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A formulation of finite-memory information processing problems is presented. The total state space of the system, including the "memory" of the source and processor, is assumed to be finite. A cost functional is specified over the trajectories of the system and a variational approach is used to minimize cost. There results a two-point boundary value problem and an associated improvement algorithm. Special attention is given to two types of cost functionals: finite-time problems, and time-average problems over an infinite time interval. Several examples are included.