4 Hierarchical Learning of Boolean Functions

The learning of Boolean functions has been the focus of a number of papers in computer science in the last few years, many of which have been stimulated by the work of Valiant [3, 41. We also mention especially the recent long article by Blumer et al. [l] on learnability for various classes of concepts including especially Boolean classes. The thrust of most of this theoretical work has essentially been to show under what conditions concepts can be learned in polynomial time or space. It is of course important to show that learning is feasible for as wide a class of concepts as possible. However, it is also important to focus on detailed results of a very finitistic kind, which are characteristic of learning theory as applied to human behavior, and are of intrinsic interest also in machine learning. In this paper we consider conditions under which a hierarchy of Boolean functions can increase in significant ways the rate of learning of a given Boolean function of n variables. In this context we also compare serial and parallel hierarchical learning, with, as might be expected, the most dramatic effects being produced by parallel hierarchical learning. Because in general the results are better for parallel than for serial learning the reader might wonder why we include a systematic study of serial learning. The reason is that even under serial learning conditions hierarchical learning can often lead to dramatic improvement in the rate of learning. There is some literature on hierarchical learning, but the results are not closely related to ours. We refer the reader to Rivest and Sloan [2], and the references given there. There are three distinct psychological processes involved in our theoretical approach to hierarchical learning. They are (i) stimulus sampling, often given in the form of a pattern; (ii) concept sampling; and (iii) function formation-meaning forming some function of the concepts sampled. In some contexts it may be usefu1 to refer to function formation as hypothesis sampling. In any case, process (i) brings in information from the world outside, but (ii) and (iii) are essentially internal.