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We show the surprising result that the cutpoint isolation problem isdecidable for Probabilistic Finite Automata (PFA) where input words are takenfrom a letter-bounded context-free language. A context-free language$\mathcal{L}$ is letter-bounded when $\mathcal{L} \subseteq a_1^*a_2^* \cdotsa_\ell^*$ for some finite $\ell > 0$ where each letter is distinct. A cutpointis isolated when it cannot be approached arbitrarily closely. The decidabilityof this problem is in marked contrast to the situation for the (strict)emptiness problem for PFA which is undecidable under the even more severerestrictions of PFA with polynomial ambiguity, commutative matrices and inputover a letter-bounded language as well as to the injectivity problem which isundecidable for PFA over letter-bounded languages. We provide a constructivenondeterministic algorithm to solve the cutpoint isolation problem, which holdseven when the PFA is exponentially ambiguous. We also show that the problem isat least NP-hard and use our decision procedure to solve several relatedproblems.

Decidability of cutpoint isolation for probabilistic finite automata on letter-bounded inputs