COMPLEXITY THEORETIC HARDNESS RESULTS FOR QUERY LEARNING

We investigate the complexity of learning for the well-studied model i n which the learning algorithm may ask membership and equivalence quer ies. While complexity theoretic techniques have previously been used t o prove hardness results in various learning models, these techniques typically are not strong enough to use when a learning algorithm may m ake membership queries. We develop a general technique for proving har dness results for learning with membership and equivalence queries lan d for more general query models). We apply the technique to show that, assuming NP not equal co-NP, no polynomial-time membership and (prope r) equivalence query algorithms exist for exactly learning read-thrice DNF formulas, unions of k greater than or equal to 3 halfspaces over the Boolean domain, or some other related classes. Our hardness result s are representation dependent, and do not preclude the existence of r epresentation independent algorithms. The general technique introduces the representation, problem for a class F of representations (e.g., f ormulas), which is naturally associated with the learning problem for F. This problem is related to the structural question of how to charac terize functions representable by formulas in F, and is a generalizati on of standard complexity problems such as SATISFIABILITY. While in ge neral the representation problem is in Sigma(2)(P), we present a theor em demonstrating that for ''reasonable'' classes F, the existence of a polynomial-time membership and equivalence query algorithm for exactl y learning F implies that the representation problem for F is in fact in co-NP. The theorem is applied to prove hardness results such as the ones mentioned above, by showing that the representation problem for specific classes of formulas is NP-hard.