A “true” cognitive radio is a sophisticated devise that is aware of, can adapt to, and even learn from its environment. To do so, such device would need to perform many complex (parallel) tasks in minuscule amounts of time, and within tight size, weight, heat, and energy limitations. In any sophisticated computing system, communication and computation limit each other. Here we discuss a formal model that abstract an information-processing limited rational entity (human or not) seeking to maximise a performance function that depends on both an action under its control, and and an exogenous input. The entity finds and optimal probabilistic response such that the Shannon-theoretic mutual information between input and output does not exceed an information capacity constraint. The possible application to secondary dynamic spectrum access receives special attention, but the framework is of interest whenever the practical limits on a cognitive radio information processing capacity cannot be neglected. Rather than proposing a particular algorithm and showing numerical results, our aims are to (i) introduce an important practical problem affecting “true” cognitive radios, (ii) introduce a body of existing scientific literature not widely known in engineering, and (iii) discuss potential applications of this literature to cognitive radio research.