Advertisement

Approximative Methods for Monotone Systems of Min-Max-Polynomial Equations

  • Javier Esparza
  • Thomas Gawlitza
  • Stefan Kiefer
  • Helmut Seidl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)

Abstract

A monotone system of min-max-polynomial equations (min-max-MSPE) over the variables X 1,...,X n has for every i exactly one equation of the form X i  = f i (X 1,...,X n ) where each f i (X 1,...,X n ) is an expression built up from polynomials with non-negative coefficients, minimum- and maximum-operators. The question of computing least solutions of min-max-MSPEs arises naturally in the analysis of recursive stochastic games [5,6,14]. Min-max-MSPEs generalize MSPEs for which convergence speed results of Newton’s method are established in [11,3]. We present the first methods for approximatively computing least solutions of min-max-MSPEs which converge at least linearly. Whereas the first one converges faster, a single step of the second method is cheaper. Furthermore, we compute ε-optimal positional strategies for the player who wants to maximize the outcome in a recursive stochastic game.

Keywords

Approximative Method Markov Decision Process Strategy Iteration Stochastic Game Linear Convergence 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Condon, A.: The complexity of stochastic games. Inf. and Comp. 96(2), 203–224 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Esparza, J., Gawlitza, T., Kiefer, S., Seidl, H.: Approximative methods for monotone systems of min-max-polynomial equations. Technical report, Technische Universität München, Institut für Informatik (February 2008)Google Scholar
  3. 3.
    Esparza, J., Kiefer, S., Luttenberger, M.: Convergence thresholds of Newton’s method for monotone polynomial equations. In: Proceedings of STACS, pp. 289–300 (2008)Google Scholar
  4. 4.
    Etessami, K., Yannakakis, M.: Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 340–352. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Etessami, K., Yannakakis, M.: Recursive Markov decision processes and recursive stochastic games. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580. Springer, Heidelberg (2005)Google Scholar
  6. 6.
    Etessami, K., Yannakakis, M.: Efficient qualitative analysis of classes of recursive Markov decision processes and simple stochastic games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 634–645. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Filar, J., Vrieze, K.: Competitive Markov Decision processes. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  8. 8.
    Gawlitza, T., Seidl, H.: Precise fixpoint computation through strategy iteration. In: De Nicola, R. (ed.) ESOP 2007. LNCS, vol. 4421, pp. 300–315. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Gawlitza, T., Seidl, H.: Precise relational invariants through strategy iteration. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 23–40. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Harris, T.E.: The Theory of Branching Processes. Springer, Heidelberg (1963)CrossRefzbMATHGoogle Scholar
  11. 11.
    Kiefer, S., Luttenberger, M., Esparza, J.: On the convergence of Newton’s method for monotone systems of polynomial equations. In: STOC, pp. 217–226. ACM, New York (2007)Google Scholar
  12. 12.
    Neyman, A., Sorin, S.: Stochastic Games and Applications. Kluwer Academic Press, Dordrecht (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Ortega, J., Rheinboldt, W.: Iterative solution of nonlinear equations in several variables. Academic Press, London (1970)zbMATHGoogle Scholar
  14. 14.
    Wojtczak, D., Etessami, K.: PReMo: An analyzer for probabilistic recursive models. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 66–71. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Javier Esparza
    • 1
  • Thomas Gawlitza
    • 1
  • Stefan Kiefer
    • 1
  • Helmut Seidl
    • 1
  1. 1.Institut für InformatikTechnische Universität MünchenGermany

Personalised recommendations