Measuring and Synthesizing Systems in Probabilistic Environments

  • Krishnendu Chatterjee
  • Thomas A. Henzinger
  • Barbara Jobstmann
  • Rohit Singh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6174)

Abstract

Often one has a preference order among the different systems that satisfy a given specification. Under a probabilistic assumption about the possible inputs, such a preference order is naturally expressed by a weighted automaton, which assigns to each word a value, such that a system is preferred if it generates a higher expected value. We solve the following optimal-synthesis problem: given an omega-regular specification, a Markov chain that describes the distribution of inputs, and a weighted automaton that measures how well a system satisfies the given specification under the given input assumption, synthesize a system that optimizes the measured value.

For safety specifications and measures that are defined by mean-payoff automata, the optimal-synthesis problem amounts to finding a strategy in a Markov decision process (MDP) that is optimal for a long-run average reward objective, which can be done in polynomial time. For general omega-regular specifications, the solution rests on a new, polynomial-time algorithm for computing optimal strategies in MDPs with mean-payoff parity objectives. We present some experimental results showing optimal systems that were automatically generated in this way.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    de Alfaro, L.: Temporal logics for the specification of performance and reliability. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 165–176. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  2. 2.
    Alur, R., Degorre, A., Maler, O., Weiss, G.: On omega-languages defined by mean-payoff conditions. In: de Alfaro, L. (ed.) FOSSACS 2009. LNCS, vol. 5504, pp. 333–347. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Baier, C., Größer, M., Leucker, M., Bollig, B., Ciesinski, F.: Controller synthesis for probabilistic systems. In: IFIP TCS, pp. 493–506 (2004)Google Scholar
  4. 4.
    Baier, C., Katoen, J.-P.: Principles of Model Checking. Representation and Mind Series. The MIT Press, Cambridge (2008)MATHGoogle Scholar
  5. 5.
    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
  6. 6.
    Bloem, R., Chatterjee, K., Henzinger, T.A., Jobstmann, B.: Better quality in synthesis through quantitative objectives. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 140–156. Springer, Heidelberg (2009)Google Scholar
  7. 7.
    Bloem, R., Greimel, K., Henzinger, T.A., Jobstmann, B.: Synthesizing robust systems. In: FMCAD’09 (2009)Google Scholar
  8. 8.
    Chakrabarti, A., Chatterjee, K., Henzinger, T.A., Kupferman, O., Majumdar, R.: Verifying quantitative properties using bound functions. In: Borrione, D., Paul, W. (eds.) CHARME 2005. LNCS, vol. 3725, pp. 50–64. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Chakrabarti, A., de Alfaro, L., Henzinger, T.A., Stoelinga, M.: Resource interfaces. In: Alur, R., Lee, I. (eds.) EMSOFT 2003. LNCS, vol. 2855, pp. 117–133. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Chatterjee, K., de Alfaro, L., Faella, M., Henzinger, T.A., Majumdar, R., Stoelinga, M.: Compositional quantitative reasoning. In: QEST, pp. 179–188 (2006)Google Scholar
  11. 11.
    Chatterjee, K., Doyen, L., Henzinger, T.A.: Quantitative languages. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 385–400. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Chatterjee, K., Henzinger, T., Jobstmann, B., Singh, R.: Measuring and synthesizing systems in probabilistic environments. In: CoRR, arXiv:1004.0739 (2010)Google Scholar
  13. 13.
    Chatterjee, K., Henzinger, T.A., Jurdzinski, M.: Mean-payoff parity games. In: LICS, pp. 178–187 (2005)Google Scholar
  14. 14.
    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
  15. 15.
