Probabilistic Abstractions with Arbitrary Domains

  • Javier Esparza
  • Andreas Gaiser
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6887)


Recent work by Hermanns et al. and Kattenbelt et al. has extended counterexample-guided abstraction refinement (CEGAR) to probabilistic programs. These approaches are limited to predicate abstraction. We present a novel technique, based on the abstract reachability tree recently introduced by Gulavani et al., that can use arbitrary abstract domains and widening operators (in the sense of Abstract Interpretation). We show how suitable widening operators can deduce loop invariants difficult to find for predicate abstraction, and propose refinement techniques.


Span Tree Markov Decision Process Stochastic Game Abstract Interpretation Concrete State 
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.
  2. 2.
    Bagnara, R., Dobson, K., Hill, P.M., Mundell, M., Zaffanella, E.: Grids: A domain for analyzing the distribution of numerical values. In: Puebla, G. (ed.) LOPSTR 2006. LNCS, vol. 4407, pp. 219–235. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Bagnara, R., Hill, P.M., Zaffanella, E.: The Parma Polyhedra Library: Toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems. Science of Computer Programming 72(1-2), 3–21 (2008)Google Scholar
  4. 4.
    Beyer, D., Henzinger, T.A., Jhala, R., Majumdar, R.: The software model checker BLAST. Proc. of STTT 9(5-6), 505–525 (2007)CrossRefGoogle Scholar
  5. 5.
    Blanchet, B., Cousot, P., Cousot, R., Feret, J., Mauborgne, L., Miné, A., Monniaux, D., Rival, X.: Design and implementation of a special-purpose static program analyzer for safety-critical real-time embedded software. In: Mogensen, T.Æ., Schmidt, D.A., Sudborough, I.H. (eds.) The Essence of Computation. LNCS, vol. 2566, pp. 85–108. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Condon, A.: The complexity of stochastic games. Inf. Comput. 96(2), 203–224 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Condon, A.: On algorithms for simple stochastic games. DIMACS Series in Discr. Math. and Theor. Comp. Sci., vol. 13, pp. 51–73. AMS (1993)Google Scholar
  8. 8.
    Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proc. of POPL, pp. 238–252 (1977)Google Scholar
  9. 9.
    Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: POPL, San Antonio, Texas, pp. 269–282. ACM Press, New York (1979)Google Scholar
  10. 10.
    Esparza, J., Gaiser, A.: Probabilistic abstractions with arbitrary domains. Technical report, Technische Universität München (2011),
  11. 11.
    Gulavani, B.S., Chakraborty, S., Nori, A.V., Rajamani, S.K.: Automatically refining abstract interpretations. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 443–458. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Gulavani, B.S., Rajamani, S.K.: Counterexample driven refinement for abstract interpretation. In: Hermanns, H. (ed.) TACAS 2006. LNCS, vol. 3920, pp. 474–488. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Hahn, E.M., Hermanns, H., Wachter, B., Zhang, L.: PASS: Abstraction refinement for infinite probabilistic models. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 353–357. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Hermanns, H., Wachter, B., Zhang, L.: Probabilistic CEGAR. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 162–175. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. 15.
    Jeannet, B., Miné, A.: apron: A library of numerical abstract domains for static analysis. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 661–667. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Kattenbelt, M., Kwiatkowska, M.Z., Norman, G., Parker, D.: Abstraction refinement for probabilistic software. In: Jones, N.D., Müller-Olm, M. (eds.) VMCAI 2009. LNCS, vol. 5403, pp. 182–197. Springer, Heidelberg (2009)Google Scholar
  17. 17.
    Kattenbelt, M., Kwiatkowska, M.Z., Norman, G., Parker, D.: A game-based abstraction-refinement framework for markov decision processes. Form. Methods Syst. Des. 36, 246–280 (2010)CrossRefzbMATHGoogle Scholar
  18. 18.
    Monniaux, D.: Abstract interpretation of probabilistic semantics. In: SAS 2000. LNCS, vol. 1824, pp. 322–340. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. 19.
    Monniaux, D.: Abstract interpretation of programs as markov decision processes. In: Proc. of SAS, pp. 237–254 (2003)Google Scholar
  20. 20.
    Di Pierro, A., Hankin, C., Wiklicky, H.: On probabilistic techniques for data flow analysis. Electr. Notes Theor. Comput. Sci. 190(3), 59–77 (2007)CrossRefzbMATHGoogle Scholar
  21. 21.
    Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley Interscience, Hoboken (1994)CrossRefzbMATHGoogle Scholar
  22. 22.
    Wachter, B.: Refined Probabilistic Abstraction. PhD thesis, Universität des Saarlandes (2011)Google Scholar
  23. 23.
    Wachter, B., Zhang, L.: Best probabilistic transformers. In: Barthe, G., Hermenegildo, M. (eds.) VMCAI 2010. LNCS, vol. 5944, pp. 362–379. Springer, Heidelberg (2010)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Javier Esparza
    • 1
  • Andreas Gaiser
    • 1
  1. 1.Fakultät für InformatikTechnische Universität MünchenGermany

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