In the context of two-player games over graphs, a language $L$ is called positional if, in all games using $L$ as winning objective, the protagonist can play optimally using positional strategies, that is, strategies that do not depend on the history of the play. In this work, we describe the class of parity automata recognising positional languages, providing a complete characterisation of positionality for $ω$-regular languages. As corollaries, we establish decidability of positionality in polynomial time, finite-to-infinite and 1-to-2-players lifts, and show the closure under union of prefix-independent positional objectives, answering a conjecture by Kopczyński in the $ω$-regular case.

翻译：在图上的双人博弈语境中，若所有以语言$L$作为获胜目标的博弈中，主角均可采用位置策略（即不依赖于对局历史的策略）实现最优博弈，则称该语言$L$具有位置性。本文通过描述可识别位置性语言的奇偶自动机类别，完整刻画了ω-正则语言的位置性特征。作为推论，我们证明了位置性判定问题可在多项式时间内求解，建立了有限到无限及单人到双人博弈的推广关系，并证明了前缀无关位置性目标在并运算下的封闭性，从而在ω-正则情形下证实了Kopczyński猜想。