Learning how to separate Authors: Sanjay Jain and Frank Stephan Email: sanjay@comp.nus.edu.sg Source: Theoretical Computer Science Vol. 313, Issue 2, 17 February 2004, pp. 209-228. Abstract. 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 when one can find better learners which satisfy additional constraints. Such learners are the following: confident learners which converge on all data-sequences; conservative learners which abandon only definitely wrong hypotheses; 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 a complete picture of the relations between these notions: the only implications are that whenever one has a learner which only outputs programs of total recursive functions as hypotheses, then one can also find learners which are conservative and set-driven. The following two major results need a nontrivial proof: (1) There is a class for which one can find, in the limit, recursive functions separating the sets in a confident and conservative way, but one cannot find even partial-recursive functions separating the sets in a set-driven way. (2) There is a class for which one can find, in the limit, recursive functions separating the sets in a confident and set-driven way, but one cannot find even partial-recursive functions separating the sets in a conservative way.

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