In the context of two-player games over graphs, a language $L$ is called half-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 half-positional languages, providing a complete characterisation of half-positionality for $\omega$-regular languages. As corollaries, we establish decidability of half-positionality in polynomial time, finite-to-infinite and 1-to-2-players lifts, and show the closure under union of prefix-independent half-positional objectives, answering a conjecture by Kopczy\'nski.

翻译：在图上的双人博弈语境中，若在所有以语言L为获胜目标的博弈中，主角可使用位置策略（即不依赖于博弈历史的策略）达到最优选择，则称语言L为半位置性的。本文描述了识别半位置性语言的奇偶自动机类别，从而完整刻画了ω-正则语言的半位置性。作为推论，我们建立了多项式时间内半位置性的可判定性、有限到无限以及1到2玩家策略的升维性质，并证明前缀无关半位置性目标在并运算下保持封闭，从而回答了Kopczyński提出的一个猜想。