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 intoMfixed 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(M - 1)M^{n-1}subsystems to receive thenth 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).