Probabilistic Abstract Interpretation

  • Patrick Cousot
  • Michael Monerau
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7211)


Abstract interpretation has been widely used for verifying properties of computer systems. Here, we present a way to extend this framework to the case of probabilistic systems.

The probabilistic abstraction framework that we propose allows us to systematically lift any classical analysis or verification method to the probabilistic setting by separating in the program semantics the probabilistic behavior from the (non-)deterministic behavior. This separation provides new insights for designing novel probabilistic static analyses and verification methods.

We define the concrete probabilistic semantics and propose different ways to abstract them. We provide examples illustrating the expressiveness and effectiveness of our approach.


Markov Decision Process Abstract Interpretation Probabilistic Program Abstract Domain Semantic Domain 
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.


  1. 1.
    Camporesi, F., Feret, J., Koeppl, H., Petrov, T.: Automatic reduction of stochastic rules-based models in a nutshell. Amer. Inst. of Physics, AIP 1281(2) (2010)Google Scholar
  2. 2.
    Chadha, R., Viswanathan, M., Viswanathan, R.: Least Upper Bounds for Probability Measures and Their Applications to Abstractions. In: van Breugel, F., Chechik, M. (eds.) CONCUR 2008. LNCS, vol. 5201, pp. 264–278. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Coletta, A., Gori, R., Levi, F.: Approximating probabilistic behaviors of biological systems using abstract interpretation  229(1), 165–182 (2009)Google Scholar
  4. 4.
    Cousot, P.: Constructive design of a hierarchy of semantics of a transition system by abstract interpretation. TCS 277(1-2), 47–103 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: POPL, pp. 238–252 (1977)Google Scholar
  6. 6.
    Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: POPL, pp. 269–282 (1979)Google Scholar
  7. 7.
    Cousot, P., Cousot, R.: Abstract interpretation frameworks. J. Logic and Comp. 2(4), 511–547 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cousot, P., Cousot, R.: Comparing the Galois Connection and Widening/Narrowing Approaches to Abstract Interpretation. In: Bruynooghe, M., Wirsing, M. (eds.) PLILP 1992. LNCS, vol. 631, pp. 269–295. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  9. 9.
    Cousot, P., Cousot, R., Feret, J., Mauborgne, L., Miné, A., Rival, X.: Why does Astrée scale up? FMSD 35(3), 229–264 (2009)zbMATHGoogle Scholar
  10. 10.
    D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reduction and Refinement Strategies for Probabilistic Analysis. In: Hermanns, H., Segala, R. (eds.) PAPM-PROBMIV 2002. LNCS, vol. 2399, pp. 57–76. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Forejt, V., Kwiatkowska, M., Norman, G., Parker, D.: Automated Verification Techniques for Probabilistic Systems. In: Bernardo, M., Issarny, V. (eds.) SFM 2011. LNCS, vol. 6659, pp. 53–113. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. FAC 6(5), 512–535 (1994)zbMATHGoogle Scholar
  13. 13.
    Hehner, E.: Probabilistic Predicative Programming. In: Kozen, D. (ed.) MPC 2004. LNCS, vol. 3125, pp. 169–185. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Hehner, E.: A probability perspective. FAC 23(4), 391–419 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Katoen, J.-P., McIver, A.K., Meinicke, L.A., Morgan, C.C.: Linear-Invariant Generation for Probabilistic Programs: Automated Support for Proof-Based Methods. In: Cousot, R., Martel, M. (eds.) SAS 2010. LNCS, vol. 6337, pp. 390–406. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Klenke, A.: Probability Theory: A Comprehensive Course. Springer, Heidelberg (2007)Google Scholar
  17. 17.
    Kozen, D.: Semantics of probabilistic programs. JCSS 22, 328–350 (1981)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kwiatkowska, M., Norman, G., Parker, D.: Using probabilistic model checking in systems biology. PER 35(4), 14–21 (2008)Google Scholar
  19. 19.
    McIver, A., Morgan, C.: Abstraction, Refinement and Proof for Probabilistic Systems. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  20. 20.
    Meyn, S.: Control Techniques for Complex Networks. CUP (2007)Google Scholar
  21. 21.
    Monniaux, D.: Abstract Interpretation of Probabilistic Semantics. In: SAS 2000. LNCS, vol. 1824, pp. 322–340. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  22. 22.
    Monniaux, D.: Abstract interpretation of programs as Markov decision processes. SCP 58(1–2), 179–205 (2005)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Morgan, C., McIver, A., Seidel, K., Sanders, J.: Probabilistic predicate transformers. TOPLAS 18(3), 325–353 (1996)CrossRefGoogle Scholar
  24. 24.
    Di Pierro, A., Hankin, C., Wiklicky, H.: Probabilistic lambda-calculus and quantitative program analysis. JLC 15(2), 159–179 (2005)zbMATHGoogle Scholar
  25. 25.
    Di Pierro, A., Wiklicky, H.: Concurrent constraint programming: towards probabilistic abstract interpretation. In: PPDP, pp. 127–138. ACM (2000)Google Scholar
  26. 26.
    Di Pierro, A., Wiklicky, H.: Probabilistic Abstract Interpretation and Statistical Testing (Extended Abstract). In: Hermanns, H., Segala, R. (eds.) PAPM-PROBMIV 2002. LNCS, vol. 2399, pp. 211–212. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  27. 27.
    Roy, P., Parker, D., Norman, G., de Alfaro, L.: Symbolic magnifying lens abstraction in Markov decision processes. In: QEST 2008, pp. 103–112. IEEE (2008)Google Scholar
  28. 28.
    Smith, M.: Probabilistic abstract interpretation of imperative programs using truncated normal distributions 220(3), 43–59 (2008)Google Scholar
  29. 29.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Patrick Cousot
    • 1
  • Michael Monerau
    • 1
  1. 1.NYU and École Normale SupérieureCourant InstituteFrance

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