    Chatterjee, K., Jurdziński, M., Henzinger, T.A.: Quantitative stochastic parity games. In: SODA’04, pp. 121–130. SIAM, Philadelphia (2004)Google Scholar
  16. 16.
    Courcoubetis, C., Yannakakis, M.: Markov decision processes and regular events. In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 336–349. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  17. 17.
    Cuninghame-Green, R.A.: Minimax algebra. Lecture Notes in Economics and Mathematical Systems, vol. 166. Springer, Heidelberg (1979)MATHGoogle Scholar
  18. 18.
    de Alfaro, L.: Formal Verification of Probabilistic Systems. PhD thesis, Stanford University (1997)Google Scholar
  19. 19.
    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
  20. 20.
    de Alfaro, L., Henzinger, T.A., Majumdar, R.: Discounting the future in systems theory. In: ICALP’03 (2003)Google Scholar
  21. 21.
    de Alfaro, L., Majumdar, R., Raman, V., Stoelinga, M.: Game relations and metrics. In: LICS, pp. 99–108. IEEE Computer Society Press, Los Alamitos (2007)Google Scholar
  22. 22.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labelled markov systems. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 258–273. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  23. 23.
    Droste, M., Gastin, P.: Weighted automata and weighted logics. Theoretical Computer Science 380, 69–86 (2007)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Droste, M., Kuich, W., Rahonis, G.: Multi-valued MSO logics over words and trees. Fundamenta Informaticae 84, 305–327 (2008)MATHMathSciNetGoogle Scholar
  25. 25.
    Droste, M., Kuich, W., Vogler, H.: Handbook of Weighted Automata. Springer Publishing Company, Incorporated, Heidelberg (2009)MATHCrossRefGoogle Scholar
  26. 26.
    Filar, J., Vrieze, K.: Competitive Markov Decision Processes. Springer, Heidelberg (1996)Google Scholar
  27. 27.
    Fortune, S., Hopcroft, J.E., Wyllie, J.: The directed subgraph homeomorphism problem. Theor. Comput. Sci., 111–121 (1980)Google Scholar
  28. 28.
    Gaubert, S.: Methods and applications of (max, +) linear algebra. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 261–282. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  29. 29.
    Glpk (gnu linear programming kit), http://www.gnu.org/software/glpk/
  30. 30.
    Haverkort, B.R.: Performance of Computer Communication Systems: A Model-Based Approach. John Wiley & Sons, Inc., New York (1998)CrossRefGoogle Scholar
  31. 31.
    Katz, G., Peled, D.: Code mutation in verification and automatic code correction. In: TACAS 2010 (to appear, 2010)Google Scholar
  32. 32.
    Kupferman, O., Lustig, Y.: Lattice automata. In: Cook, B., Podelski, A. (eds.) VMCAI 2007. LNCS, vol. 4349, pp. 199–213. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  33. 33.
    Kwiatkowska, M., Norman, G., Parker, D.: PRISM: Probabilistic model checking for performance and reliability analysis. ACM SIGMETRICS Perform. Evaluation Review (2009)Google Scholar
  34. 34.
    Niebert, P., Peled, D., Pnueli, A.: Discriminative model checking. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 504–516. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  35. 35.
    Odersky, M., Spoon, L., Venners, B.: Programming in Scala. Artima (2008), http://www.scala-lang.org/
  36. 36.
    Puterman, M.L.: Markov Decision Processes. John Wiley and Sons, Chichester (1994)MATHCrossRefGoogle Scholar
  37. 37.
    Tsay, Y.-K., Chen, Y.-F., Tsai, M.-H., Wu, K.-N., Chan, W.-C.: GOAL: A graphical tool for Büchi automata and temporal formulae. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 466–471. Springer, Heidelberg (2007), http://goal.im.ntu.edu.tw CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Krishnendu Chatterjee
    • 1
  • Thomas A. Henzinger
    • 1
    • 2
  • Barbara Jobstmann
    • 3
  • Rohit Singh
    • 4
  1. 1.ISTAustria
  2. 2.EPFLSwitzerland
  3. 3.CNRS/VerimagFrance
  4. 4.IIT BombayIndia

Personalised recommendations