Decidability of cutpoint isolation for letter-monotonic probabilistic finite automata
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We show the surprising result that the cutpoint isolation problem is decidable for probabilistic finite automata where input words are taken from a letter-monotonic context-free language. A context-free language L is letter-monotonic when L ⊆ a_1^*a_2^* ... a_ℓ^* for some finite ℓ > 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-monotonic language as well as the injectivity problem which is undecidable for PFA over letter-monotonic languages. We provide a constructive nondeterministic algorithm to solve the cutpoint isolation problem, even for exponentially ambiguous PFA, and we also show that the problem is at least NP-hard.
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