Learning orthogonal F-Horn formulas

Abstract

In the PAC-learning, or the query learning model, it has been an important open problem to decide whether the class of DNF and CNF formulas is learnable. Recently, it was pointed out that the problem of PAC-learning for these classes with membership queries can be reduced to that of learning for the class of k-quasi Horn formulas with membership and equivalence queries. A k-quasi Horn formula is a CNF formula with each clause containing at most k unnegated literals. In this paper, notions of F-Horn formulas and l- F-Horn formulas, which are extensions of k-quasi formulas, are introduced, and it is shown that the problem of query learning for DNF and CNF formulas with membership and equivalence queries can be reduced to that for l- F-Horn formulas for an appropriate choice of F. It is shown that under a condition on F, the class of orthogonal F-Horn formulas is learnable with membership, equivalence and subset queries. Moreover, it is shown that under the same condition the class of orthogonal l- F-Horn formulas is learnable with membership and equivalence queries. For the latter result, the condition of orthogonality of F-Horn formulas is crucial because, if the statement held without the condition, then the result would imply that DNF and CNF are exactly learnable with membership and equivalence queries.