The most probable string: an algorithmic study

The problem of finding the consensus (most probable string) for a distribution generated by a weighted finite automaton or a probabilistic grammar is related to a number of important questions: computing the distance between two distributions or finding the best translation (the most probable one) given a probabilistic finite state transducer. The problem is undecidable with general weights and is NP-hard if the automaton is probabilistic. We give a pseudo-polynomial algorithm that solves a decision problem directly associated with the consensus string and answers if there is a (reasonably short) string whose probability is larger than a given bound in time polynomial in the the size of this bound, both for probabilistic finite automata and probabilistic context-free grammars. We also study a randomized algorithm solving the same problem. Finally, we report links between the length of the consensus string and the probability of this string.