A central topic in query learning is to determine which classes of Boolean formulas are efficiently learnable with membership and equivalence queries. We consider the class Rkconsisting of conjunctions ofkunate DNF formulas. This class generalizes the class ofk-clause CNF formulas and the class of unate DNF formulas, both of which are known to be learnable in polynomial time with membership and equivalence queries. We prove that R2can be properly learned with a polynomial number of polynomial-size membership and equivalence queries, but can be properly learned in polynomial time with such queries if and only if P=NP. Thus the barrier to properly learning R2with membership and equivalence queries is computational rather than informational. Few results of this type are known. In our proofs, we use recent results of Hellersteinet al.(1997,J. Assoc. Comput. Mach.43(5), 840–862), characterizing the classes that are polynomial-query learnable, together with work of Bshouty on the monotone dimension of Boolean functions. We extend some of our results to Rkand pose open questions on learning DNF formulas of small monotone dimension. We also prove structural results for Rk. We construct, for any fixedk⩾2, a class of functionsfthat cannot be represented by any formula in Rk, but which cannot be “easily” shown to have this property. More precisely, for any functionfonnvariables in the class, the value offon any polynomial-size set of points in its domain is not a witness thatfcannot be represented by a formula in Rk. Our construction is based on BCH codes.

We consider the class 77,2, consisting of conjunctions of two unate DNF formulas.This class is a generalization of the class of 2-clause CNF formulas, and of the class of unate DNF formulas, both of which are properly learnable in polynomial time with membership and equivalence queries.We show that 7?2 can be properly learned with a polynomial number of polynomial-size membership and equivalence queries, but that it cannot be learned in polynomial time unless P = NP.Thus the barrier to learning 7?,2is computational rather than informational,In proving our results, we use recent techniques developed for the membership and equivalence query model, as well as Bshout y's work on the monotone dimension.We pose some related open questions on learning DNF formulas of small monotone dimension.queries to learn the class) or whether it is computational (a