Deciding the value 1 problem for probabilistic leaktight automata

TL;DR

This paper introduces leaktight automata, a new class of probabilistic automata, and proves that the value 1 problem is decidable and PSPACE-complete for them, extending to infinite words.

Contribution

It defines leaktight automata, proves the value 1 problem is decidable and PSPACE-complete for this class, and extends results to infinite words.

Findings

Value 1 problem is decidable for leaktight automata.

The problem is PSPACE-complete.

Decidability extends to infinite words for parity automata.

Abstract

The value 1 problem is a decision problem for probabilistic automata over finite words: given a probabilistic automaton, are there words accepted with probability arbitrarily close to 1? This problem was proved undecidable recently; to overcome this, several classes of probabilistic automata of different nature were proposed, for which the value 1 problem has been shown decidable. In this paper, we introduce yet another class of probabilistic automata, called leaktight automata, which strictly subsumes all classes of probabilistic automata whose value 1 problem is known to be decidable. We prove that for leaktight automata, the value 1 problem is decidable (in fact, PSPACE-complete) by constructing a saturation algorithm based on the computation of a monoid abstracting the behaviours of the automaton. We rely on algebraic techniques developed by Simon to prove that this abstraction is complete. Furthermore, we adapt this saturation algorithm to decide whether an automaton is leaktight. Finally, we show a reduction allowing to extend our decidability results from finite words to infinite ones, implying that the value 1 problem for probabilistic leaktight parity automata is decidable.