Semantic Importance Sampling for Statistical Model Checking

  • Jeffery P. HansenEmail author
  • Lutz Wrage
  • Sagar Chaki
  • Dionisio de Niz
  • Mark Klein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9035)


Statistical Model Checking (SMC) is a technique, based on Monte-Carlo simulations, for computing the bounded probability that a specific event occurs during a stochastic system’s execution. Estimating the probability of a “rare” event accurately with SMC requires many simulations. To this end, Importance Sampling (IS) is used to reduce the simulation effort. Commonly, IS involves “tilting” the parameters of the original input distribution, which is ineffective if the set of inputs causing the event (i.e., input-event region) is disjoint. In this paper, we propose a technique called Semantic Importance Sampling (SIS) to address this challenge. Using an SMT solver, SIS recursively constructs an abstract indicator function that over-approximates the input-event region, and then uses this abstract indicator function to perform SMC with IS. By using abstraction and SMT solving, SIS thus exposes a new connection between the verification of non-deterministic and stochastic systems. We also propose two optimizations that reduce the SMT solving cost of SIS significantly. Finally, we implement SIS and validate it on several problems. Our results indicate that SIS reduces simulation effort by multiple orders of magnitude even in systems with disjoint input-event regions.


Importance Sampling Recursive Call Random Input Symbolic Execution Input Distribution 
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.
    Barbot, B., Haddad, S., Picaronny, C.: Coupling and importance sampling for statistical model checking. In: Flanagan, C., König, B. (eds.) TACAS 2012. LNCS, vol. 7214, pp. 331–346. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Clarke, E.M., Zuliani, P.: Statistical model checking for cyber-physical systems. In: Bultan, T., Hsiung, P.-A. (eds.) ATVA 2011. LNCS, vol. 6996, pp. 1–12. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Reijsbergen, D., et al.: Rare event simulation for highly dependable systems with fast repairs. In: Proceedings of the 7th International Conference on Quantitative Evaluation of Systems (2010)Google Scholar
  4. 4.
    David, A., Du, D., Guldstrand Larsen, K., Legay, A., Mikučionis, M.: Optimizing Control Strategy Using Statistical Model Checking. In: Brat, G., Rungta, N., Venet, A. (eds.) NFM 2013. LNCS, vol. 7871, pp. 352–367. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    David, A., Larsen, K.G., Legay, A., Mikučionis, M., Wang, Z.: Time for Statistical Model Checking of Real-Time Systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 349–355. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Gao, S., Kong, S., Clarke, E.M.: dReal: An SMT Solver for Nonlinear Theories over the Reals. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 208–214. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  7. 7.
    Gurfinkel, A., Chaki, S.: Boxes: A Symbolic Abstract Domain of Boxes. In: Cousot, R., Martel, M. (eds.) SAS 2010. LNCS, vol. 6337, pp. 287–303. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Hansson, H., Jonsson, B.: A Logic for Reasoning about Time and Reliability. Formal Aspects of Computing (FACJ) 6(5), 512–535 (1994)CrossRefzbMATHGoogle Scholar
  9. 9.
    Jegourel, C., Legay, A., Sedwards, S.: Cross-entropy optimisation of importance sampling parameters for statistical model checking. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 327–342. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  10. 10.
    Jegourel, C., Legay, A., Sedwards, S.: Importance Splitting for Statistical Model Checking Rare Properties. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 576–591. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Kahn, H.: Stochastic (monte carlo) attenuation analysis. Tech. Rep. P-88, Rand Corp. (1949)Google Scholar
  12. 12.
    Luckow, K.S., Pasareanu, C.S., Dwyer, M.B., Filieri, A., Visser, W.: Exact and approximate probabilistic symbolic execution for nondeterministic programs. In: Proc. of ASE (2014)Google Scholar
  13. 13.
    Borges, M., et al.: Compositional solution space quantification for probabilistics software analysis. In: Proceedings of PLDI: Programming Language Design and Implementation (June 2014)Google Scholar
  14. 14.
    Srinivasan, R.: Importance Sampling: Applications in Communications and Detection. Engineering online library, Springer (2002)Google Scholar
  15. 15.
    Stoelinga, M.: Alea jacta est: verification of probabilistic, real-time and parametric systems. Ph.D. thesis, University of Nijmegen, the Netherlands (2002)Google Scholar
  16. 16.
    Younes, H.L.S.: Verification and planning for stochastic processes with asynchronous events. Ph.D. thesis, Carnegie Mellon University (2004)Google Scholar
  17. 17.
    Younes, H.L.S., Kwiatkowska, M.Z., Norman, G., Parker, D.: Numerical vs. statistical probabilistic model checking. STTT 8(3), 216–228 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Jeffery P. Hansen
    • 1
    Email author
  • Lutz Wrage
    • 1
  • Sagar Chaki
    • 1
  • Dionisio de Niz
    • 1
  • Mark Klein
    • 1
  1. 1.Carnegie Mellon UniversityPittsburghUSA

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