Positional Determinacy of Games with Infinitely Many Priorities

Abstract

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.

The Descriptive Complexity of Parity Games

“…The result is extended to infinite game graphs with finitely many priorities in [25]. This has been further extended in [14] to parity games with rng(Ω) = ω.…”

Back and Forth Between Logic and Games

“…This is generalised by Zielonka [1998] to infinite game graphs with a finite number of priorities. Finally positional determinacy has been extended by Grädel and Walukiewicz [2006] to parity games with rng(Ω) = ω.…”

Transfinite Extension of the Mu-Calculus

“…When looking at a formula in the transfinite mu-calculus, we need to play a parity game with infinitely many priorities: for each set variable we need a distinct priority. If we take the binary tree and attach to each node a priority in an arbitrary fashion, then, when playing a parity game on this tree, we might end up having a "wild" payoff set for player I, and we might also lose the nice property of having a memoryless winning strategy [6]. Furthermore, it might be that there is no maximum among the priorities seen infinitely often, and infinite runs might even meet each priority only finitely many times.…”