During the last few years there has been considerable interest in the use of trainable controllers based upon the use of neuron-like elements, with the expectation being that these controllers can be trained, with relatively little effort, to achieve good performance. Good performance, however, hinges on the ability of the neural net to generate a "good" control law even when the input does not belong to the training set, and it has been shown that neural nets do not necessarily generalize well. It has been proposed that this problem can be solved by essentially quantizing the state space and then using a neural net to implement a table lookup procedure. There is little information on the effect of this quantization upon the controllability properties of the system. In this paper we address this problem by extending the theory of control of constrained systems to the case where the controls and measured states are restricted to finite or countably infinite sets. These results provide the theoretical framework for recently suggested neuromorphic controllers, but they are also valuable for analyzing the controllability properties of computer-based control systems.