We study alternating automata with qualitative semantics over infinite binary trees: Alternation means that two opposing players construct a decoration of the input tree called a run, and the qualitative semantics says that a run of the automaton is accepting if almost all branches of the run are accepting. In this article, we prove a positive and a negative result for the emptiness problem of alternating automata with qualitative semantics. The positive result is the decidability of the emptiness problem for the case of Büchi acceptance condition. An interesting aspect of our approach is that we do not extend the classical solution for solving the emptiness problem of alternating automata, which first constructs an equivalent non-deterministic automaton. Instead, we directly construct an emptiness game making use of imperfect information. The negative result is the undecidability of the emptiness problem for the case of co-Büchi acceptance condition. This result has two direct consequences: the undecidability of monadic second-order logic extended with the qualitative path-measure quantifier and the undecidability of the emptiness problem for alternating tree automata with non-zero semantics, a recently introduced probabilistic model of alternating tree automata.

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具有定性语义的交替树自动机

我们在无限二叉树上研究具有定性语义的交替自动机：交替意味着两个对立的玩家构建输入树的装饰，称为运行，而定性语义表示，如果运行的几乎所有分支都是自动机的运行，则该运行正在接受接受。在本文中，我们证明了具有定性语义的交替自动机的空性问题的正面和负面结果。积极的结果是对于 Büchi 接受条件的空洞问题的可判定性。我们方法的一个有趣方面是我们没有扩展解决交替自动机空性问题的经典解决方案，它首先构造了一个等效的非确定性自动机。相反，我们直接利用不完全信息构建了一个空虚游戏。否定的结果是在 co-Büchi 接受条件的情况下空性问题的不可判定性。这个结果有两个直接的后果：用定性路径测量量词扩展的一元二阶逻辑的不可判定性和具有非零语义的交替树自动机的空性问题的不可判定性，这是最近引入的交替树自动机的概率模型。