Partial Order Reduction for Markov Decision Processes: A Survey

  • Marcus Groesser
  • Christel Baier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4111)

Abstract

In the past, several model checking algorithms have been proposed to verify probabilistic reactive systems. In contrast to the non-probabilistic setting where various techniques have been suggested and successfully applied to combat the state space-explosion problem in the context of model checking the techniques used for probabilistic systems have mainly concentrated on symbolic methods with variants of decision diagrams or abstraction methods. Only recently results have been published that give criteria on applying partial order reduction for verifying quantitative linear time properties as well as branching time properties for probabilistic systems. This paper summarizes the results that have been established so far about partial order reduction for Markov decision processes. We present the different reduction conditions and provide a comparison of the corresponding results.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baier, C., Ciesinski, F., Groesser, M.: Quantitative analysis of distributed randomized protocols. In: Proc. of the tenth International Workshop on Formal Methods for Industrial Critical Systems (FMICS 2005) (2005)Google Scholar
  2. 2.
    Baier, C., Clarke, E., Hartonas-Garmhausen, V., Kwiatkowska, M., Ryan, M.: Symbolic model checking for probabilistic processes. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 430–440. Springer, Heidelberg (1997)Google Scholar
  3. 3.
    Baier, C., D’Argenio, P., Größer, M.: Partial order reduction for probabilistic branching time. In: Proc. QAPL (2005)Google Scholar
  4. 4.
    Baier, C., Engelen, B., Majster-Cederbaum, M.: Deciding bisimularity and similarity for probabilistic processes. Jounal of Computer and System Sciences 60, 187–231 (2000)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Baier, C., Größer, M., Ciesinski, F.: Partial order reduction for probabilistic systems. In: QEST 2004 [37], pp. 230–239 (2004)Google Scholar
  6. 6.
    Baier, C., Haverkort, B., Hermanns, H., Katoen, J.-P., Siegle, M.: Validation of Stochastic Systems. LNCS, vol. 2925. Springer, Heidelberg (2004)MATHCrossRefGoogle Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Bozga, M., Maler, O.: On the Representation of Probabilities over Structured Domains. In: Halbwachs, N., Peled, D.A. (eds.) CAV 1999. LNCS, vol. 1633, pp. 261–273. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  9. 9.
    Cattani, S., Segala, R.: Decision algorithms for probabilistic bisimulation. In: Brim, L., Jančar, P., Křetínský, M., Kucera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 371–385. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Clarke, E., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (1999)Google Scholar
  11. 11.
    Courcoubetis, C., Yannakakis, M.: Markov decision processes and regular events (extended abstract). In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 336–349. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  12. 12.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. Journal of the ACM 42(4), 857–907 (1995)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    d’ Argenio, P., Jeannet, B., Jensen, H., Larsen, K.: Reachability analysis of probabilistic systems by successive refinements. In: [17], pp. 57–76 (2001)Google Scholar
  14. 14.
    D’Argenio, P.R., Niebert, P.: Partial order reduction on concurrent probabilistic programs. In: QEST 2004 [37], pp. 240–249 (2004)Google Scholar
  15. 15.
    de Alfaro, L.: Formal Verification of Probabilistic Systems. PhD thesis, Stanford University, Department of Computer Science (1997)Google Scholar
  16. 16.
    de Alfaro, L.: Stochastic transition systems. In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 423–438. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  17. 17.
    de Alfaro, L., Gilmore, S. (eds.): PROBMIV 2002, PAPM-PROBMIV 2002, and PAPM 2002. LNCS, vol. 2399. Springer, Heidelberg (2001)Google Scholar
  18. 18.
    Fecher, H., Leuker, M., Wolf, V.: Don’t know in probabilistic systems. In: Valmari, A. (ed.) SPIN 2006. LNCS, vol. 3925. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Gerth, R., Kuiper, R., Peled, D., Penczek, W.: A partial order approach to branching time logic model checking. In: Proc. 3rd Israel Symposium on the Theory of Computing Systems (ISTCS 1995), pp. 130–139. IEEE Press, Los Alamitos (1995)CrossRefGoogle Scholar
  20. 20.
    Godefroid, P.: Partial-Order Methods for the Verification of Concurrent Systems. LNCS, vol. 1032. Springer, Heidelberg (1996)Google Scholar
