Advertisement

A Simple Algorithm for Solving Qualitative Probabilistic Parity Games

  • Ernst Moritz Hahn
  • Sven Schewe
  • Andrea Turrini
  • Lijun ZhangEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9780)

Abstract

In this paper, we develop an approach to find strategies that guarantee a property in systems that contain controllable, uncontrollable, and random vertices, resulting in probabilistic games. Such games are a reasonable abstraction of systems that comprise partial control over the system (reflected by controllable transitions), hostile nondeterminism (abstraction of the unknown, such as the behaviour of an attacker or a potentially hostile environment), and probabilistic transitions for the abstraction of unknown behaviour neutral to our goals. We exploit a simple and only mildly adjusted algorithm from the analysis of non-probabilistic systems, and use it to show that the qualitative analysis of probabilistic games inherits the much celebrated sub-exponential complexity from 2-player games. The simple structure of the exploited algorithm allows us to offer tool support for finding the desired strategy, if it exists, for the given systems and properties. Our experimental evaluation shows that our technique is powerful enough to construct simple strategies that guarantee the specified probabilistic temporal properties.

Keywords

Recursive Call Winning Strategy Weak Attractor Winning Region Winning Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgement

This work is supported by the National Natural Science Foundation of China (Grants No. 61472473, 61532019, 61550110249, 61550110506), by the National 973 Program (No. 2014CB340701), by the CDZ project CAP (GZ 1023), by the Chinese Academy of Sciences Fellowship for International Young Scientists, by by the CAS/SAFEA International Partnership Program for Creative Research Teams, and by the Engineering and Physical Sciences Research Council (EPSRC) through grant EP/M027287/1 (Energy Efficient Control).

