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Abstract

Monolithic finite-state probabilistic programs have been abstractly modeled by finite Markov chains, and the algorithmic verification problems for them have been investigated very extensively. In this paper we survey recent work conducted by the authors together with colleagues on he algorithmic verification of probabilistic procedural programs ([BKS,EKM04,EY04]). Probabilistic procedural programs can more naturally be modeled by recursive Markov chains ([EY04)], or equivalently, probabilistic pushdown automata ([EKM04)]. A very rich theory emerges for these models. While our recent work solves a number of verification problems for these models, many intriguing questions remain open.

Keywords

Markov Chain Model Check Atomic Proposition Probabilistic Program Reachability Problem 
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. [AEY01]
    Alur, R., Etessami, K., Yannakakis, M.: Analysis of recursive state machines. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 304–313. Springer, Heidelberg (2001)Google Scholar
  2. [BEM97]
    Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of pushdown automata: Applications to model checking. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997)Google Scholar
  3. [BGR01]
    Benedikt, M., Godefroid, P., Reps, T.: Model checking of unrestricted hierarchical state machines. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 652–666. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. [BJMT01]
    Besson, F., Jensen, T., Métayer, D.L., Thorn, T.: Model checking security properties of control flow graphs. Journal of Computer Security 9, 217–250 (2001)Google Scholar
  5. [BKS]
    Brázdil, T., Kučera, A., Stražovský, O.: Decidability of temporal properties of probabilistic pushdown automata. Technical report (in preparation)Google Scholar
  6. [BPR96]
    Basu, S., Pollack, R., Roy, M.F.: On the combinatorial and algebraic complexity of quantifier elimination. Journal of the ACM 43(6), 1002–1045 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [BT73]
    Booth, T.L., Thompson, R.A.: Applying probability measures to abstract languages. IEEE Transactions on Computers 22(5), 442–450 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Can88]
    Canny, J.: Some algebraic and geometric computations in pspace. In: Proceedings of 20th ACM STOC, pp. 460–467 (1988)Google Scholar
  9. [CY95]
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. Journal of the ACM 42(4), 857–907 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  10. [EHRS00]
    Esparza, J., Hansel, D., Rossmanith, P., Schwoon, S.: Efficient algorithms for model checking pushdown systems. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 232–247. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. [EKM04]
    Esparza, J., Kučera, A., Mayr, R.: Model checking probabilistic pushdown automata. In: Proceedings of LICS 2004, pp. 12–21. IEEE Computer Society, Los Alamitos (2004), Full version: Tech. report FIMU-RS-2004-03, Masaryk University, Brno, available online at http://www.fmi.uni-stuttgart.de/szs/publications/info/esparza.EKM04rep.shtml.
  12. [EY04]
    Etessami, K., Yannakakis, M.: Recursive markov chains, stochastic grammars, and monotone systems of non-linear equations. Technical report, School of Informatics, University of Edinburgh (2004)Google Scholar
  13. [GGJ76]
    Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. In: Proceedings of 8th ACM STOC, pp. 10–22 (1976)Google Scholar
  14. [Har63]
    Harris, T.E.: The Theory of Branching Processes. Springer, Heidelberg (1963)zbMATHGoogle Scholar
  15. [HJ94]
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6, 512–535 (1994)zbMATHCrossRefGoogle Scholar
  16. [Kuč04]
    Kučera, A.: Private communication (2004)Google Scholar
  17. [Kwi03]
    Kwiatkowska, M.: Model checking for probability and time: From theory to practice. In: Proceedings of LICS 2003, pp. 351–360. IEEE Computer Society Press, Los Alamitos (2003)Google Scholar
  18. [May04]
    Mayr, R.: Private communication (2004)Google Scholar
  19. [MS99]
    Manning, C., Schütze, H.: Foundations of Statistical Natural Language Processing. MIT Press, Cambridge (1999)zbMATHGoogle Scholar
  20. [Ren92]
    Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. Parts I,II, III. Journal of Symbolic Computation, 255–352 (1992)Google Scholar
  21. [SB93]
    Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, Heidelberg (1993)zbMATHGoogle Scholar
  22. [Tiw92]
    Tiwari, P.: A problem that is easier to solve on the unit-cost algebraic RAM. Journal of Complexity, 393–397 (1992)Google Scholar
  23. [Var85]
    Vardi, M.: Automatic verification of probabilistic concurrent finite-state programs. In: Proceedings of FOCS 1985, pp. 327–338. IEEE Computer Society Press, Los Alamitos (1985)Google Scholar
  24. [Wal00]
    Walukiewicz, I.: Model checking CTL properties of pushdown systems. In: Kapoor, S., Prasad, S. (eds.) FST TCS 2000. LNCS, vol. 1974, pp. 127–138. Springer, Heidelberg (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Javier Esparza
    • 1
  • Kousha Etessami
    • 2
  1. 1.Institute for Formal Methods in Computer ScienceUniversity of Stuttgart 
  2. 2.School of InformaticsUniversity of Edinburgh 

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