Numberings and learnability

Abstract

For a computable family A of computably enumerable sets there are two properties that indicate that the sets in A can be sufficiently easily distinguished: first, learnability of the class A where two models of learning may be considered, explanatory learning (EX) and behaviorally correct learning (BC); and, second, equivalence of all computable numberings of the family A under computable functions (computable equivalence) or under functions computable relative to the halting problem (∅′-equivalence). Ambos-Spies, Badaev and Goncharov (2011) have studied the relations among these properties. They have shown that EX-learnability of A implies that all computable numberings of A are ∅′-equivalent but that the converse is not true in general, and that the properties of BC-learnability of A and of ∅′-equivalence of the computable numberings of A are independent. They left open the question whether there is a computable family A of c.e. sets such that all computable numberings of A are computably equivalent and A is not BC-learnable. Such a family has been recently constructed by Ambos-Spies and Badaev. The above results are presented in this paper.