Nilsson, M., Touili, T.: Regular model checking. In: Emerson, A.E., Sistla, A.P. (eds.) Proceedings of the 12th International Conference on Computer Aided Verification (CAV 2000). Lecture Notes in Computer Science, vol. 1855, pp. 403–418. Springer, Chicago, IL, USA (2000)

    Google Scholar 

  6. Bräuner, T., Ghilardi, S.: First-order modal logic. In: van Benthem, J., Blackburn, P., Wolter, F. (eds.) Handbook of Modal Logic, pp. 549–620. Elsevier, Amsterdam (2007)

    Chapter  Google Scholar 

  7. Chang, C.-C., Keisler, J.H.: Model Theory, 3rd edn. North Holland, Amsterdam, The Netherlands (1990)

    MATH  Google Scholar 

  8. Cimatti, A., Clarke, E.M., Giunchiglia, E., Giunchiglia, F., Pistore, M., Roveri, M., Sebastiani, R., Tacchella, A.: NuSMV 2: an opensource tool for symbolic model checking. In: Brinksma, E., Larsen, K.G. (eds.) Proceedings of the 14th International Conference on Computer Aided Verification (CAV 2002). Lecture Notes in Computer Science, vol. 2404, pp. 359–364. Springer, Copenhagen, Denmark (2002)

    Google Scholar 

  9. Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press, Cambridge (1999)

    Google Scholar 

  10. D’Agostino, G., Hollenberg, M.: Logical questions concerning the μ-calculus: interpolation, Lyndon and Los-Tarski. J. Symb. Log. 65(1), 310–332 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Demri, S., Finkel, A., Goranko, V., van Drimmelen, G.: Towards a model-checker for counter systems. In: Graf, S., Zhang, W. (eds.) Proceedings of the 4th International Symposium on Automated Technology for Verification and Analysis (ATVA 2006). Lecture Notes in Computer Science, vol. 4218, pp. 493–507. Springer, Beijing, ROC (2006)

    Chapter  Google Scholar 

  12. Demri, S.: Linear-time temporal logics with Presburger constraints: an overview. J. Appl. Non-Class. Log. 16(3–4), 311–347 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. de Moura, L.M., Rueß, H., Sorea, M.: Lazy theorem proving for bounded model checking over infinite domains. In: Voronkov, A. (ed.) Proceedings of the 18th International Conference on Automated Deduction (CADE 2002). Lecture Notes in Computer Science, vol. 2392, pp. 438–455. Springer, Copenhagen, Denmark (2002)

    Google Scholar 

  14. Ebbinghaus, H.-D., Flum, J., Thomas, W.: Mathematical logic. In: Undergraduate Texts in Mathematics, 2nd edn. Springer, New York (1994)

    Google Scholar 

  15. Ghilardi, S.: Model theoretic methods in combined constraint satisfiability. J. Autom. Reason. 33(3–4), 221–249 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Combination methods for satisfiability and model-checking of infinite-state systems. In: Pfenning, F. (ed.) Proceedings of the 21st Conference on Automated Deduction (CADE 2007). Lecture Notes in Computer Science, vol. 4603, pp. 362–378. Springer, Bremen, Germany (2007)

    Chapter  Google Scholar 

  17. Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Combination methods for satisfiability and model-checking of infinite-state systems. Rapporto Interno DSI 313-07, Università degli Studi di Milano, Milano, Italy (2007)

  18. Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Noetherianity and combination problems. In: Konev, B., Wolter, F. (eds.) Proceedings of the 6th International Workshop on Frontiers of Combining Systems (FroCoS 2007). Lecture Notes in Computer Science, vol. 4720, pp. 206–220. Springer, Liverpool, UK (2007)

    Google Scholar 

  19. Ghilardi, S., Nicolini, E., Zucchelli, D.: A comprehensive combination framework. ACM Trans. Comput. Log. 9(2), 1–54 (2008)

    Article  MathSciNet  Google Scholar 

  20. Graf, S., Saïdi, H.: Construction of abstract state graphs with PVS. In: Grumberg, O. (ed.) Proceedings of the 9th International Conference on Computer Aided Verification (CAV 1997). Lecture Notes in Computer Science, vol. 1254, pp. 72–83. Springer, Haifa, Israel (1997)

    Google Scholar 

  21. Hodges, W.: Model theory. In: Encyclopedia of Mathematics and its Applications, vol. 42. Cambridge University Press, Cambridge, UK (1993)

    Google Scholar 

  22. Kröger, F.: On the interpretability of arithmetic in temporal logic. Theor. Comp. Sci. 73(1), 47–60 (1990)

    Article  MATH  Google Scholar 

  23. Maidl, M.: A unifying model checking approach for safety properties of parameterized systems. In: Berry, G., Comon, H., Finkel, A. (eds.) Proceedings of the 13th International Conference on Computer Aided Verification (CAV 2001). Lecture Notes in Computer Science, vol. 2102, pp. 311–323. Springer, Paris, France (2001)

