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 paper 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 B\"uchi 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-B\"uchi 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

我们研究了在无限二叉树上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造了一个称为运行的输入树的装饰，并且定性语义说如果运行的几乎所有分支都在，则自动机的运行正在接受接受。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。积极的结果是在 B\"uchi 接受条件的情况下空问题的可判定性。我们方法的一个有趣的方面是我们没有扩展解决交替自动机空问题的经典解决方案，它首先构造了一个等价的非确定性自动机。相反，我们利用不完美信息直接构建了一个空性博弈。