Deciding the Value 1 Problem of Probabilistic Leaktight Automata

Nathanaël Fijalkow (1, 2) , Hugo Gimbert (3) , Youssouf Oualhadj (3)

1 ENS Cachan - École normale supérieure - Cachan
2 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
3 LaBRI - Laboratoire Bordelais de Recherche en Informatique

The value 1 problem is a decision problem for probabilistic automata over finite words: given a probabilistic automaton A, are there words accepted by A with probability arbitrarily close to 1? This problem was proved undecidable recently. We sharpen this result, showing that the undecidability result holds even if the probabilistic automata have only one probabilistic transition. Our main contribution is to introduce a new class of probabilistic automata, called leaktight automata, for which the value 1 problem is shown decidable (and PSPACE-complete). We construct an algorithm based on the computation of a monoid abstracting the behaviours of the automaton, and rely on algebraic techniques developed by Simon for the correctness proof. The class of leaktight automata is decidable in PSPACE, subsumes all subclasses of probabilistic automata whose value 1 problem is known to be decidable (in particular deterministic automata), and is closed under two natural composition operators.

Keywords: Probabilistic automata; Value 1 problem; Decidability

Domains: Computer Science and Game Theory [cs.GT]; Formal Languages and Automata Theory [cs.FL]

Nathanaël Fijalkow, Hugo Gimbert, Youssouf Oualhadj. Deciding the Value 1 Problem of Probabilistic Leaktight Automata. Logic in Computer Science (LICS), 2012 27th Annual IEEE Symposium on, Jun 2012, Croatia. pp.295-304, ⟨10.1109/LICS.2012.40⟩. ⟨hal-00585835v5⟩