PAC-Learning Prolog Clauses With or Without Errors

Abstract

Recently researchers have been interested in trying to expand the domain of learnability to subsets of first-order logic, in particular Prolog programs. This new research area has been named Inductive Logic Programming (ILP). In a nutshell we can describe a generic ILP problem as following: given a set E of (positive and negative) examples of a target predicate, and some background knowledge B about the world (usually a logic program including facts and auxiliary predicates), the task is to find a logic program H (our hypothesis) such that all positive examples can be deduced from B and H, while no negative example can. In this paper we review some of the results achieved in this area and discuss the techniques used. Moreover we prove the following new results: (1) Predicates described by non-recursive, local clauses of at most k literals are PAC-learnable under any distribution. This generalizes a previous result that was valid only for constrained clauses. (2) Predicates that are described by k non-recursive local clauses are PAC-learnable under any distribution. This generalizes a previous result that was non constructive and valid only under some class of distributions. Finally we introduce what we believe is the first theoretical framework for learning Prolog clauses in the presence of errors. To this purpose we introduce a new noise model, that we call the fixed attribute noise model, for learning propositional concepts over the Boolean domain. This new noise model can be of its own interest.