We show the following: (a) For any > 0, log (3+ ) 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 polynomialquery learned with membership and equivalence queries that use t 1+ -term DNF formulas as hypotheses, for some < 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

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... y 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.