Stochastic Differential Dynamic Logic for Stochastic Hybrid Programs

  • André Platzer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6803)


Logic is a powerful tool for analyzing and verifying systems, including programs, discrete systems, real-time systems, hybrid systems, and distributed systems. Some applications also have a stochastic behavior, however, either because of fundamental properties of nature, uncertain environments, or simplifications to overcome complexity. Discrete probabilistic systems have been studied using logic. But logic has been chronically underdeveloped in the context of stochastic hybrid systems, i.e., systems with interacting discrete, continuous, and stochastic dynamics. We aim at overcoming this deficiency and introduce a dynamic logic for stochastic hybrid systems. Our results indicate that logic is a promising tool for understanding stochastic hybrid systems and can help taming some of their complexity. We introduce a compositional model for stochastic hybrid systems. We prove adaptivity, càdlàg, and Markov time properties, and prove that the semantics of our logic is measurable. We present compositional proof rules, including rules for stochastic differential equations, and prove soundness.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abate, A., Prandini, M., Lygeros, J., Sastry, S.: Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems. Automatica 44(11), 2724–2734 (2008)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bujorianu, M.L., Lygeros, J.: Towards a general theory of stochastic hybrid systems. In: Blom, H.A.P., Lygeros, J. (eds.) Stochastic Hybrid Systems: Theory and Safety Critical Applications. Lecture Notes Contr. Inf., vol. 337, pp. 3–30. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Cassandras, C.G., Lygeros, J. (eds.): Stochastic Hybrid Systems. CRC, Boca Raton (2006)Google Scholar
  4. 4.
    Dutertre, B.: Complete proof systems for first order interval temporal logic. In: LICS, pp. 36–43. IEEE Computer Society, Los Alamitos (1995)Google Scholar
  5. 5.
    Dynkin, E.B.: Markov Processes. Springer, Heidelberg (1965)CrossRefMATHGoogle Scholar
  6. 6.
    Feldman, Y.A., Harel, D.: A probabilistic dynamic logic. J. Comput. Syst. Sci. 28(2), 193–215 (1984)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Fränzle, M., Teige, T., Eggers, A.: Engineering constraint solvers for automatic analysis of probabilistic hybrid automata. J. Log. Algebr. Program. 79(7), 436–466 (2010)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Ghosh, M.K., Arapostathis, A., Marcus, S.I.: Ergodic control of switching diffusions. SIAM J. Control Optim. 35(6), 1952–1988 (1997)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Hu, J., Lygeros, J., Sastry, S.: Towards a theory of stochastic hybrid systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 160–173. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  10. 10.
    Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Springer, Heidelberg (1991)MATHGoogle Scholar
  11. 11.
    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, New York (2010)MATHGoogle Scholar
  12. 12.
    Koutsoukos, X.D., Riley, D.: Computational methods for verification of stochastic hybrid systems. IEEE T. Syst. Man, Cy. A 38(2), 385–396 (2008)CrossRefGoogle Scholar
  13. 13.
    Kozen, D.: Semantics of probabilistic programs. J. Comput. Syst. Sci. 22(3), 328–350 (1981)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Kozen, D.: A probabilistic PDL. J. Comput. Syst. Sci. 30(2), 162–178 (1985)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Kushner, H.J. (ed.): Stochastic Stability and Control. Academic Press, New York (1967)MATHGoogle Scholar
  16. 16.
    Kwiatkowska, M.Z., Norman, G., Parker, D., Qu, H.: Assume-guarantee verification for probabilistic systems. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 23–37. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Kwiatkowska, M.Z., Norman, G., Sproston, J., Wang, F.: Symbolic model checking for probabilistic timed automata. Inf. Comput. 205(7), 1027–1077 (2007)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Bevilacqua, V., Sharykin, R.: Specification and analysis of distributed object-based stochastic hybrid systems. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 460–475. Springer, Heidelberg (2006)Google Scholar
  19. 19.
    Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer, Heidelberg (2007)MATHGoogle Scholar
  20. 20.
    Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Log. Comput. 20(1), 309–352 (2010)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Platzer, A.: Quantified differential dynamic logic for distributed hybrid systems. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 469–483. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  22. 22.
    Platzer, A.: Stochastic differential dynamic logic for stochastic hybrid systems. Tech. Rep. CMU-CS-11-111, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA (2011)Google Scholar
  23. 23.
    Prajna, S., Jadbabaie, A., Pappas, G.J.: A framework for worst-case and stochastic safety verification using barrier certificates. IEEE T. Automat. Contr. 52(8), 1415–1429 (2007)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Pratt, V.R.: Semantical considerations on Floyd-Hoare logic. In: FOCS, pp. 109–121. IEEE, Los Alamitos (1976)Google Scholar
  25. 25.
    Richardson, M., Domingos, P.: Markov logic networks. Machine Learning 62(1-2), 107–136 (2006)CrossRefGoogle Scholar
  26. 26.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1951)MATHGoogle Scholar
  27. 27.
    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
  28. 28.
    Zuliani, P., Platzer, A., Clarke, E.M.: Bayesian statistical model checking with application to Simulink/Stateflow verification. In: Johansson, K.H., Yi, W. (eds.) HSCC, pp. 243–252. ACM, New York (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • André Platzer
    • 1
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations