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.
翻译:我们研究在无限的二进制树上交替自动图与定性语义交替:交替意味着两个对立的玩家建了一个称为运行的输入树的装饰,而定性语义则表示,如果几乎所有运行的分支都接受,自动图的运行就会被接受。在本文中,我们证明自动图与定性语义交替的空洞性问题产生了积极和消极的结果。积极的结果是B\"uchi"接受条件情况下的空洞性问题的下降。我们的方法的一个有趣的方面是,我们不扩展解决交替自动图的纯洁性问题的典型解决方案,而后者首先构建了相等的非非非定性自动图案。相反,我们直接构建了一个空洞性游戏,利用了不完善的信息。负面的结果是,对于 com- B\\" 无效性接受条件的情况来说,自容性问题是不可减损的。结果有两个直接后果:单调二进制逻辑的不衰败性与定性路径定性模型相延延延延缩,以及不稳性树的交替性树本性。