Trees, automata, and games

Gurevich, Yuri; Harrington, Leo

doi:10.1145/800070.802177

Cited by 221 publications

(139 citation statements)

References 9 publications

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“…The conditions that we provide look also quite "natural", the examples of payoff mappings that are interesting in the context of the perfect information deterministic games over finite graphs but which do not satisfy our conditions are rare: the payoff mapping mentioned above is rather artificial, another prominent example are the games with Muller condition [7] but these games need memory.…”

Section: Introductionmentioning

confidence: 99%

When Can You Play Positionally?

Gimbert

Zielonka

2004

Lecture Notes in Computer Science

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Section: Introductionmentioning

confidence: 99%

When Can You Play Positionally?

Gimbert

Zielonka

2004

Lecture Notes in Computer Science

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“…[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. An important special case of Muller games are parity games where the least (or greatest) priority occurring infinitely often determines the winner.…”

Section: 2mentioning

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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“…Theorem 4. [15,7] Given an LTL specification ϕ, we can construct a deterministic word automaton D ϕ that accepts exactly the models of ϕ. The number of states of D ϕ is doubly exponential in the length of ϕ.…”

Section: Temporal Logicsmentioning

confidence: 99%

Synthesis for Probabilistic Environments

Schewe

2006

Automated Technology for Verification and Analysis

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Abstract. In synthesis we construct finite state systems from temporal specifications. While this problem is well understood in the classical setting of non-probabilistic synthesis, this paper suggests the novel approach of open synthesis under the assumptions of an environment that chooses its actions randomized rather than nondeterministically. Assuming a randomized environment inspires alternative semantics both for linear-time and branching-time logics. For linear-time, natural acceptance criteria are almost-sure and observable acceptance, where it suffices if the probability measure of accepting paths is 1 and greater than 0, respectively. We distinguish 0-environments, which can freely assign probabilities to each environment action, from ε-environments, where the probabilities assigned by the environment are bound from below by some ε > 0. While the results in case of 0-environments are essentially the same as for nondeterministic environments, the languages occurring in case of ε-environments are topologically different from the results for nondeterministic and 0-environments (in case of LTL, recognizable by weak alternating automata vs. recognizable by deterministic automata). The complexity of open synthesis is, in both cases, EXPTIME and 2EXPTIME-complete for CTL and LTL specifications, respectively.

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Trees, automata, and games

Cited by 221 publications

References 9 publications

When Can You Play Positionally?

When Can You Play Positionally?

Positional Determinacy of Games with Infinitely Many Priorities

Synthesis for Probabilistic Environments

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