Probabilistic Automata with Monitored Final State Sets

Probabilistic automata with time-variant final state sets monitored, in an intuitive manner, by autonomous probabilistic automata are considered. It is shown that they include a scheme suggested by Turakainen as a particular case, and that they accept all the stochastic languages. If the monitors are started in stationary distributions, it is proved that such schemes will accept all and only the stochastic languages; other cases in which the languages can be shown to be stochastic follow, including the case in which the state transition matrix of the monitor is deterministic.