Oracles for learning programs

The most elementary study of man-machine communication deals with functions that are computable by an oracle machine. Such a machine is not supposed to work just by itself-it may interact with the external environment too. As far as learning programs are concerned, we may consider that the environment which provides data to the learner plays the role of an oracle. According to the various learning paradigms, the oracle can be used in quite different ways. It is fruitful to look at the paradigms from this point of view. Indeed, the various concepts of reducibility of one function or one set to another (e.g. Turing, truth-table and bounded truth-table reducibilities), that have been extensively studied in the theory of recursive functions, bare some relation to this aspect of learning. For most of the existing learning paradigms, the type of interaction between a learner and his environment is rather plain, but we describe a paradigm, dedicated to learning to control physical systems, for which this interaction is more complex. The reducibility notions are then extremely useful. We state a few results which illuminate the strength of reducibility concepts for studying learnability