Learnability and automatizability

We consider the complexity of properly learning concept classes, i.e. when the learner must output a hypothesis of the same form as the unknown concept. We present the following upper and lower bounds on well-known concept classes: 1) We show that unless NP = RP, there is no polynomial-time PAC learning algorithm for DNF formulae where the hypothesis is an OR-of-thresholds. Note that as special cases, we show that neither DNF nor OR-of-thresholds are properly learnable unless NP = RP. Previous hardness results have required strong restrictions on the size of the output DNF formula. We also prove that it is NP-hard to learn the intersection of ℓ ≥ 2 halfspaces by the intersection of k halfspaces for any constant k > 0. Previous work held for the case when k = ℓ; 2) Assuming that NP