Algorithms for game metrics Journal Article

Author(s): Chatterjee, Krishnendu; de Alfaro, Luca; Majumdar, Ritankar S; Raman, Vishwanath
Article Title: Algorithms for game metrics
Affiliation IST Austria
Abstract: Simulation and bisimulation metrics for stochastic systems provide a quantitative generalization of the classical simulation and bisimulation relations. These metrics capture the similarity of states with respect to quantitative specifications written in the quantitative mu-calculus and related probabilistic logics. We first show that the metrics provide a bound for the difference in long-run average and discounted average behavior across states, indicating that the metrics can be used both in system verification, and in performance evaluation. For turn-based games and MDPs, we provide a polynomial-time algorithm for the computation of the one-step metric distance between states. The algorithm is based on linear programming; it improves on the previous known exponential-time algorithm 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 algorithms for Markov chains. For the bisimulation kernel of the metric our algorithm works in time O(n(4)) for both turn-based games and MDPs; improving the previously best known O(n(9).log(n)) time algorithm for MDPs. For a concurrent game G, we show that computing the exact distance be tween 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 done via a reduction to the theory of real closed fields, involving a formula with three quantifier alternations, yielding O(vertical bar G vertical bar(O(vertical bar G vertical bar 5))) time complexity, improving the previously known reduction, which yielded O(vertical bar G vertical bar(O(vertical bar G vertical bar 7))) time complexity. These algorithms can be iterated to approximate the metrics using binary search
Keywords: Metrics; game semantics; minimax theorem; ω-regular properties; quantitative μ-calculus; probabilistic choice; equivalence of states; refinement of states
Journal Title: Logical Methods in Computer Science
Volume: 6
Issue 3:13
ISSN: 1860-5974
Publisher: International Federation of Computational Logic  
Date Published: 2010-09-01
Start Page: 1
End Page: 27
Copyright Statement: CC-BY-ND 2.0
Sponsor: The first, second and fourth author were supported in part by the National Science Foundation grants CNS-0720884 and CCR-01 32780. The third author was supported in part by the National Science Foundation grants CCF-0427202, CCF-0546170.
DOI: 10.2168/LMCS-6(3:13)2010
Notes: A preliminary version of this paper titled “Algorithms for Game Metrics” appeared in the IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, December 2008. This is a version with full proofs and extensions.
Open access: yes (OA journal)
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