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This paper describes the results of an investigation of some communication receivers whose response to an input signal changes in a manner determined by the input signal. The problem considered is the design of a communication receiver to receive a message which is coded into fixed unknown signal waveforms and transmitted through a noisy channel. An optimal (minimum probability of error at each time interval) receiver is derived which has an exponentially growing structure. It requires subsystems to receive the th message symbol. The derivation suggests forms of adaptive receivers which need a more practical amount of equipment to implement, which we call the gremlin and the decision-directed adaptive receiver. The gremlin receiver is a taught-learning machine since, after it makes a decision, a gremlin tells it what the correct decision was. The decision-directed receiver is a self-taught learning machine, using its own output instead of a gremlin's. It is shown that the gremlin receiver converges to a matched filter for the unknown signal and that, in any practical case, the decision-directed receiver performs almost as well. Finally, some results of an experimental simulation of the decision-directed receiver are presented. A plot of the relative frequency of error vs. time is given for a number of different signal-to-noise ratio's (SNR's).