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 it is k-ambiguous for some $k \in \mathbb{N}$. 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 that are not k - 1 ambiguous; and there are finitely (respectively countably, uncountably) ambiguous languages that are not boundedly (respectively finitely, countably) ambiguous.
翻译：自动图如果对它每个输入的计算都具有一定的( 相对的、 相当的) 接受计算, 自动图如果对每个输入的计算都具有模糊性。 如果对每个输入的计算, 自动图如果对一些 $k 来说是模糊的, 则自动图如果对一些 $k\ in\ mathbb{ n} 美元来说是模糊的。 如果对每个输入的计算都具有一定的( 相对的、 相当的) 模糊性, 则自动图是模糊的。 正常语言的模糊性程度以自然的方式界定。 如果对某个输入的计算是模糊的( 相对的、 相当的、 相当的、 有限的、 相当的、 相当的 模糊性), 自动图解说是模糊性的( $\ omga $ ), 普通语言的模糊性程度以自然的方式界定。 一种不清晰的 B\\\\ 的、 约束性、 限定性的语言是无限的、 直观的树。
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