    Chapter  Google Scholar 

  24. Manna, Z., Pnueli, A.: Temporal Verification of Reactive Systems: Safety. Springer, New York (1995)

    Book  Google Scholar 

  25. McMillan, K.L.: Applications of Craig interpolants in model checking. In: Halbwachs, N., Zuck, L.D. (eds.) Proceedings of the 11th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS 2005). Lecture Notes in Computer Science, vol. 3440, pp. 1–12. Springer, Edinburgh, UK (2005)

    Chapter  Google Scholar 

  26. Merz, S.: Decidability and incompleteness results for first-order temporal logics of linear time. J. Appl. Non-Class. Log. 2(2), 139–156 (1992)

    MATH  MathSciNet  Google Scholar 

  27. Minsky, M.L.: Recursive unsolvability of Post’s problem of “tag” and other topics in the theory of Turing machines. Ann. Math. 74(3), 437–455 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  28. Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. Program. Lang. Syst. 1(2), 245–257 (1979)

    Article  MATH  Google Scholar 

  29. Nicolini, E.: Combined decision procedures for constraint satisfiability. Ph.D. thesis, Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy (2007)

  30. Pitts, A.M.: On an interpretation of second order quantification in first order intuitionistic propositional logic. J. Symb. Log. 57(1), 33–52 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  31. Plaisted, D.A.: A decision procedure for combination of propositional temporal logic and other specialized theories. J. Autom. Reason. 2(2), 171–190 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  32. Rybina, T., Voronkov, A.: A logical reconstruction of reachability. In: Broy, M., Zamulin, A.V. (eds.) 5th International Andrei Ershov Memorial Conference (PSI 2003). Lecture Notes in Computer Science, vol. 2890, pp. 222–237. Springer, Akademgorodok, Novosibirsk, Russia (2003)

    Google Scholar 

  33. Sipma, H.B., Uribe, T.E., Manna, Z.: Deductive model checking. Form. Methods Syst. Des. 15(1), 49–74 (1999)

    Article  Google Scholar 

  34. Sofronie-Stokkermans, V.: Interpolation in local theory extensions. In: Furbach, U., Shankar, N. (eds.) Proceedings of the 3rd International Joint Conference on Automated Reasoning (IJCAR 2006). Lecture Notes in Computer Science, vol. 4130, pp. 235–250. Springer, Seattle, WA, USA (2006)

    Google Scholar 

  35. Szalas, A.: Concerning the semantic consequence relation in first-order temporal logic. Theor. Comp. Sci. 47(3), 329–334 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  36. Szalas, A., Holenderski, L.: Incompleteness of first-order temporal logic with until. Theor. Comp. Sci. 57, 317–325 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  37. Tinelli, C., Harandi, M.T.: A new correctness proof of the Nelson-Oppen combination procedure. In: Baader, F., Schulz, K. (eds.) Proceedings of the 1st International Workshop on Frontiers of Combining Systems (FroCoS 1996), Applied Logic, pp. 103–120. Kluwer, Munich, Germany (1996)

    Google Scholar 

  38. Tinelli, C., Ringeissen, C.: Unions of non-disjoint theories and combinations of satisfiability procedures. Theor. Comp. Sci. 290(1), 291–353 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  39. Visser, A.: Uniform interpolation and layered bisimulation. In: Hájek, P. (ed.) Proceedings of Gödel96: Logical Foundations of Mathematics, Computer Science, and Physics. Lecture Notes Logic, vol. 6, , pp. 139–164. Springer, Brno, Czech Republic (1996)

    Google Scholar 

  40. Zucchelli, D.: Combination methods for software verification. Ph.D. thesis, Università degli Studi di Milano and Université Henri Poincaré - Nancy 1, Milano, Italy (2007)

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Authors and Affiliations

  1. Dipartimento di Scienze dell’Informazione, Università degli Studi di Milano, Via Comelico 39/41, 20135, Milano, Italy

    Daniele Zucchelli

  2. LORIA & INRIA Nancy-Grand Est, 615, rue du Jardin Botanique, 54602, Villers-les-Nancy, France

    Enrica Nicolini

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  1. Daniele Zucchelli
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  2. Enrica Nicolini
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Correspondence to Enrica Nicolini.

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Zucchelli, D., Nicolini, E. A Decidability Result for the Model Checking of Infinite-State Systems. J Autom Reasoning 48, 1–42 (2012). https://doi.org/10.1007/s10817-010-9192-z

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  • DOI: https://doi.org/10.1007/s10817-010-9192-z

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