Borel Determinacy

Martin, Donald A.

doi:10.2307/1971035

Cited by 684 publications

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“…randomized almost-winning) observation-based strategy in G ℓ for the objective φ. Theorem 2.2 (Determinacy). [16] For all perfect-information game structures G and all Borel objectives φ, either there exists a deterministic sure-winning strategy for Player 1 for the objective φ, or there exists a deterministic sure-winning strategy for Player 2 for the complementary objective Plays(G) \ φ.…”

Section: Alg For ω-Regular Gamesmentioning

confidence: 99%

Algorithms for Omega-Regular Games with Imperfect Information

Chatterjee¹,

Doyen²,

Henzinger³

et al. 2007

Logical Methods in Computer Science

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Abstract. We study observation-based strategies for two-player turn-based games on graphs with omega-regular objectives. An observation-based strategy relies on imperfect information about the history of a play, namely, on the past sequence of observations. Such games occur in the synthesis of a controller that does not see the private state of the plant. Our main results are twofold. First, we give a fixed-point algorithm for computing the set of states from which a player can win with a deterministic observation-based strategy for any omega-regular objective. The fixed point is computed in the lattice of antichains of state sets. This algorithm has the advantages of being directed by the objective and of avoiding an explicit subset construction on the game graph. Second, we give an algorithm for computing the set of states from which a player can win with probability 1 with a randomized observation-based strategy for a Büchi objective. This set is of interest because in the absence of perfect information, randomized strategies are more powerful than deterministic ones. We show that our algorithms are optimal by proving matching lower bounds.

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Section: Alg For ω-Regular Gamesmentioning

confidence: 99%

Algorithms for Omega-Regular Games with Imperfect Information

Chatterjee¹,

Doyen²,

Henzinger³

et al. 2007

Logical Methods in Computer Science

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“…for games with plays of infinite length. A research line that began in the 1950s with the work of Gale and Stewart [5] on open games culminated with the work of Martin [6] who showed that all two-player zero-sum sequential games with perfect information in which the plays (winning sets) of the game form a Borel set are determined.…”

Section: Introductionmentioning

confidence: 99%

Borel Determinacy of Concurrent Games

Gutiérrez

Winskel

2013

CONCUR 2013 – Concurrency Theory

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Abstract-Just as traditional games are represented by trees, so distributed/concurrent games are represented by event structures. We show the determinacy of such concurrent games with Borel sets of configurations as winning conditions, provided the games are race-free and bounded-concurrent. Both restrictions are shown necessary. The determinacy proof proceeds via a reduction to the determinacy of tree games, and the determinacy of these in turn reduces to the determinacy of Gale-Stewart games.

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“…The determinacy of parity games follows from a far more general game-theoretic result concerning Borel games, due to Martin [17]. The fact that winning strategies in parity games can always be taken to be history free, was independently proved in Mostowski [20] and Emerson & Jutla [8].…”

mentioning

confidence: 99%

Coalgebraic Automata Theory: Basic Results

Kupke¹,

Venema²

2008

Logical Methods in Computer Science

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Abstract. We generalize some of the central results in automata theory to the abstraction level of coalgebras and thus lay out the foundations of a universal theory of automata operating on infinite objects.Let F be any set functor that preserves weak pullbacks. We show that the class of recognizable languages of F-coalgebras is closed under taking unions, intersections, and projections. We also prove that if a nondeterministic F-automaton accepts some coalgebra it accepts a finite one of the size of the automaton. Our main technical result concerns an explicit construction which transforms a given alternating F-automaton into an equivalent nondeterministic one, whose size is exponentially bound by the size of the original automaton.

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Borel Determinacy

Cited by 684 publications

References 0 publications

Algorithms for Omega-Regular Games with Imperfect Information

Algorithms for Omega-Regular Games with Imperfect Information

Borel Determinacy of Concurrent Games

Coalgebraic Automata Theory: Basic Results

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