On Abstraction of Probabilistic Systems

  • Christian Dehnert
  • Daniel Gebler
  • Michele Volpato
  • David N. Jansen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8453)

Abstract

Probabilistic model checking extends traditional model checking by incorporating quantitative information about the probability of system transitions. However, probabilistic models that describe interesting behavior are often too complex for straightforward analysis. Abstraction is one way to deal with this complexity: instead of analyzing the (“concrete”) model, a simpler (“abstract”) model that preserves the relevant properties is built and analyzed. This paper surveys various abstraction techniques proposed in the past decade. For each abstraction technique we identify in what sense properties are preserved or provide alternatively suitable boundaries.

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References

  1. 1.
    de Alfaro, L., Roy, P.: Magnifying-lens abstraction for Markov decision processes. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 325–338. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Aljazzar, H., Leue, S.: Directed explicit state-space search in the generation of counterexamples for stochastic model checking. IEEE Trans. Software Eng. 36(1), 37–60 (2010)CrossRefGoogle Scholar
  3. 3.
    Aziz, A., Singhal, V., Balarin, F., Brayton, R.K., Sangiovanni-Vincentelli, A.L.: It usually works: The temporal logic of stochastic systems. In: Wolper, P. (ed.) CAV 1995. LNCS, vol. 939, pp. 155–165. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  4. 4.
    Baier, C., Katoen, J.-P.: Principles of model checking. MIT Press, Cambridge (2008)Google Scholar
  5. 5.
    Baier, C., Katoen, J.-P., Hermanns, H., Wolf, V.: Comparative branching-time semantics for Markov chains. Information and Computation 200(2), 149–214 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    Bozzano, M., Cimatti, A., Katoen, J.-P., Nguyen, V.Y., Noll, T., Roveri, M.: Safety, dependability and performance analysis of extended aadl models. Comput. J. 54(5), 754–775 (2011)CrossRefGoogle Scholar
  8. 8.
    Chadha, R., Viswanathan, M.: A counterexample-guided abstraction-refinement framework for markov decision processes. ACM Trans. Comput. Log. 12(1), 1 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chen, T., Forejt, V., Kwiatkowska, M., Parker, D., Simaitis, A.: Prism-games: A model checker for stochastic multi-player games. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 185–191. Springer, Heidelberg (2013)Google Scholar
  10. 10.
    Clarke, E., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 154–169. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. 11.
    Condon, A.: The complexity of stochastic games. Information and Computation 96(2), 203–224 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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.) PAPM-PROBMIV 2001. LNCS, vol. 2165, pp. 39–56. Springer, Heidelberg (2001)Google Scholar
  13. 13.
    D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reduction and refinement strategies for probabilistic analysis. In: Hermanns, H., Segala, R. (eds.) PAPM-PROBMIV 2002. LNCS, vol. 2399, pp. 57–76. Springer, Heidelberg (2002)Google Scholar
  14. 14.
    Dehnert, C., Katoen, J.-P., Parker, D.: SMT-based bisimulation minimisation of markov models. In: Giacobazzi, R., Berdine, J., Mastroeni, I. (eds.) VMCAI 2013. LNCS, vol. 7737, pp. 28–47. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  15. 15.
    Delahaye, B., Katoen, J.-P., Larsen, K.G., Legay, A., Pedersen, M.L., Sher, F., Wąsowski, A.: Abstract probabilistic automata. In: Jhala, R., Schmidt, D. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 324–339. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Approximating labelled Markov processes. Information and Computation 184(1), 160–200 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Donaldson, A.F., Miller, A.: Symmetry reduction for probabilistic model checking using generic representatives. In: Graf, S., Zhang, W. (eds.) ATVA 2006. LNCS, vol. 4218, pp. 9–23. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Eisentraut, C., Hermanns, H., Schuster, J., Turrini, A., Zhang, L.: The quest for minimal quotients for probabilistic automata. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 16–31. Springer, Heidelberg (2013)Google Scholar
  19. 19.
    Emerson, E.A., Clarke, E.M.: Using branching time temporal logic to synthesize synchronization skeletons. Science of Computer Programming 2(3), 241–266 (1982)CrossRefMATHGoogle Scholar
  20. 20.
    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
  21. 21.
    Feng, L., Kwiatkowska, M.Z., Parker, D.: Compositional verification of probabilistic systems using learning. In: QEST, pp. 133–142. IEEE Computer Society (2010)Google Scholar
  22. 22.
    Ferrer Fioriti, L.M., Hahn, E.M., Hermanns, H., Wachter, B.: Variable probabilistic abstraction refinement. In: Chakraborty, S., Mukund, M. (eds.) ATVA 2012. LNCS, vol. 7561, pp. 300–316. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  23. 23.
    Hahn, E.M., Hermanns, H., Wachter, B., Zhang, L.: Pass: Abstraction refinement for infinite probabilistic models. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 353–357. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  24. 24.
