Learning Recursive Functions Refutably?

Abstract. Learning of recursive functions refutably means that for ev-ery recursive function, the learning machine has either to learn this func-tion or to refute it, i.e., to signal that it is not able to learn it. Three modi of making precise the notion of refuting are considered. We show that the corresponding types of learning refutably are of strictly increas-ing power, where already the most stringent of them turns out to be of remarkable topological and algorithmical richness. All these types are closed under union, though in different strengths. Also, these types are shown to be different with respect to their intrinsic complexity; two of them do not contain function classes that are “most difficult ” to learn, while the third one does. Moreover, we present characterizations for these types of learning refutably. Some of these characterizations make clea