Quantitative Analysis under Fairness Constraints

  • Christel Baier
  • Marcus Groesser
  • Frank Ciesinski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5799)

Abstract

It is well-known that fairness assumptions can be crucial for verifying progress, reactivity or other liveness properties for interleaving models. This also applies to Markov decision processes as an operational model for concurrent probabilistic systems and the task to establish tight lower or upper probability bounds for events that are specified by liveness properties. In this paper, we study general notions of strong and weak fairness constraints for Markov decision processes, formalized in an action- or state-based setting. We present a polynomially time-bounded algorithm for the quantitative analysis of an MDP against ω-automata specifications under fair worst- or best-case scenarios. Furthermore, we discuss the treatment of strong and weak fairness and process fairness constraints in the context of partial order reduction techniques for Markov decision processes that have been realized in the model checker LiQuor and rely on a variant of Peled’s ample set method.

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References

  1. 1.
    Alur, R., Henzinger, T.A.: Reactive modules. Formal Methods in System Design: An International Journal 15(1), 7–48 (1999)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Arons, T., Pnueli, A., Zuck, L.: Parameterized verification by probabilistic abstraction. In: Gordon, A.D. (ed.) FOSSACS 2003. LNCS, vol. 2620, pp. 87–102. Springer, Heidelberg (2003)Google Scholar
  3. 3.
    Baier, C., Ciesinski, F., Größer, M.: Probmela: a modeling language for communicating probabilistic systems. In: Proc. MEMOCODE (2004)Google Scholar
  4. 4.
    Baier, C., Ciesinski, F., Grösser, M., Klein, J.: Reduction techniques for model checking markov decision processes. In: Proc.QEST 2008. IEEE CS Press, Los Alamitos (2008)Google Scholar
  5. 5.
    Baier, C., D’Argenio, P., Größer, M.: Partial order reduction for probabilistic branching time. In: Proc. QAPL. ENTCS, vol. 153(2) (2006)Google Scholar
  6. 6.
    Baier, C., Größer, M., Ciesinski, F.: Partial order reduction for probabilistic systems. In: Proc. QEST 2004. IEEE CS Press, Los Alamitos (2004)Google Scholar
  7. 7.
    Baier, C., Katoen, J.-P.: Principles of Model Checking. MIT Press, Cambridge (2008)MATHGoogle Scholar
  8. 8.
    Baier, C., Kwiatkoswka, M.: Model checking for a probabilistic branching time logic with fairness. Distributed Computing 11(3) (1998)Google Scholar
  9. 9.
    Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026. Springer, Heidelberg (1995)Google Scholar
  10. 10.
    Chrobak, M., Gasieniec, L., Rytter, W.: A randomized algorithm for gossiping in radio networks. In: Wang, J. (ed.) COCOON 2001. LNCS, vol. 2108, p. 483. Springer, Heidelberg (2001)Google Scholar
  11. 11.
    Ciesinski, F., Baier, C.: LiQuor: a tool for qualitative and quantitative linear time analysis of reactive systems. In: Proc. QEST 2007. IEEE CS Press, Los Alamitos (2007)Google Scholar
  12. 12.
    Clarke, E., Emerson, E., Sistla, A.: Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM TOPLAS 8(2) (1986)Google Scholar
  13. 13.
    Clarke, E., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (1999)Google Scholar
  14. 14.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. Journal of the ACM 42(4) (1995)Google Scholar
  15. 15.
    D’Argenio, P.R., Niebert, P.: Partial order reduction on concurrent probabilistic programs. In: Proc. QEST 2004. IEEE CS Press, Los Alamitos (2004)Google Scholar
  16. 16.
    de Alfaro, L.: Formal Verification of Probabilistic Systems. PhD thesis (1997)Google Scholar
  17. 17.
    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
  18. 18.
    de Alfaro, L.: From fairness to chance. In: Proc. PROBMIV. ENTCS, vol. 22 (1999)Google Scholar
  19. 19.
    Dijkstra, E.W.: Guarded commands, non-determinacy and the formal derivation of programs. Comm. ACM 18 (1975)Google Scholar
  20. 20.
    Allen Emerson, E., Lei, C.-L.: Modalities for model checking: branching time logic strikes back. Sci. Comput. Program 8(3) (1987)Google Scholar
  21. 21.
    Francez, N.: Fairness. Springer, Heidelberg (1986)MATHGoogle Scholar
  22. 22.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)MATHGoogle Scholar
  23. 23.
    Größer, M.: Reduction Methods for Probabilistic Model Checking. PhD thesis (2008)Google Scholar
  24. 24.
    Hart, S., Sharir, M., Pnueli, A.: Termination of probabilistic concurrent programs. ACM TOPLAS 5(3) (1983)Google Scholar
  25. 25.
    Holzmann, G., Peled, D.: An improvement in formal verification. In: Proc. FORTE. Chapman & Hall, Boca Raton (1994)Google Scholar
  26. 26.
    Holzmann, G.: The model checker SPIN. Software Engineering 23(5) (1997)Google Scholar
  27. 27.
    Klein, J., Baier, C.: On-the-fly stuttering in the construction of deterministic omega-automata. In: Holub, J., Žďárek, J. (eds.) CIAA 2007. LNCS, vol. 4783, pp. 51–61. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  28. 28.
    Kwiatkowska, M.: Survey of fairness notions. Inf. and Softw.Techn. 31(7) (1989)Google Scholar
  29. 29.
    Kwiatkowska, M., Norman, G., Parker, D.: PRISM: Probabilistic symbolic model checker. In: Field, T., Harrison, P.G., Bradley, J., Harder, U. (eds.) TOOLS 2002. LNCS, vol. 2324, p. 200. Springer, Heidelberg (2002)Google Scholar
  30. 30.
    Lamport, L.: Specifying concurrent program modules. TOPLAS 5(2) (1983)Google Scholar
  31. 31.
    Lehmann, D., Pnueli, A., Stavi, J.: Impartiality, justice and fairness: the ethics of concurrent termination. In: Even, S., Kariv, O. (eds.) ICALP 1981. LNCS, vol. 115, Springer, Heidelberg (1981)Google Scholar
  32. 32.
    Lehmann, D., Rabin, M.O.: On the advantage of free choice: A symmetric and fully distributed solution to the Dining Philosophers problem (extended abstract). In: Proc. POPL (1981)Google Scholar
  33. 33.
    Peled, D.: All from one, one for all: On model checking using representatives. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697. Springer, Heidelberg (1993)Google Scholar
  34. 34.
    Peled, D.: Partial order reduction: Linear and branching time logics and process algebras. In: Partial Order Methods in Verification, DIMACS, vol. 29(10) (1997)Google Scholar
  35. 35.
    Pnueli, A., Zuck, L.: Probabilistic verification. Information and Computation 103(1) (March 1993)Google Scholar
  36. 36.
    Puterman, M.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley & Sons, Inc., New York (1994)MATHGoogle Scholar
  37. 37.
    Rosier, L.E., Yen, H.C.: On the complexity of deciding fair termination of probabilistic concurrent finite-state programs. Theoretical Computer Science (1988)Google Scholar
  38. 38.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003)MATHGoogle Scholar
  39. 39.
    Vardi, M.: Automatic verification of probabilistic concurrent finite-state programs. In: Proc. FOCS (1985)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Christel Baier
    • 1
  • Marcus Groesser
    • 1
  • Frank Ciesinski
    • 1
  1. 1.Technische Universtät DresdenDresdenGermany

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