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Mean-Payoff Automaton Expressions

  • Krishnendu Chatterjee
  • Laurent Doyen
  • Herbert Edelsbrunner
  • Thomas A. Henzinger
  • Philippe Rannou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6269)

Abstract

Quantitative languages are an extension of boolean languages that assign to each word a real number. Mean-payoff automata are finite automata with numerical weights on transitions that assign to each infinite path the long-run average of the transition weights. When the mode of branching of the automaton is deterministic, nondeterministic, or alternating, the corresponding class of quantitative languages is not robust as it is not closed under the pointwise operations of max, min, sum, and numerical complement. Nondeterministic and alternating mean-payoff automata are not decidable either, as the quantitative generalization of the problems of universality and language inclusion is undecidable.

We introduce a new class of quantitative languages, defined by mean-payoff automaton expressions, which is robust and decidable: it is closed under the four pointwise operations, and we show that all decision problems are decidable for this class. Mean-payoff automaton expressions subsume deterministic mean-payoff automata, and we show that they have expressive power incomparable to nondeterministic and alternating mean-payoff automata. We also present for the first time an algorithm to compute distance between two quantitative languages, and in our case the quantitative languages are given as mean-payoff automaton expressions.

Keywords

Decision Problem Linear Constraint Expressive Power Closure Property Simple Cycle 
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.

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References

  1. 1.
    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
  2. 2.
    Bojanczyk, M.: Beyond omega-regular languages. In: Proc. of STACS. LIPIcs, vol. 3. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany (2010)Google Scholar
  3. 3.
    Chatterjee, K., Doyen, L., Edelsbrunner, H., Henzinger, T.A., Rannou, P.: Mean-payoff automaton expressions. CoRR, abs/1006.1492 (2010)Google Scholar
  4. 4.
    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
  5. 5.
    Chatterjee, K., Doyen, L., Henzinger, T.A.: Alternating weighted automata. In: Gȩbala, M. (ed.) FCT 2009. LNCS, vol. 5699, pp. 3–13. Springer, Heidelberg (2009)Google Scholar
  6. 6.
    Chatterjee, K., Doyen, L., Henzinger, T.A.: Expressiveness and closure properties for quantitative languages. In: Proc. of LICS, pp. 199–208. IEEE, Los Alamitos (2009)Google Scholar
  7. 7.
    Chatterjee, K., Ghosal, A., Henzinger, T.A., Iercan, D., Kirsch, C., Pinello, C., Sangiovanni-Vincentelli, A.: Logical reliability of interacting real-time tasks. In: Proc. of DATE, pp. 909–914. ACM, New York (2008)CrossRefGoogle Scholar
  8. 8.
    de Alfaro, L.: How to specify and verify the long-run average behavior of probabilistic systems. In: Proc. of LICS, pp. 454–465. IEEE, Los Alamitos (1998)Google Scholar
  9. 9.
    de Alfaro, L., Majumdar, R., Raman, V., Stoelinga, M.: Game relations and metrics. In: Proc. of LICS, pp. 99–108. IEEE, Los Alamitos (2007)Google Scholar
  10. 10.
    Degorre, A., Doyen, L., Gentilini, R., Raskin, J.-F., Toruńczyk, S.: Energy and mean-payoff games with imperfect information. In: Proc. of CSL. LNCS, Springer, Heidelberg (to appear, 2010) Google Scholar
  11. 11.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labeled markov systems. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 258–273. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  12. 12.
    Droste, M., Gastin, P.: Weighted automata and weighted logics. Theor. Comput. Sci. 380(1-2), 69–86 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Droste, M., Kuich, W., Vogler, H.: Handbook of Weighted Automata. Springer, Heidelberg (2009)zbMATHCrossRefGoogle Scholar
  14. 14.
    Droste, M., Kuske, D.: Skew and infinitary formal power series. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 426–438. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Ehrenfeucht, A., Mycielski, J.: Positional strategies for mean payoff games. Int. Journal of Game Theory 8(2), 109–113 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    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
  17. 17.
    Rabin, M.O.: Probabilistic automata. Information and Control 6(3), 230–245 (1963)CrossRefGoogle Scholar
  18. 18.
    Schützenberger, M.P.: On the definition of a family of automata. Information and Control 4(2-3), 245–270 (1961)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Vidal, E., Thollard, F., de la Higuera, C., Casacuberta, F., Carrasco, R.C.: Probabilistic finite-state machines-part I. IEEE Trans. Pattern Anal. Mach. Intell. 27(7), 1013–1025 (2005)CrossRefGoogle Scholar
  20. 20.
    Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theor. Comput. Sci. 158(1&2), 343–359 (1996)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Krishnendu Chatterjee
    • 1
  • Laurent Doyen
    • 2
  • Herbert Edelsbrunner
    • 1
  • Thomas A. Henzinger
    • 1
    • 3
  • Philippe Rannou
    • 2
    • 3
    • 4
  1. 1.IST Austria Institute of Science and Technology Austria 
  2. 2.LSV, ENS Cachan & CNRSFrance
  3. 3.EPFL LausanneSwitzerland
  4. 4.ENS Cachan BretagneRennesFrance

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