Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Komplexität und Kryptografie

Johannes Köbler and Wolfgang Lindner

Abstract:

We use the notion of general dimension to show that any p-evaluatable concept class with polynomial query complexity can be learned in polynomial time with the help of an oracle in the polynomial hierarchy, where the complexity of the required oracle depends on the query-types used by the learning algorithm. In particular, we show that for subset and superset queries an oracle in \Sigma^P_3 suffices. Since the concept class of DNF formulas has polynomial query complexity with respect to subset and superset queries with DNF formulas as hypotheses, it follows that DNF formulas are properly learnable in polynomial time with subset and superset queries and the help of an oracle in $\Sigma^{\P}_3$. We also show that the required oracle in our main theorem cannot be replaced by an oracle in a lower level of the polynomial-time hierarchy, unless the hierarchy collapses.