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he study of epsilon-machines occurs at the intersection of many fields: physics and dynamical systems; computer science and automata; and information theory and modeling. We provide novel characterizations of stationary stochastic processes by making use of minimal representations. Some recent work has provided interesting insights into the nature of stochastic processes in terms of "memory", "irreversibility", and "information overhead". We will review these ideas, develop some intuitive understanding, and reveal a new perspective on the nature of stochastic processes.