References

  1. 1.
    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
  2. 2.
    Chatterjee, K.: Stochastic \(\omega \)-regular games. Ph.D. thesis, University of California at Berkeley (2007)Google Scholar
  3. 3.
    Chatterjee, K.: Qualitative concurrent parity games: bounded rationality. In: Baldan, P., Gorla, D. (eds.) CONCUR 2014. LNCS, vol. 8704, pp. 544–559. Springer, Heidelberg (2014)Google Scholar
  4. 4.
    Chatterjee, K., de Alfaro, L., Henzinger, T.A.: Strategy improvement for concurrent reachability games. In: QEST, pp. 291–300. IEEE Computer Society (2006)Google Scholar
  5. 5.
    Chatterjee, K., de Alfaro, L., Henzinger, T.A.: Qualitative concurrent parity games. ACM Trans. Comput. Log. 12(4), 28 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chatterjee, K., Doyen, L., Gimbert, H., Oualhadj, Y.: Perfect-information stochastic mean-payoff parity games. In: Muscholl, A. (ed.) FOSSACS 2014 (ETAPS). LNCS, vol. 8412, pp. 210–225. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  7. 7.
    Chatterjee, K., Henzinger, M.: An O(n\({}^{\text{2}}\)) time algorithm for alternating Büchi games. In: SODA, pp. 1386–1399. SIAM (2012)Google Scholar
  8. 8.
    Chatterjee, K., Jurdziński, M., Henzinger, T.A.: Simple stochastic parity games. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 100–113. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Chatterjee, K., Jurdziński, M., Henzinger, T.A.: Quantitative stochastic parity games. In: SODA 2004, pp. 121–130 (2004)Google Scholar
  10. 10.
    Chen, T., Forejt, V., Kwiatkowska, M.Z., Parker, D., Simaitis, A.: Automatic verification of competitive stochastic systems. Formal Methods Syst. Des. 43(1), 61–92 (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    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 (ETAPS 2013). LNCS, vol. 7795, pp. 185–191. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  12. 12.
    Duret-Lutz, A.: LTL translation improvements in Spot. In: VECoS, pp. 72–83 (2011)Google Scholar
  13. 13.
    Emerson, E.A., Lei, C.: Efficient model checking in fragments of the propositional \(\mu \)-calculus. In: Proceedings of LICS, pp. 267–278. IEEE Computer Society Press (1986)Google Scholar
  14. 14.
    Friedmann, O., Lange, M.: Solving parity games in practice. In: Liu, Z., Ravn, A.P. (eds.) ATVA 2009. LNCS, vol. 5799, pp. 182–196. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Gawlitza, T.M., Seidl, H.: Games through nested fixpoints. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 291–305. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Gimbert, H., Zielonka, W.: Perfect information stochastic priority games. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 850–861. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Hahn, E.M., Li, G., Schewe, S., Turrini, A., Zhang, L.: Lazy probabilistic model checking without determinisation. In: CONCUR. LIPIcs, vol. 42, pp. 354–367 (2015)Google Scholar
  18. 18.
    Hahn, E.M., Li, Y., Schewe, S., Turrini, A., Zhang, L.: iscasMc: a web-based probabilistic model checker. In: Jones, C., Pihlajasaari, P., Sun, J. (eds.) FM 2014. LNCS, vol. 8442, pp. 312–317. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  19. 19.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. FAC 6(5), 512–535 (1994)zbMATHGoogle Scholar
  20. 20.
    Hespanha, J.P., Kim, H.J., Sastry, S.: Multiple-agent probabilistic pursuit-evasion games. In: CDC, pp. 2432–2437 (1999)Google Scholar
  21. 21.
    Jurdziński, M.: Small progress measures for solving parity games. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 290–301. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  22. 22.
    Jurdziński, M., Paterson, M., Zwick, U.: A deterministic subexponential algorithm for solving parity games. SIAM J. Comput. 38(4), 1519–1532 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Katz, G., Peled, D.A.: Genetic programming and model checking: synthesizing new mutual exclusion algorithms. In: Cha, S.S., Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 33–47. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  24. 24.
    Kremer, S., Raskin, J.: A game-based verification of non-repudiation and fair exchange protocols. J. Comput. Secur. 11(3), 399–430 (2003)CrossRefGoogle Scholar
  25. 25.
    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
  26. 26.
    McIver, A., Morgan, C.: Results on the quantitative mu-calculus qMu. ACM Trans. Comput. Logic 8(1) (2007). Article No.3Google Scholar
  27. 27.
    McNaughton, R.: Infinite games played on finite graphs. Ann. Pure Appl. Logic 65(2), 149–184 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Park, T., Shin, K.G.: Lisp: a lightweight security protocol for wireless sensor networks. ACM Trans. Embed. Comput. Syst. 3, 634–660 (2004)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Perrig, A., Szewczyk, R., Wen, V., Culler, D., Tygar, J.D.: SPINS: Security protocols for sensor networks. Wirel. Netw. 189–199 (2001)Google Scholar
  30. 30.
    Pnueli, A.: The temporal logic of programs. In: Proceedings of FOCS, pp. 46–57. IEEE Computer Society Press (1977)Google Scholar
  31. 31.
    Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, New York (1994)CrossRefzbMATHGoogle Scholar
  32. 32.
    Schewe, S.: Solving parity games in big steps. In: Arvind, V., Prasad, S. (eds.) FSTTCS 2007. LNCS, vol. 4855, pp. 449–460. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  33. 33.
    Schewe, S., Varghese, T.: Tight bounds for the determinisation and complementation of generalised Büchi automata. In: Chakraborty, S., Mukund, M. (eds.) ATVA 2012. LNCS, vol. 7561, pp. 42–56. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  34. 34.
    van der Hoek, W., Wooldridge, M.: Model checking cooperation, knowledge, and time - a case study. Res. Econ. 57(3), 235–265 (2003)CrossRefGoogle Scholar
  35. 35.
    Wu, L., Su, K., Chen, Q.: Model checking temporal logics of knowledge and its application in security verification. In: Hao, Y., Liu, J., Wang, Y.-P., Cheung, Y., Yin, H., Jiao, L., Ma, J., Jiao, Y.-C. (eds.) CIS 2005. LNCS (LNAI), vol. 3801, pp. 349–354. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  36. 36.
    Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. TCS 200(1–2), 135–183 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Ernst Moritz Hahn
    • 1
  • Sven Schewe
    • 2
  • Andrea Turrini
    • 1
  • Lijun Zhang
    • 1
    Email author
  1. 1.State Key Laboratory of Computer ScienceInstitute of Software, CASBeijingChina
  2. 2.University of LiverpoolLiverpoolUK

Personalised recommendations