Abstract
This paper investigates relative precision and optimality of analyses for concurrent probabilistic systems. Aiming at the problem at the heart of probabilistic model checking – computing the probability of reaching a particular set of states – we leverage the theory of abstract interpretation. With a focus on predicate abstraction, we develop the first abstract-interpretation framework for Markov decision processes which admits to compute both lower and upper bounds on reachability probabilities. Further, we describe how to compute and approximate such abstractions using abstraction refinement and give experimental results.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bianco, A., de Alfaro, L.: Model Checking of Probabilistic and Nondeterministic Systems. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 499–513. Springer, Heidelberg (1995)
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)
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)
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)
Ball, T., Podelski, A., Rajamani, S.K.: Boolean and Cartesian Abstraction for Model Checking C Programs. In: Margaria, T., Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 268–283. Springer, Heidelberg (2001)
Cousot, P., Cousot, R., Feret, J., Mauborgne, L., Miné, A., Monniaux, D., Rival, X.: The ASTREÉ Analyzer. In: Sagiv, M. (ed.) ESOP 2005. LNCS, vol. 3444, pp. 21–30. Springer, Heidelberg (2005)
Alt, M., Ferdinand, C., Martin, F., Wilhelm, R.: Cache Behavior Prediction by Abstract Interpretation. In: Cousot, R., Schmidt, D.A. (eds.) SAS 1996. LNCS, vol. 1145, pp. 52–66. Springer, Heidelberg (1996)
Reps, T.W., Sagiv, S., Yorsh, G.: Symbolic Implementation of the Best Transformer. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 252–266. Springer, Heidelberg (2004)
Cousot, P., Cousot, R.: Systematic Design of Program Transformation Frameworks by Abstract Interpretation. In: POPL 2002, pp. 178–190. ACM, New York (2002)
Monniaux, D.: Abstract Interpretation of Programs as Markov Decision Processes. Sci. Comput. Program. 58, 179–205 (2005)
Pierro, A.D., Wiklicky, H.: Concurrent Constraint Programming: Towards Probabilistic Abstract Interpretation. In: PPDP, pp. 127–138 (2000)
Smith, M.J.A.: Probabilistic Abstract Interpretation of Imperative Programs using Truncated Normal Distributions. ENTCS 220, 43–59 (2008)
Coletta, A., Gori, R., Levi, F.: Approximating Probabilistic Behaviors of Biological Systems Using Abstract Interpretation. ENTCS 229, 165–182 (2009)
Monniaux, D.: Backwards Abstract Interpretation of Probabilistic Programs. In: Sands, D. (ed.) ESOP 2001. LNCS, vol. 2028, pp. 367–382. Springer, Heidelberg (2001)
Hinton, A., Kwiatkowska, M.Z., Norman, G., Parker, D.: PRISM: A Tool for Automatic Verification of Probabilistic Systems. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 441–444. Springer, Heidelberg (2006)
Kwiatkowska, M.Z., Norman, G., Parker, D.: Game-based Abstraction for Markov Decision Processes. In: QEST, pp. 157–166 (2006)
Wachter, B., Zhang, L., Hermanns, H.: Probabilistic model checking modulo theories. In: QEST, pp. 129–140 (2007)
Kattenbelt, M., Kwiatkowska, M.Z., Norman, G., Parker, D.: Game-Based Probabilistic Predicate Abstraction in PRISM. In: QAPL (2008)
D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reachability Analysis of Probabilistic Systems by Successive Refinements. In: de Luca, L., Gilmore, S. (eds.) PROBMIV 2001, PAPM-PROBMIV 2001, and PAPM 2001. LNCS, vol. 2165, pp. 39–56. Springer, Heidelberg (2001)
Tarski, A.: A Lattice-Theoretical Fixpoint Theorem and Its Applications. Pacific Journal of Mathematics 5(2), 285–309 (1955)
Baier, C.: On Algorithmic Verification Methods for Probabilistic Systems, Habilitationsschrift, Universität Mannheim (1998)
Condon, A.: The Complexity of Stochastic Games. Inf. Comput. 96, 203–224 (1992)
Nielson, F., Nielson, H.R., Hankin, C.: Principles of Program Analysis. Springer-Verlag New York, Inc., Secaucus (1999)
Murawski, A.S., Ouaknine, J.: On Probabilistic Program Equivalence and Refinement. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 156–170. Springer, Heidelberg (2005)
Legay, A., Murawski, A.S., Ouaknine, J., Worrell, J.: On Automated Verification of Probabilistic Programs. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 173–187. Springer, Heidelberg (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wachter, B., Zhang, L. (2010). Best Probabilistic Transformers. In: Barthe, G., Hermenegildo, M. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2010. Lecture Notes in Computer Science, vol 5944. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11319-2_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-11319-2_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11318-5
Online ISBN: 978-3-642-11319-2
eBook Packages: Computer ScienceComputer Science (R0)