Learnability of relatively quantified generalized formulas

In this paper we study the learning complexity of a vast class of quantifed formulas called Relatively Quantified Generalized Formulas. This class of formulas is parameterized by a set of predicates, called a basis. We give a complete classification theorem, showing that every basis gives rise to quantified formulas that are either polynomially learnable with equivalence queries, or not polynomially predictable with membership queries under some cryptographic assumption. We also provide a simple criteria distinguishing the learnable cases from the non-learnable cases.