Topological Separations in Inductive Inference

Abstract

A major question asked by learning in the limit from positive data is about what classes of languages are learnable with respect to a given learning criterion. We are particularly interested in the reasons for a class of languages to be un learnable. We consider two types of reasons. One type is called topological (as an example, Gold has shown that no class containing an infinite language and all its finite sub-languages is learnable). Another reason is called computational (as the learners are required to be algorithmic). In particular, two learning criteria might allow for learning different classes of languages because of different topological restrictions, or because of different computational restrictions. In this paper we formalize the idea of two learning criteria separating topologically in learning power. This allows us to study more closely why two learning criteria separate in learning power. For a variety of learning criteria (concerning Fex, monotone, iterative and feedback learning) we show that certain learning criteria separate topologically, and certain others, which are known to separate, are shown not to separate topologically. Showing that learning criteria do not separate topologically implies that any known separation must necessarily exploit some computational restrictions.