Structural analysis of polynomial-time query learnability

Abstract

In this paper we study the computational power of polynomialtime query learning systems for several query types, and the computational complexity of a “learning problem” for several representation classes. As corollaries of our results, we prove some polynomial-time nonlearnability results, and relate polynomial-time learnability of some representation classes to the complexity of representation finding problems of P/poly oracles. For example, forCIR, a representation class by logical circuits, it is shown that P(NP ⁽⁾₁ ) is an upper bound of power of query learning systems forCIR, and that P(NP ⁽⁾₁ ) is also a lower bounds of power of query learning systems forCIR when they are used to learn a certain subclassR ofCIR. It is also shown that the problem of learningCIR is P(NP(NP ⁽⁾₁ ))-solvable. Then, using these results, the following relations are proved: (1) If, for someA ε P/poly, the representation finding problem ofA is not in P(NP ^A₁ ), thenCIR is not polynomial-time query learnable even by using queries such as membership, equivalence, subset, superset, etc. (2) On the other hand, if the above-mentioned subclassR ofCIR is not polynomial-time query learnable by using subset and superset queries, then some Bε P/poly exists such that its representation finding problem is not in P(NP ^B₁ ).