Equivalence of History and Generator e-Machines

e-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 e-machines are defined by a process, whereas generator e-machines define a process. We show here that these two definitions are equivalent.