Krishnendu Chatterjee ; Luca de Alfaro ; Rupak Majumdar ; Vishwanath Raman - Algorithms for Game Metrics

lmcs:783 - Logical Methods in Computer Science, September 1, 2010, Volume 6, Issue 3 - https://doi.org/10.2168/LMCS-6(3:13)2010
Algorithms for Game MetricsArticle

Authors: Krishnendu Chatterjee ; Luca de Alfaro ; Rupak Majumdar ; Vishwanath Raman

    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\cdot 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 done 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, which yielded O(|G|^O(|G|^7)) time complexity. These algorithms can be iterated to approximate the metrics using binary search.


    Volume: Volume 6, Issue 3
    Published on: September 1, 2010
    Imported on: October 20, 2009
    Keywords: Computer Science - Computer Science and Game Theory,F.4.1, F.1.1
    Funding:
      Source : OpenAIRE Graph
    • CAREER: Structured Design of Embedded Software; Funder: National Science Foundation; Code: 0132780
    • CAREER: Modular Verification of Software; Funder: National Science Foundation; Code: 0546170
    • CSR---EHS: Collaborative: Directed Real-Time Testing; Funder: National Science Foundation; Code: 0720884
    • ITR - ASE - int: Event Driven Software Quality; Funder: National Science Foundation; Code: 0427202

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