| This paper tackles three algorithmic problems for probabilis- tic automata on finite words: the Emptiness Problem, the Isolation Prob- lem and the Value 1 Problem. The Emptiness Problem asks, given some probability 0 ≤ λ ≤ 1, whether there exists a word accepted with prob- ability greater than λ, and the Isolation Problem asks whether there exist words whose acceptance probability is arbitrarily close to λ. Both these problems are known to be undecidable [8, 2, 3]. About the Empti- ness problem, we provide a new simple undecidability proof and prove that it is decidable for automata with one probabilistic transition and undecidable for automata with as few as two probabilistic transitions. The Value 1 Problem is the special case of the Isolation Problem when λ = 1 or λ = 0. The decidability of the Value 1 Problem was an open question. We show that the Value 1 Problem is undecidable. Moreover, we introduce a new class of probabilistic automata, ♯-acyclic automata, for which the Value 1 Problem is decidable. |
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