Equivalence of History and Generator Epsilon-Machines

Epsilon-machines are minimal, unifilar representations of stationary stochastic processes. They were originally defined in the history machine sense---as machines whose states are the equivalence classes of infinite histories with the same probability distribution over futures. In analyzing synchronization, though, an alternative generator definition was given: unifilar edge-label hidden Markov models with probabilistically distinct states. The key difference is that history epsilon-machines are defined by a process, whereas generator epsilon-machines define a process. We show here that these two definitions are equivalent.