A Theorem on the State Reduction of Synthesized Stochastic Machines

Stochastic sequential machines produced by the insertion of noise in a deterministic machine are considered; the question of the relationship between the state reduction of the machine so formed and that of the underlying deterministic machine is raised. The theorem states that, for the case of observational noise only, if the stochastic machine has stochastically distinguishable outputs, then it is reduced iff the underlying machine is reduced. Machines with additive noise are used as an example of application.