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Qualitative concurrent parity games

Published: 22 July 2011 Publication History

Abstract

We consider two-player games played on a finite state space for an infinite number of rounds. The games are concurrent: in each round, the two players (player 1 and player 2) choose their moves independently and simultaneously; the current state and the two moves determine the successor state. We consider ω-regular winning conditions specified as parity objectives. Both players are allowed to use randomization when choosing their moves. We study the computation of the limit-winning set of states, consisting of the states where the sup-inf value of the game for player 1 is 1: in other words, a state is limit-winning if player 1 can ensure a probability of winning arbitrarily close to 1. We show that the limit-winning set can be computed in O(n2d+2) time, where n is the size of the game structure and 2d is the number of priorities (or colors). The membership problem of whether a state belongs to the limit-winning set can be decided in NP ∩ coNP. While this complexity is the same as for the simpler class of turn-based parity games, where in each state only one of the two players has a choice of moves, our algorithms are considerably more involved than those for turn-based games. This is because concurrent games do not satisfy two of the most fundamental properties of turn-based parity games. First, in concurrent games limit-winning strategies require randomization; and second, they require infinite memory.

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      cover image ACM Transactions on Computational Logic
      ACM Transactions on Computational Logic  Volume 12, Issue 4
      July 2011
      295 pages
      ISSN:1529-3785
      EISSN:1557-945X
      DOI:10.1145/1970398
      Issue’s Table of Contents
      Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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      Publication History

      Published: 22 July 2011
      Accepted: 01 October 2010
      Revised: 01 December 2009
      Received: 01 April 2008
      Published in TOCL Volume 12, Issue 4

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      Author Tags

      1. ω-regular objectives
      2. Concurrent games
      3. games on graphs
      4. parity objectives
      5. temporal logics

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