State-calculable stochastic sequential machines, equivalences, and events

The observer/state-calculable stochastic machine is such that the present state, input, and output determine the next state. It is shown that since, for such machines, an initial state leads to degenerate terminal distributions, and reduction by merging equivalent states leads to unique reduced forms, several interrelated difficulties in stochastic machine theory are suppressed when attention is restricted to these structures. In particular, their input-output relations possess finitely many generalized states in an appropriate sense, and system realizations can be obtained from initial segments of such i-o relations. However, the need for considering input-output events in the stochastic case (rather than input-events only) is indicated even for observer/state-calculable structures.