An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is k-ambiguous (for k>0) if for every input it has at most k accepting computations. An automaton is boundedly ambiguous if there is k, such that for every input it has at most k accepting computations. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. The degree of ambiguity of a regular language is defined in a natural way. A language is k-ambiguous (respectively, boundedly, finitely, countably ambiguous) if it is accepted by a k-ambiguous (respectively, boundedly, finitely, countably ambiguous) automaton. Over finite words, every regular language is accepted by a deterministic automaton. Over finite trees, every regular language is accepted by an unambiguous automaton. Over $\omega$-words every regular language is accepted by an unambiguous B\"uchi automaton and by a deterministic parity automaton. Over infinite trees, Carayol et al. showed that there are ambiguous languages. We show that over infinite trees there is a hierarchy of degrees of ambiguity: For every k>1 there are k-ambiguous languages which are not k-1 ambiguous; and there are finitely (respectively countably, uncountably) ambiguous languages which are not boundedly (respectively finitely, countably) ambiguous.

中文翻译：

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正则无限树语言的歧义层次

如果自动机对于每个输入最多有一个接受计算，则它是明确的。如果自动机对于每个输入最多有 k 个接受计算，则它是 k-歧义的（对于 k>0）。如果存在 k，则自动机是有界模糊的，因此对于每个输入，它最多有 k 个接受计算。如果自动机对于每个输入最多有有限（分别，可数）许多接受计算，则自动机是有限（分别，可数）模糊的。正则语言的歧义程度是以自然的方式定义的。如果语言被 k-歧义（分别、有界、有限、可数歧义）自动机接受，则该语言是 k-歧义（分别、有界、有限、可数歧义）。在有限的词上，每个常规语言都被确定性自动机接受。在有限树上，每一种常规语言都被一个明确的自动机接受。在 $\omega$-words 上，每个常规语言都被一个明确的 B\"uchi 自动机和一个确定性奇偶自动机接受。在无限树上，Carayol 等人证明了有歧义的语言。我们证明在无限树上有歧义度等级：对于每 k>1 有 k-歧义语言不是 k-1 歧义的；并且有有限（分别是可数、不可数）歧义语言不是有界（分别是有限、可数）歧义。我们证明了在无限树上有一个歧义度等级：对于每 k>1 有 k-歧义语言不是 k-1 歧义的；并且存在有限（分别是可数、不可数）歧义的语言，它们不是有界（分别是有限、可数）歧义的。我们证明了在无限树上有一个歧义度等级：对于每 k>1 有 k-歧义语言不是 k-1 歧义的；并且存在有限（分别是可数、不可数）歧义的语言，它们不是有界（分别是有限、可数）歧义的。