Learning How to Separate

The main question addressed in the present work is how to find effectively a recursive function separating two sets drawn arbitrarily from a given collection of disjoint sets. In particular, it is investigated in which cases it is possible to satisfy the following additional constraints: confidence where the learner converges on all data-sequences; conservativeness where the learner abandons only definitely wrong hypotheses; consistency where also every intermediate hypothesis is consistent with the data seen so far; set-driven learners whose hypotheses are independent of the order and the number of repetitions of the data-items supplied; learners where either the last or even all hypotheses are programs of total recursive functions. The present work gives an overview of the relations between these notions and succeeds to answer many questions by finding ways to carry over the corresponding results from other scenarios within inductive inference. Nevertheless, the relations between conservativeness and set-driven inference needed a novel approach which enabled to show the following two major results: (1) There is a class for which recursive separators can be found in a confident and set-driven way, but no conservative learner finds a (not necessarily total) separator for this class. (2) There is a class for which recursive separators can be found in a confident and conservative way, but no set-driven learner finds a (not necessarily total) separator for this class.