On Computing Fixpoints in Well-Structured Regular Model Checking, with Applications to Lossy Channel Systems

  • Christel Baier
  • Nathalie Bertrand
  • Philippe Schnoebelen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4246)


We prove a general finite convergence theorem for “upward-guarded” fixpoint expressions over a well-quasi-ordered set. This has immediate applications in regular model checking of well-structured systems, where a main issue is the eventual convergence of fixpoint computations. In particular, we are able to directly obtain several new decidability results on lossy channel systems.


Operational Semantic Transition Rule Regular Language Message Loss Nullary Operator 
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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  1. 1.
    Abdulla, P.A., Bertrand, N., Rabinovich, A., Schnoebelen, P.: Verification of probabilistic systems with faulty communication. Information and Computation 202(2), 141–165 (2005)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abdulla, P.A., Bouajjani, A., d’Orso, J.: Deciding Monotonic Games. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 1–14. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Abdulla, P.A., Čerāns, K., Jonsson, B., Tsay, Y.-K.: Algorithmic analysis of programs with well quasi-ordered domains. Information and Computation 160(1/2), 109–127 (2000)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Abdulla, P.A., Jonsson, B.: Undecidable verification problems for programs with unreliable channels. Information and Computation 130(1), 71–90 (1996)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Abdulla, P.A., Jonsson, B.: Verifying programs with unreliable channels. Information and Computation 127(2), 91–101 (1996)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Abdulla, P.A., Jonsson, B.: Channel representations in protocol verification. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 1–15. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Arnold, A., Niwiński, D.: Rudiments of μ-Calculus. Studies in Logic and the Foundations of Mathematics, vol. 146. Elsevier Science, Amsterdam (2001)CrossRefGoogle Scholar
  8. 8.
    Baier, C., Bertrand, N., Schnoebelen, P.: A note on the attractor-property of infinite-state Markov chains. Information Processing Letters 97(2), 58–63 (2006)MathSciNetMATHGoogle Scholar
  9. 9.
    Baier, C., Bertrand, N., Schnoebelen, Ph.: Verifying nondeterministic probabilistic channel systems against ω-regular linear-time properties. ACM Transactions on Computational Logic (to appear, 2006),
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Bouajjani, A., Jonsson, B., Nilsson, M., Touili, T.: Regular model checking. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 403–418. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  12. 12.
    Cécé, G., Finkel, A., Purushothaman Iyer, S.: Unreliable channels are easier to verify than perfect channels. Information and Computation 124(1), 20–31 (1996)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chatterjee, K., de Alfaro, L., Henzinger, T.A.: The Complexity of Stochastic Rabin and Streett Games,. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 878–890. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    de Alfaro, L., Henzinger, T.A., Majumdar, R.: Symbolic algorithms for infinite-state games. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 536–550. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Finkel, A.: Decidability of the termination problem for completely specificied protocols. Distributed Computing 7(3), 129–135 (1994)CrossRefGoogle Scholar
  16. 16.
    Finkel, A., Schnoebelen, P.: Well-structured transition systems everywhere! Theoretical Computer Science 256(1–2), 63–92 (2001)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games: A Guide to Current Research. LNCS, vol. 2500. Springer, Heidelberg (2002)MATHGoogle Scholar
  18. 18.
    Henzinger, T.A., Majumdar, R., Raskin, J.-F.: A classification of symbolic transition systems. ACM Trans. Computational Logic 6(1), 1–32 (2005)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Kesten, Y., Maler, O., Marcus, M., Pnueli, A., Shahar, E.: Symbolic model checking with rich assertional languages. Theoretical Computer Science 256(1–2), 93–112 (2001)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kruskal, J.B.: The theory of well-quasi-ordering: A frequently discovered concept. Journal of Combinatorial Theory, Series A 13(3), 297–305 (1972)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kučera, A., Schnoebelen, P.: A general approach to comparing infinite-state systems with their finite-state specifications. Theoretical Computer Science 358(2-3), 315–333 (2006)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Martin, D.A.: The determinacy of Blackwell games. The Journal of Symbolic Logic 63(4), 1565–1581 (1998)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Mayr, R.: Undecidable problems in unreliable computations. Theoretical Computer Science 297(1–3), 337–354 (2003)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Milner, E.C.: Basic WQO- and BQO-theory. In: Rival, I. (ed.) Graphs and Order. The Role of Graphs in the Theory of Ordered Sets and Its Applications, pp. 487–502. D.Reidel Publishing (1985)Google Scholar
  25. 25.
    Ouaknine, J., Worrell, J.B.: On Metric Temporal Logic and Faulty Turing Machines. In: Aceto, L., Ingólfsdóttir, A. (eds.) FOSSACS 2006. LNCS, vol. 3921, pp. 217–230. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  26. 26.
    Perrin, D.: Finite automata. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, ch. 1, vol. B, pp. 1–57. Elsevier, Amsterdam (1990)Google Scholar
  27. 27.
    Raskin, J.-F., Samuelides, M., Van Begin, L.: Petri games are monotonic but difficult to decide. Tech. Report, 2003.21, Centre Fédéré en Vérification (2003), Available at:
  28. 28.
    Raskin, J.-F., Samuelides, M., Van Begin, L.: Games for counting abstractions. In: Proc. 4th Int. Workshop on Automated Verification of Critical Systems (AVoCS 2004), London, UK, September. Electronic Notes in Theor. Comp. Sci, vol. 128(6), pp. 69–85. Elsevier Science, Amsterdam (2005)Google Scholar
  29. 29.
    Schnoebelen, P.: The verification of probabilistic lossy channel systems. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 445–465. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  30. 30.
    Sifakis, J.: A unified approach for studying the properties of transitions systems. Theoretical Computer Science 18, 227–258 (1982)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Christel Baier
    • 1
  • Nathalie Bertrand
    • 2
  • Philippe Schnoebelen
    • 2
  1. 1.Institut für Informatik IUniversität BonnGermany
  2. 2.LSV, ENS de Cachan & CNRSFrance

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