Probabilistic Hybrid Systems Verification via SMT and Monte Carlo Techniques

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10028)

Abstract

We develop numerically rigorous Monte Carlo approaches for computing probabilistic reachability in hybrid systems subject to random and nondeterministic parameters. Instead of standard simulation we use \(\delta \)-complete SMT procedures, which enable formal reasoning for nonlinear systems up to a user-definable numeric precision. Monte Carlo approaches for probability estimation assume that sampling is possible for the real system at hand. However, when using \(\delta \)-complete simulation one instead samples from an overapproximation of the real random variable. In this paper, we introduce a Monte Carlo-SMT approach for computing probabilistic reachability confidence intervals that are both statistically and numerically rigorous. We apply our technique to hybrid systems involving nonlinear differential equations.

References

  1. 1.
    Alur, R., Courcoubetis, C., Henzinger, T.A., Ho, P.-H.: Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. In: Grossman, R.L., Nerode, A., Ravn, A.P., Rischel, H. (eds.) HS 1991-1992. LNCS, vol. 736, pp. 209–229. Springer, Heidelberg (1993). doi:10.1007/3-540-57318-6_30 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). doi:10.1007/978-3-642-24372-1_1 CrossRefGoogle Scholar
  3. 3.
    Ellen, C., Gerwinn, S., Fränzle, M.: Statistical model checking for stochastic hybrid systems involving nondeterminism over continuous domains. Int. J. Softw. Tools Technol. Transf. (STTT) 17(4), 485–504 (2015)CrossRefGoogle Scholar
  4. 4.
    Gao, S., Avigad, J., Clarke, E.M.: Delta-decidability over the reals. In: LICS, pp. 305–314 (2012)Google Scholar
  5. 5.
    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 (LNAI), vol. 7898, pp. 208–214. Springer, Heidelberg (2013). doi:10.1007/978-3-642-38574-2_14 CrossRefGoogle Scholar
  6. 6.
    Gao, S., Kong, S., Clarke, E.M.: Satisfiability modulo ODEs. In: FMCAD, pp. 105–112 (2013)Google Scholar
  7. 7.
    Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions, 1st edn. Cambridge University Press, Cambridge (2010)MATHGoogle Scholar
  8. 8.
    Platzer, A., Clarke, E.M.: The image computation problem in hybrid systems model checking. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 473–486. Springer, Heidelberg (2007). doi:10.1007/978-3-540-71493-4_37 CrossRefGoogle Scholar
  9. 9.
    Richardson, D.: Some undecidable problems involving elementary functions of a real variable. J. Symb. Log. 33(4), 514–520 (1968)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Rubinstein, R.Y., Kroese, D.: Simulation and the Monte Carlo Method. Wiley, Hoboken (2008)MATHGoogle Scholar
  11. 11.
    Shmarov, F., Zuliani, P.: ProbReach: verified probabilistic \(\delta \)-reachability for stochastic hybrid systems. In: HSCC, pp. 134–139. ACM (2015)Google Scholar
  12. 12.
    Song, B., Thomas, D.: Dynamics of starvation in humans. J. Math. Biol. 54(1), 27–43 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Wang, Q., Zuliani, P., Kong, S., Gao, S., Clarke, E.M.: SReach: a probabilistic bounded delta-reachability analyzer for stochastic hybrid systems. In: Roux, O., Bourdon, J. (eds.) CMSB 2015. LNCS, vol. 9308, pp. 15–27. Springer, Heidelberg (2015). doi:10.1007/978-3-319-23401-4_3 CrossRefGoogle Scholar
  14. 14.
    Younes, H.L.S., Kwiatkowska, M.Z., Norman, G., Parker, D.: Numerical vs. statistical probabilistic model checking. STTT 8(3), 216–228 (2006)CrossRefMATHGoogle Scholar
  15. 15.
    Younes, H.L.S., Simmons, R.G.: Statistical probabilistic model checking with a focus on time-bounded properties. Inf. Comput. 204(9), 1368–1409 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Zhang, H., Hou, W., Henrot, L., Schnebert, S., Dumas, M., Heusèle, C., Yang, J.: Modelling epidermis homoeostasis and psoriasis pathogenesis. J. R. Soc. Interface 12(103), 20141071 (2015)CrossRefGoogle Scholar
  17. 17.
    Zuliani, P., Platzer, A., Clarke, E.M.: Bayesian statistical model checking with application to Stateflow/Simulink verification. Formal Methods Syst. Des. 43(2), 338–367 (2013)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.School of Computing ScienceNewcastle UniversityNewcastle upon TyneUK

Personalised recommendations