Javier Esparza, Thomas Gawlitza, Stefan Kiefer and Helmut Seidl. Approximative Methods for Monotone Systems of Min-Max-Polynomial Equations. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir and Igor Walukiewicz, editors, Automata, Languages and Programming(1)Tack A: Algorithms, Automata, Complexity,and Games, volume 5125 of Lecture Notes in Computer Science, pages 698-710, Reykjavik, Iceland, July 2008. Springer.

A monotone system of min-max-polynomial equations (min-max-MSPE) over the variables $\mathnormal {X_1},...,{X_n}$ has for every $i$ exactly one equation of the form $\mathnormal {X_i = f_i(X_1,...,X_n)}$ where each $\mathnormal {f_i(X_1,...,X_n)}$ 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. This work was in part supported by the DFG project Algorithms for Software Model Checking.

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