Algorithms for game metrics Conference Paper

Author(s): Chatterjee, Krishnendu; de Alfaro, Luca; Majumdar, Ritankar S; Raman, Vishwanath
Title: Algorithms for game metrics
Title Series: LIPIcs
Abstract: Simulation and bisimulation metrics for stochastic systems provide a quantitative gen- eralization of the classical simulation and bisimulation relations. These metrics capture the similarity of states with respect to quantitative specifications written in the quantitative μ-calculus and related probabilistic logics. We present algorithms for computing the metrics on Markov decision processes (MDPs), turn- based stochastic games, and concurrent games. For turn-based games and MDPs, we provide a polynomial-time algorithm based on linear programming for the computation of the one-step metric distance between states. The algorithm improves on the previously known exponential-time algo- rithm based on a reduction to the theory of reals. We then present PSPACE algorithms for both the decision problem and the problem of approximating the metric distance between two states, matching the best known bound for Markov chains. For the bisimulation kernel of the metric, which corresponds to probabilistic bisimulation, our algorithm works in time O(n4) for both turn-based games and MDPs; improving the previously best known O(n9 · log(n)) time algorithm for MDPs. For a concurrent game G, we show that computing the exact distance between states is at least as hard as computing the value of concurrent reachability games and the square-root-sum problem in computational geometry. We show that checking whether the metric distance is bounded by a rational r, can be accomplished via a reduction to the theory of real closed fields, involving a formula with three quantifier alternations, yielding O(|G|O(|G|5)) time complexity, improving the previously known reduction with O(|G|O(|G|7)) time complexity. These algorithms can be iterated to approximate the metrics using binary search.
Keywords: simulation; Algorithms; bisimulation; Metrics; Kernel; Linear Programming; Theory of Reals
Conference Title: FSTTCS: Foundations of Software Technology and Theoretical Computer Science
Volume: 2
Conference Dates: December 9-11, 2008
Conference Location: Bangalore, India
Publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik  
Date Published: 2008-12-05
Start Page: 107
End Page: 118
Copyright Statement: This work is licensed under the Creative Commons Attribution- NonCommercial-No Derivative Works 3.0 License.
Sponsor: This research was supported in part by the NSF grants CCR-0132780 and CNS-0720884.
DOI: 10.4230/LIPIcs.FSTTCS.2008.1745
Open access: no
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