  21. 21.
    Godefroid, P., Peled, D., Staskauskas, M.: Using partial-order methods in the formal validation of industrial concurrent programs. In: Proc. International Symposium on Software Testing and Analysis, pp. 261–269. ACM Press, New York (1996)Google Scholar
  22. 22.
    Hachtel, G., Macii, E., Pardo, A., Somenzi, F.: Probabilistic Analysis of Large Finite State Machines. In: 31st ACM/IEEE Design Automation Conference (DAC). San Diego Convention Center (1994)Google Scholar
  23. 23.
    Hermanns, H., Kwiatkowska, M., Norman, G., Parker, D., Siegle, M.: On the use of MTBDDs for performability analysis and verification of stochastic systems. Journal of Logic and Algebraic Programming: Special Issue on Probabilistic Techniques for the Design and Analysis of Systems 56, 23–67 (2003)MATHMathSciNetGoogle Scholar
  24. 24.
    Hermanns, H., Segala, R. (eds.): PROBMIV 2002, PAPM-PROBMIV 2002, and PAPM 2002. LNCS, vol. 2399. Springer, Heidelberg (2002)MATHGoogle Scholar
  25. 25.
    Holzmann, G., Peled, D.: An improvement in formal verification. In: Proc. Formal Description Techniques, FORTE 1994, Berne, Switzerland, pp. 197–211. Chapman & Hall, Boca Raton (1994)Google Scholar
  26. 26.
    Huth, M.: Possibilistic and probabilistic abstraction-based model checking. In: [24], pp. 115–134 (2002)Google Scholar
  27. 27.
    Huth, M.: Abstraction and probabilities for hybrid logics. In: Proc. 2nd workshop on Quantitative Aspects of Programming Languages (2004)Google Scholar
  28. 28.
    Jonsson, B., Larsen, K.: Specification and refinement of probabilistic processes. In: Proc. LICS, pp. 266–277. IEEE CS Press, Los Alamitos (1991)Google Scholar
  29. 29.
    Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model checking with PRISM: A hybrid approach. International Journal on Software Tools for Technology Transfer (STTT) (2004)Google Scholar
  30. 30.
    Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Information and Computation 94(1), 1–28 (1991)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Miner, A., Parker, D.: Symbolic representations and analysis of large probabilistic systems.In: [6] (2003)Google Scholar
  32. 32.
    Peled, D.: All from one, one for all: On model checking using representatives. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 409–423. Springer, Heidelberg (1993)Google Scholar
  33. 33.
    Peled, D.: Partial order reduction: Linear and branching time logics and process algebras. In: [34], pp. 79–88 (1996)Google Scholar
  34. 34.
    Peled, D., Pratt, V., Holzmann, G. (eds.): Partial Order Methods in Verification. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 29(10). American Mathematical Society (1997)Google Scholar
  35. 35.
    Pnueli, A., Zuck, L.D.: Probabilistic verification. Information and Computation 103(1), 1–29 (1993)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Puterman, M.L.: Markov Decision Processes—Discrete Stochastic Dynamic Programming. John Wiley & Sons, Inc., New York (1994)MATHGoogle Scholar
  37. 37.
    Proceedings of the 1st International Conference on Quantitative Evaluation of SysTems (QEST 2004), Enschede, The Netherlands. IEEE Computer Society Press, Los Alamitos (2004)Google Scholar
  38. 38.
    Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, Massachusetts Institute of Technology (1995)Google Scholar
  39. 39.
    Valmari, A.: A stubborn attack on state explosion. Formal Methods in System Design 1, 297–322 (1992)MATHCrossRefGoogle Scholar
  40. 40.
    Valmari, A.: State of the art report: Stubborn sets. Petri-Net Newsletters 46, 6–14 (1994)Google Scholar
  41. 41.
    Valmari, A.: Stubborn set methods for process algebras. In: [34], pp. 79–88 (1996)Google Scholar
  42. 42.
    van Glabbeek, R., Smolka, S., Steffen, B., Tofts, C.: Reactive, generative, and stratified models of probabilistic processes. In: Proc. 5th Annual Symposium on Logic in Computer Science (LICS), pp. 130–141. IEEE Computer Society Press, Los Alamitos (1990)CrossRefGoogle Scholar
  43. 43.
    Vardi, M.: Automatic verification of probabilistic concurrent finite-state programs. In: Proc. 26th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 327–338 (1985)Google Scholar
  44. 44.
    Vardi, M., Wolper, P.: An automata-theoretic approach to automatic program verification (preliminary report). In: Proc. 1st Annual Symposium on Logic in Computer Science (LICS), pp. 332–344. IEEE Computer Society Press, Los Alamitos (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Marcus Groesser
    • 1
  • Christel Baier
    • 1
  1. 1.Institut für Informatik IBonn

Personalised recommendations