Efficient Probabilistic Model Checking of Systems with Ranged Probabilities

  • Khalil Ghorbal
  • Parasara Sridhar Duggirala
  • Vineet Kahlon
  • Franjo Ivančić
  • Aarti Gupta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7550)


We introduce a new technique to model check reachability properties on Interval Discrete-Time Markov Chains (IDTMC). We compute a sound over-approximation of the probabilities of satisfying a given property where the accuracy is characterized in terms of error bounds. We leverage affine arithmetic to propagate the first-order error terms. Higher-order error terms are bounded using interval arithmetic.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bleich, C., Overton, M.L.: A linear-time algorithm for the weighted median problem. Courant Institute of Mathematical Sciences, New York University, New York (1983)Google Scholar
  2. 2.
    Comba, J.L.D., Stolfi, J.: Affine arithmetic and its applications to computer graphics. In: SIBGRAPI 1993 (1993)Google Scholar
  3. 3.
    Corsaro, S., Marino, M.: Interval linear systems: the state of the art. Computational Statistics 21, 365–384 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    de Figueiredo, L.H., Stolfi, J.: Self-Validated Numerical Methods and Applications. Brazilian Mathematics Colloquium monographs. IMPA/CNPq, Rio de Janeiro, Brazil (1997)Google Scholar
  5. 5.
    Efron, B., Tibshirani, R.J.: An Introduction to the Bootstrap. CRC Press (1993)Google Scholar
  6. 6.
    Fecher, H., Leucker, M., Wolf, V.: Don’t Know in Probabilistic Systems. In: Valmari, A. (ed.) SPIN 2006. LNCS, vol. 3925, pp. 71–88. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Ghorbal, K., Goubault, E., Putot, S.: A Logical Product Approach to Zonotope Intersection. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 212–226. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Girard, A.: Reachability of Uncertain Linear Systems Using Zonotopes. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 291–305. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Goubault, É., Putot, S.: Static Analysis of Numerical Algorithms. In: Yi, K. (ed.) SAS 2006. LNCS, vol. 4134, pp. 18–34. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Hansen, E., Sengupta, S.: Bounding solutions of systems of equations using interval analysis. BIT Numerical Mathematics 21, 203–211 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6, 512–535 (1994)zbMATHCrossRefGoogle Scholar
  12. 12.
    Hérault, T., Lassaigne, R., Magniette, F., Peyronnet, S.: Approximate Probabilistic Model Checking. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 73–84. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Jiri, Rohn: Systems of linear interval equations. Linear Algebra and its Applications 126, 39–78 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jonsson, B., Larsen, K.: Specification and refinement of probabilistic processes. In: LICS, pp. 266–277 (July 1991)Google Scholar
  15. 15.
    Katoen, J.-P., Klink, D., Leucker, M., Wolf, V.: Three-valued abstraction for probabilistic systems. Journal of Logic and Algebraic Programming 81(4), 356–389 (2012)zbMATHCrossRefGoogle Scholar
  16. 16.
    Kozine, I.O., Utkin, L.V.: Interval-valued finite markov chains. Reliable Computing 8, 97–113 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Makhorin, A.: The GNU Linear Programming Kit (GLPK) (2000)Google Scholar
  18. 18.
    Moore, R.E., Yang, C.T.: Interval analysis I. Technical Report LMSD-285875, Lockheed Missiles and Space Division, Sunnyvale, CA, USA (1959)Google Scholar
  19. 19.
    Rump, S.M.: Profil/biasGoogle Scholar
  20. 20.
    Sen, K., Viswanathan, M., Agha, G.: On Statistical Model Checking of Stochastic Systems. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 266–280. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  21. 21.
    Younes, H.L.S., Simmons, R.G.: Probabilistic Verification of Discrete Event Systems Using Acceptance Sampling. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 223–235. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  22. 22.
    Zuliani, P., Platzer, A., Clarke, E.M.: Bayesian statistical model checking with application to Simulink/Stateflow verification. In: HSCC, pp. 243–252 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Khalil Ghorbal
    • 1
  • Parasara Sridhar Duggirala
    • 1
    • 2
  • Vineet Kahlon
    • 1
  • Franjo Ivančić
    • 1
  • Aarti Gupta
    • 1
  1. 1.NEC Laboratories America, Inc.USA
  2. 2.University of Illinois at Urbana ChampaignUSA

Personalised recommendations