    Han, T., Katoen, J.-P.: Counterexamples in probabilistic model checking. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 72–86. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  25. 25.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994)CrossRefMATHGoogle Scholar
  26. 26.
    Henzinger, T.A., Mateescu, M., Wolf, V.: Sliding window abstraction for infinite Markov chains. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 337–352. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  27. 27.
    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)CrossRefGoogle Scholar
  28. 28.
    Huth, M., Jagadeesan, R., Schmidt, D.: Modal transition systems: A foundation for three-valued program analysis. In: Sands, D. (ed.) ESOP 2001. LNCS, vol. 2028, pp. 155–169. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  29. 29.
    Jansen, D.N., Song, L., Zhang, L.: Revisiting weak simulation for substochastic Markov chains. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 209–224. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  30. 30.
    Jansen, N., Ábrahám, E., Katelaan, J., Wimmer, R., Katoen, J.-P., Becker, B.: Hierarchical counterexamples for discrete-time markov chains. In: Bultan, T., Hsiung, P.-A. (eds.) ATVA 2011. LNCS, vol. 6996, pp. 443–452. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  31. 31.
    Jonsson, B., Larsen, K.G.: Specification and refinement of probabilistic processes. In: Proc. LICS 1991, pp. 266–277. IEEE Comp. Soc. Pr. (1991)Google Scholar
  32. 32.
    Katoen, J.-P., Kemna, T., Zapreev, I., Jansen, D.N.: Bisimulation minimisation mostly speeds up probabilistic model checking. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 87–101. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  33. 33.
    Katoen, J.-P., Klink, D., Leucker, M., Wolf, V.: Three-valued abstraction for probabilistic systems. JLAP 81(4), 356–389 (2012)MathSciNetMATHGoogle Scholar
  34. 34.
    Kattenbelt, M., Kwiatkowska, M., Norman, G., Parker, D.: A game-based abstraction-refinement framework for markov decision processes. Formal Methods in System Design 36(3), 246–280 (2010)CrossRefMATHGoogle Scholar
  35. 35.
    Kattenbelt, M., Kwiatkowska, M.Z., Norman, G., Parker, D.: A game-based abstraction-refinement framework for markov decision processes. Formal Methods in System Design 36(3), 246–280 (2010)CrossRefMATHGoogle Scholar
  36. 36.
    Komuravelli, A., Păsăreanu, C.S., Clarke, E.M.: Assume-guarantee abstraction refinement for probabilistic systems. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 310–326. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  37. 37.
    Kwiatkowska, M., Norman, G., Parker, D.: Symmetry reduction for probabilistic model checking. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 234–248. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  38. 38.
    Kwiatkowska, M., Norman, G., Parker, D.: Prism 4.0: Verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  39. 39.
    Kwiatkowska, M., 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
  40. 40.
    Larsen, K.G., Thomsen, B.: A modal process logic. In: Proc. LICS 1988, pp. 203–210. Los Alamitos, Calif (1988)Google Scholar
  41. 41.
    Larsen, K.G., Thomsen, B.: A modal process logic. In: LICS, pp. 203–210 (1988)Google Scholar
  42. 42.
    McMillan, K.L.: Lazy abstraction with interpolants. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 123–136. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  43. 43.
    Segala, R., Lynch, N.: Probabilistic simulations for probabilistic processes. Nordic Journal of Computing 2, 250–273 (1995)MathSciNetMATHGoogle Scholar
  44. 44.
    Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state programs. In: FOCS, pp. 327–338. IEEE Comp. Soc. Pr., Washington, DC (1985)Google Scholar
  45. 45.
    Wachter, B.: Refined probabilistic abstraction. Ph.D. thesis, Universität des Saarlandes, Saarbrücken (2011)Google Scholar
  46. 46.
    Wachter, B., Zhang, L.: Best probabilistic transformers. In: Barthe, G., Hermenegildo, M. (eds.) VMCAI 2010. LNCS, vol. 5944, pp. 362–379. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  47. 47.
    Wimmer, R., Herbstritt, M., Hermanns, H., Strampp, K., Becker, B.: Sigref- a symbolic bisimulation tool box. In: Graf, S., Zhang, W. (eds.) ATVA 2006. LNCS, vol. 4218, pp. 477–492. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  48. 48.
    Wimmer, R., Jansen, N., Ábrahám, E., Becker, B., Katoen, J.-P.: Minimal critical subsystems for discrete-time markov models. In: Flanagan, C., König, B. (eds.) TACAS 2012. LNCS, vol. 7214, pp. 299–314. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  49. 49.
    Wimmer, R., Jansen, N., Vorpahl, A., Ábrahám, E., Katoen, J.-P., Becker, B.: High-level counterexamples for probabilistic automata. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 39–54. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  50. 50.
    Zhang, L.: Decision algorithms for probabilistic simulations. Ph.D. thesis, Universität des Saarlandes, Saarbrücken (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Christian Dehnert
    • 1
  • Daniel Gebler
    • 2
  • Michele Volpato
    • 3
  • David N. Jansen
    • 3
  1. 1.Software Modeling and Verification GroupRWTH Aachen UniversityAachenGermany
  2. 2.Department of Computer ScienceVU University AmsterdamAmsterdamThe Netherlands
  3. 3.Institute for Computing and Information SciencesRadboud UniversityNijmegenThe Netherlands

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