  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.
