We show the surprising result that the cutpoint isolation problem is decidable for Probabilistic Finite Automata (PFA) where input words are taken from a letter-bounded context-free language. A context-free language $\mathcal{L}$ is letter-bounded when $\mathcal{L} \subseteq a_1^*a_2^* \cdots a_\ell^*$ for some finite $\ell > 0$ where each letter is distinct. A cutpoint is isolated when it cannot be approached arbitrarily closely. The decidability of this problem is in marked contrast to the situation for the (strict) emptiness problem for PFA which is undecidable under the even more severe restrictions of PFA with polynomial ambiguity, commutative matrices and input over a letter-bounded language as well as to the injectivity problem which is undecidable for PFA over letter-bounded languages. We provide a constructive nondeterministic algorithm to solve the cutpoint isolation problem, which holds even when the PFA is exponentially ambiguous. We also show that the problem is at least NP-hard and use our decision procedure to solve several related problems.

中文翻译：

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字母有界输入上概率有限自动机的切点隔离的可判定性

我们展示了一个令人惊讶的结果，即对于概率有限自动机 (PFA) 来说，分割点隔离问题是可判定的，其中输入词取自字母有界上下文无关语言。上下文无关语言 $\mathcal{L}$ 当 $\mathcal{L} \subseteq a_1^*a_2^* \cdots a_\ell^*$ 对于某些有限的 $\ell > 0$ 时是字母有界的，其中每个字母是不同的。当不能任意接近时，分割点是孤立的。该问题的可判定性与 PFA 的（严格）空性问题的情况形成鲜明对比，后者在具有多项式歧义、可交换矩阵和字母有界语言输入的 PFA 的更严格限制下不可判定对于字母有界语言的 PFA 无法确定的注入性问题。我们提供了一种建设性的非确定性算法来解决分割点隔离问题，即使 PFA 呈指数模糊，该算法也成立。我们还表明该问题至少是 NP 难的，并使用我们的决策程序来解决几个相关问题。