Exact learning of DNF formulas using DNF hypotheses

Abstract

We show the following: (a) For any ε > 0 , log ( 3 + ε ) n -term DNF cannot be polynomial-query learned with membership and strongly proper equivalence queries. (b) For sufficiently large t, t-term DNF formulas cannot be polynomial-query learned with membership and equivalence queries that use t 1 + ε -term DNF formulas as hypotheses, for some ε < 1 (c) Read-thrice DNF formulas are not polynomial-query learnable with membership and proper equivalence queries. (d) log n -term DNF formulas can be polynomial-query learned with membership and proper equivalence queries. (This complements a result of Bshouty, Goldman, Hancock, and Matar that log n -term DNF can be so learned in polynomial time.) Versions of (a)–(c) were known previously, but the previous versions applied to polynomial- time learning and used complexity theoretic assumptions. In contrast, (a)–(c) apply to polynomial- query learning, imply the results for polynomial-time learning, and do not use any complexity-theoretic assumptions.