Conjunctions of Unate DNF Formulas: Learning and Structure

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 ℛ^k consisting of conjunctions of k unate DNF formulas. This class generalizes the class of k-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 ℛ² can 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 ℛ² with membership and equivalence queries is computational rather than informational. Few results of this type are known. In our proofs, we use recent results of Hellerstein et 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 ℛ² and pose open questions on learning DNF formulas of small monotone dimension. We also prove structural results for ℛ^k. We construct, for any fixed k ≥ 2, a class of functions f that cannot be represented by any formula in ℛ^k, but which cannot be "easily" shown to have this property. More precisely, for any function f on n variables in the class, the value of f on any polynomial-size set of points in its domain is not a witness that f cannot be represented by a formula in ℛ^k. Our construction is based on BCH codes.