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The results presented above are part of research carried on in the area of pattern recognition in collaboration with John M. Richardson at the Hughes Research Laboratories at Malibu, California, as a consultant. Preliminary results were given in Bellman, R., System Identification, Pattern Recognition and Dynamic Programming—I: General Concepts, Hughes Research Laboratories (1964); System Identification, Pattern Recognition and Dynamic Programming—II: Endgames in Chess, Hughes Research Laboratories (1964); System Identification, Pattern Recognition and Dynamic Programming—III: The Game of Checkers, Hughes Research Laboratories (1964); System Identification, Pattern Recognition and Dynamic Programming—IV: Abstraction of Properties and Adaptive Pattern Recognition, Hughes Research Laboratories (1964).