On learning of functions refutably

Learning of recursive functions refutably informally means that for every recursive function, the learning machine has either to learn this function or to refute it, that is 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 increasing power, where already the most stringent of them turns out to be of remarkable topological and algorithmical richness. Furthermore, 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 several characterizations for these types of learning refutably. Some of these characterizations make clear where the refuting ability of the corresponding learning machines comes from and how it can be realized, in general.For learning with anomalies refutably, we show that several results from standard learning without refutation stand refutably. From this we derive some hierarchies for refutable learning. Finally, we prove that in general one cannot trade stricter refutability constraints for more liberal learning criteria.