Probabilistic Automata on Finite words: Decidable and Undecidable Problems

Hugo Gimbert (1) , Youssouf Oualhadj (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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.

Mots clés: probabilistic automata; value 1 problem

Domaines: Informatique et théorie des jeux [cs.GT]

Hugo Gimbert, Youssouf Oualhadj. Probabilistic Automata on Finite words: Decidable and Undecidable Problems. 2010. ⟨hal-00456538v1⟩