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We consider the value 1 problem for probabilistic automata over finite words:it asks whether a given probabilistic automaton accepts words with probabilityarbitrarily close to 1. This problem is known to be undecidable. However,different algorithms have been proposed to partially solve it; it has beenrecently shown that the Markov Monoid algorithm, based on algebra, is the mostcorrect algorithm so far. The first contribution of this paper is to give acharacterisation of the Markov Monoid algorithm. The second contribution is todevelop a profinite theory for probabilistic automata, called the prostochastictheory. This new framework gives a topological account of the value 1 problem,which in this context is cast as an emptiness problem. The abovecharacterisation is reformulated using the prostochastic theory, allowing us togive a simple and modular proof.

Profinite Techniques for Probabilistic Automata and the Markov Monoid Algorithm