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The Complexity of Quantitative Concurrent Parity Games
Krishnendu Chatterjee, Luca de Alfaro, Tom Henzinger

Citation
Krishnendu Chatterjee, Luca de Alfaro, Tom Henzinger. "The Complexity of Quantitative Concurrent Parity Games". SODA, 2006.

Abstract
We consider two-player infinite games played on graphs. The games are concurrent, in that at each state the players choose their moves simultaneously and independently, and stochastic, in that the moves determine a probability distribution for the successor state. The value of a game is the maximal probability with which a player can guarantee the satisfaction of her objective. We show that the values of concurrent games with \omega-regular objectives expressed as parity conditions can be decided in NP and coNP. This result substantially improves the best known previous bound of 3EXPTIME. It also shows that the full class of concurrent parity games is no harder than the special case of turn-based stochastic reachability games, for which NP and coNP is the best known bound. While the previous, more restricted NP and coNP results for graph games relied on the existence of particularly simple (pure memoryless) optimal strategies, in concurrent games with parity objectives optimal strategies may not exist, and e-optimal strategies (which achieve the value of the game within a parameter e > 0) require in general both randomization and infinite memory. Hence our proof must rely on a more detailed analysis of strategies and, in addition to the main result, yields two results that are interesting on their own. First, we show that there exist e-optimal strategies that in the limit coincide with memoryless strategies; this parallels the celebrated result of Mertens-Neyman for concurrent games with limit-average objectives. Second, we complete the characterization of the memory requirements for e-optimal strategies for concurrent games with parity conditions, by showing that memoryless strategies suffice for e-optimality for coB├╝chi conditions.

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Citation formats  
  • HTML
    Krishnendu Chatterjee, Luca de Alfaro, Tom Henzinger. <a
    href="http://chess.eecs.berkeley.edu/pubs/79.html"
    >The Complexity of Quantitative Concurrent Parity
    Games</a>, SODA, 2006.
  • Plain text
    Krishnendu Chatterjee, Luca de Alfaro, Tom Henzinger.
    "The Complexity of Quantitative Concurrent Parity
    Games". SODA, 2006.
  • BibTeX
    @inproceedings{ChatterjeeAlfaroHenzinger06_ComplexityOfQuantitativeConcurrentParityGames,
        author = {Krishnendu Chatterjee and Luca de Alfaro and Tom
                  Henzinger},
        title = {The Complexity of Quantitative Concurrent Parity
                  Games},
        booktitle = {SODA},
        year = {2006},
        abstract = {We consider two-player infinite games played on
                  graphs. The games are concurrent, in that at each
                  state the players choose their moves
                  simultaneously and independently, and stochastic,
                  in that the moves determine a probability
                  distribution for the successor state. The value of
                  a game is the maximal probability with which a
                  player can guarantee the satisfaction of her
                  objective. We show that the values of concurrent
                  games with \omega-regular objectives expressed as
                  parity conditions can be decided in NP and coNP.
                  This result substantially improves the best known
                  previous bound of 3EXPTIME. It also shows that the
                  full class of concurrent parity games is no harder
                  than the special case of turn-based stochastic
                  reachability games, for which NP and coNP is the
                  best known bound. While the previous, more
                  restricted NP and coNP results for graph games
                  relied on the existence of particularly simple
                  (pure memoryless) optimal strategies, in
                  concurrent games with parity objectives optimal
                  strategies may not exist, and e-optimal strategies
                  (which achieve the value of the game within a
                  parameter e > 0) require in general both
                  randomization and infinite memory. Hence our proof
                  must rely on a more detailed analysis of
                  strategies and, in addition to the main result,
                  yields two results that are interesting on their
                  own. First, we show that there exist e-optimal
                  strategies that in the limit coincide with
                  memoryless strategies; this parallels the
                  celebrated result of Mertens-Neyman for concurrent
                  games with limit-average objectives. Second, we
                  complete the characterization of the memory
                  requirements for e-optimal strategies for
                  concurrent games with parity conditions, by
                  showing that memoryless strategies suffice for
                  e-optimality for coBüchi conditions. },
        URL = {http://chess.eecs.berkeley.edu/pubs/79.html}
    }
    

Posted by Krishnendu Chatterjee, PhD on 9 May 2006.
Groups: chess
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