How much memory is needed to win infinite games?

Dziembowski, Stefan; Jurdziński, Marcin; Walukiewicz, Igor

doi:10.1109/lics.1997.614939

Cited by 87 publications

(140 citation statements)

References 9 publications

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“…Still we can hope to have one uniform memory for all the games of fixed size. There is a similar situation in the case of games with Muller conditions [18,20,5]. There, the size of memory also cannot in general be bounded, but there is a finite memory sufficient for all games with conditions over some fixed set of elements.…”

Section: Finding Permissive Strategiesmentioning

confidence: 99%

Permissive strategies: from parity games to safety games

Bernet

Janin

Walukiewicz

2002

RAIRO-Theor. Inf. Appl.

102

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It is proposed to compare strategies in a parity game by comparing the sets of behaviours they allow. For such a game, there may be no winning strategy that encompasses all the behaviours of all winning strategies. It is shown, however, that there always exists a permissive strategy that encompasses all the behaviours of all memoryless strategies. An algorithm for finding such a permissive strategy is presented. Its complexity matches currently known upper bounds for the simpler problem of finding the set of winning positions in a parity game. The algorithm can be seen as a reduction of a parity to a safety game and computation of the set of winning positions in the resulting game.

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Section: Finding Permissive Strategiesmentioning

confidence: 99%

Permissive strategies: from parity games to safety games

Bernet

Janin

Walukiewicz

2002

RAIRO-Theor. Inf. Appl.

102

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“…This is not the case for Streett. In order to solve Streett games we require exponential memory [DJW97,Hor05]. Applications like nondeterminization of alternating tree automata use co-determinization but require the result to be a Rabin or parity automaton.…”

Section: Determinization Of Büchi and Streett Automatamentioning

confidence: 99%

From Nondeterministic B\"uchi and Streett Automata to Deterministic Parity Automata

Piterman¹

2007

Logical Methods in Computer Science

174

232

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Abstract. In this paper we revisit Safra's determinization constructions for automata on infinite words. We show how to construct deterministic automata with fewer states and, most importantly, parity acceptance conditions. Determinization is used in numerous applications, such as reasoning about tree automata, satisfiability of CTL * , and realizability and synthesis of logical specifications. The upper bounds for all these applications are reduced by using the smaller deterministic automata produced by our construction. In addition, the parity acceptance conditions allows to use more efficient algorithms (when compared to handling Rabin or Streett acceptance conditions).

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“…In the context of finite-memory determinacy or positional determinacy of infinite games it is usually assumed that the range of the priority function is finite, and the winning condition is defined by a formula on infinite paths (from S1S or LTL, say) referring to the predicates (P c ) c∈C , or by an automata-theoretic condition like a Muller, Rabin, Streett, or parity (Mostowski) condition (see e.g. [15,9,29]). In Muller games the winner of a play depends only on the set of priorities that have been seen infinitely often; it has been proved by Gurevich and Harrington [16] that Muller games are determined and that the winner has a finite-memory winning strategy.…”

Section: 2mentioning

confidence: 99%

“…The Zielonka tree Z(F 0 , F 1 ) exists, if for every maximal Y ∈ F 1−σ the Zielonka tree Z(F 0 ∩ P(Y ), F 1 ∩ P(Y )) exists and every set in F 1−σ is a subset of some maximal set in F 1−σ . In that case Z(F 0 , F 1 ) consists of a root, labeled by (C, σ), to which we attach as subtrees the Zielonka trees Z(F 0 ∩ P(Y ), For Muller conditions over a finite set C, the Zielonka tree always exists, and it is a fundamental tool for analysing the memory that is required for solving Muller games [9]. For infinite sets C, the Zielonka tree need not exist, since there is no guarantee, that for X ∈ F σ , the set P(X) ∪ F 1−σ contains maximal elements.…”

Section: Remarkmentioning

confidence: 99%

Positional Determinacy of Games with Infinitely Many Priorities

Grädel¹,

Walukiewicz²

2006

Logical Methods in Computer Science

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Abstract. We study two-player games of infinite duration that are played on finite or infinite game graphs. A winning strategy for such a game is positional if it only depends on the current position, and not on the history of the play. A game is positionally determined if, from each position, one of the two players has a positional winning strategy.The theory of such games is well studied for winning conditions that are defined in terms of a mapping that assigns to each position a priority from a finite set C. Specifically, in Muller games the winner of a play is determined by the set of those priorities that have been seen infinitely often; an important special case are parity games where the least (or greatest) priority occurring infinitely often determines the winner. It is well-known that parity games are positionally determined whereas Muller games are determined via finite-memory strategies.In this paper, we extend this theory to the case of games with infinitely many priorities. Such games arise in several application areas, for instance in pushdown games with winning conditions depending on stack contents.For parity games there are several generalisations to the case of infinitely many priorities. While max-parity games over ω or min-parity games over larger ordinals than ω require strategies with infinite memory, we can prove that min-parity games with priorities in ω are positionally determined. Indeed, it turns out that the min-parity condition over ω is the only infinitary Muller condition that guarantees positional determinacy on all game graphs. MotivationThe problem of computing winning positions and winning strategies in infinite games has numerous applications in computing, most notably for the synthesis and verification of reactive controllers and for the model-checking of the µ-calculus and other logics. Of special importance are parity games, due to several reasons.

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How much memory is needed to win infinite games?

Cited by 87 publications

References 9 publications

Permissive strategies: from parity games to safety games

Permissive strategies: from parity games to safety games

From Nondeterministic B\"uchi and Streett Automata to Deterministic Parity Automata

Positional Determinacy of Games with Infinitely Many Priorities

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