PAC Meditation on Boolean Formulas

Abstract

We present a Probably Approximate Correct (PAC) learning paradigm for boolean formulas, which we call PAC meditation, where the class of formulas to be learnt are not known in advance. On the contrary we split the building of the hypothesis in various levels of increasing description complexity according to additional constraints received at run time. In particular, starting from atomic forms constituted by clauses and monomials learned from the examples at the 0-level, we provide a procedure for computing hypotheses in the various layers of a polynomial hierarchy including k_term-DNF formulas at the second level. Assessment of the sample complexity is based on the notion of sentry functions, introduced in a previous paper, which extends naturally to the various levels of the learning procedure. We make a distinction between meditations which waste some sample information and those which exploit all information at each description level, and propose a procedure that is free from information waste. The procedure takes only a polynomial time if we restrict us to learn an inner and outer boundary to the target formula in the polynomial hierarchy, while an access to an NP-oracle is needed if we want to fix the hypothesis in a proper representation.