Forward Analysis for WSTS, Part II: Complete WSTS

  • Alain Finkel
  • Jean Goubault-Larrecq
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5556)

Abstract

We describe a simple, conceptual forward analysis procedure for ∞-complete WSTS \(\mathfrak S\). This computes the clover of a state s0, i.e., a finite description of the closure of the cover of s0. When \(\mathfrak S\) is the completion of a WSTS \(\mathfrak X\), the clover in \(\mathfrak S\) is a finite description of the cover in \(\mathfrak X\). We show that this applies exactly when \(\mathfrak X\) is an ω2-WSTS, a new robust class of WSTS. We show that our procedure terminates in more cases than the generalized Karp-Miller procedure on extensions of Petri nets. We characterize the WSTS where our procedure terminates as those that are clover-flattable. Finally, we apply this to well-structured counter systems.

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References

  1. 1.
    Abdulla, P., Bouajjani, A., Jonsson, B.: On-the-fly analysis of systems with unbounded, lossy Fifo channels. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427, pp. 305–318. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  2. 2.
    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
  3. 3.
    Abdulla, P.A., Collomb-Annichini, A., Bouajjani, A., Jonsson, B.: Using forward reachability analysis for verification of lossy channel systems. Formal Methods in System Design 25(1), 39–65 (2004)MATHCrossRefGoogle Scholar
  4. 4.
    Abdulla, P.A., Deneux, J., Mahata, P., Nylén, A.: Forward reachability analysis of timed petri nets. In: Lakhnech, Y., Yovine, S. (eds.) FORMATS/FTRTFT 2004. LNCS, vol. 3253, pp. 343–362. Springer, Heidelberg (2004)Google Scholar
  5. 5.
    Abdulla, P.A., Nylén, A.: Better is better than well: On efficient verification of infinite-state systems. In: 14th LICS, pp. 132–140 (2000)Google Scholar
  6. 6.
    Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3, pp. 1–168. Oxford University Press, Oxford (1994)Google Scholar
  7. 7.
    Antonik, A.: Presburger monotone affine functions can be lub-accelerated. Personal communication (2009)Google Scholar
  8. 8.
    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
  9. 9.
    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
  10. 10.
    Demri, S., Finkel, A., Goranko, V., van Drimmelen, G.: Towards a model-checker for counter systems. In: Graf, S., Zhang, W. (eds.) ATVA 2006. LNCS, vol. 4218, pp. 493–507. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Dufourd, C., Finkel, A., Schnoebelen, P.: Reset nets between decidability and undecidability. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 103–115. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Emerson, E.A., Namjoshi, K.S.: On model-checking for non-deterministic infinite-state systems. In: 13th LICS, pp. 70–80 (1998)Google Scholar
  13. 13.
    Esparza, J., Finkel, A., Mayr, R.: On the verification of broadcast protocols. In: 14th LICS, pp. 352–359 (1999)Google Scholar
  14. 14.
    Finkel, A.: A generalization of the procedure of Karp and Miller to well structured transition systems. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 499–508. Springer, Heidelberg (1987)Google Scholar
  15. 15.
    Finkel, A.: Reduction and covering of infinite reachability trees. Information and Computation 89(2), 144–179 (1990)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Finkel, A.: The minimal coverability graph for Petri nets. In: Rozenberg, G. (ed.) APN 1993. LNCS, vol. 674, pp. 210–243. Springer, Heidelberg (1993)Google Scholar
  17. 17.
    Finkel, A., Goubault-Larrecq, J.: Forward analysis for WSTS, part I: Completions. In: 26th STACS, Freiburg, Germany. Springer, Heidelberg (to appear, 2009)Google Scholar
  18. 18.
    Finkel, A., McKenzie, P., Picaronny, C.: A well-structured framework for analysing Petri net extensions. Information and Computation 195(1-2), 1–29 (2004)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Finkel, A., Schnoebelen, P.: Well-structured transition systems everywhere! Theoretical Computer Science 256(1–2), 63–92 (2001)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ganty, P., Raskin, J.-F., van Begin, L.: A complete abstract interpretation framework for coverability properties of WSTS. In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 49–64. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  21. 21.
    Geeraerts, G., Raskin, J.-F., van Begin, L.: Expand, enlarge and check: New algorithms for the coverability problem of WSTS. J. Comp. and System Sciences 72(1), 180–203 (2006)MATHCrossRefGoogle Scholar
  22. 22.
    Geeraerts, G., Raskin, J.-F., van Begin, L.: On the efficient computation of the minimal coverability set for Petri nets. In: Namjoshi, K.S., Yoneda, T., Higashino, T., Okamura, Y. (eds.) ATVA 2007. LNCS, vol. 4762, pp. 98–113. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  23. 23.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous lattices and domains. In: Encyclopedia of Mathematics and its Applications, vol. 93. Cambridge University Press, Cambridge (2003)Google Scholar
  24. 24.
    Ginsburg, S., Spanier, E.H.: Bounded Algol-like languages. Trans. American Mathematical Society 113(2), 333–368 (1964)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Goubault-Larrecq, J.: On Noetherian spaces. In: 22nd LICS, Wrocław, Poland, pp. 453–462. IEEE Computer Society Press, Los Alamitos (2007)Google Scholar
  26. 26.
    Jančar, P.: A note on well quasi-orderings for powersets. Information Processing Letters 72(5–6), 155–160 (1999)MATHMathSciNetGoogle Scholar
  27. 27.
    Karp, R.M., Miller, R.E.: Parallel program schemata. J. Comp. and System Sciences 3(2), 147–195 (1969)MATHMathSciNetGoogle Scholar
  28. 28.
    Lazič, R., Newcomb, T., Ouaknine, J., Roscoe, A.W., Worrell, J.: Nets with tokens which carry data. Fundamenta Informaticae 88(3), 251–274 (2008)MATHMathSciNetGoogle Scholar
  29. 29.
    Rado, R.: Partial well-ordering of sets of vectors. Mathematika 1, 89–95 (1954)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Valk, R.: Self-modidying nets, a natural extension of Petri nets. In: Ausiello, G., Böhm, C. (eds.) ICALP 1978, vol. 62, pp. 464–476. Springer, Heidelberg (1978)Google Scholar
  31. 31.
    Verma, K.N., Goubault-Larrecq, J.: Karp-Miller trees for a branching extension of VASS. Discrete Mathematics & Theoretical Computer Science 7(1), 217–230 (2005)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Alain Finkel
    • 1
  • Jean Goubault-Larrecq
    • 1
    • 2
  1. 1.LSV, ENS Cachan, CNRSFrance
  2. 2.INRIA SaclayFrance

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