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This paper is concerned with the transformation of a single-channel (series-operated) finite-state machine into a multichannel (parallel-operated) machine, and conversely. It is shown that every single-channel machine has a multichannel analog with the same number of states. A multichannel machine has a single-channel analog if and only if a so-called "prefix condition" is satisfied. Algorithms are given for constructing both types of analogs with the minimum number of states.