5 research outputs found
Conjunctions of Unate DNF Formulas: Learning and Structure
AbstractA 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
Exact learning of subclasses of CDNF formulas with membership queries
We consider the exact learnability of subclasses of
Boolean formulas from membership queries alone.
We show how to combine known learning algorithms
that use membership and equivalence queries
to obtain new learning results only with memberships.
In particular we show the exact learnability of
read-k monotone formulas, Sat-k O(log n)-CDNF,
and O(sqrt{log n})-size CDNF from membership queries only.
Exact Learning of subclasses of CDNF formulas with membership queries
. We consider the exact learnability of subclasses of Boolean formulas from membership queries alone. We show how to combine known learning algorithms that use membership and equivalence queries to obtain new learning results only with memberships. In particular we show the exact learnability of read-k monotone CDNF formulas, Sat- k O(log n)-CDNF, and O( p log n)-size CDNF from membership queries only. 1 Introduction Learning DNF formulas has been one of the most attractive and tantalizing problems since the seminal paper of Valiant [Val84]. Although many results in the literature give evidence that the problem is hard even if we are allow to use membership queries [AK91, AHP92], it has been recently proved by Jackson [Jac94] that using membership queries, DNF are PAC learnable in polynomial time under the uniform distribution. Here we concentrate in a more restricted framework. While Jackson's algorithm is a PAC learning algorithm, we wish to have exact identification of the target..