Some Comments on Functional Self-Reducibility and the NP Hierarchy

Abstract

In Valiant [11] and Schnorr [9], concepts of "functional self-reducibility" are introduced and investigated. We concentrate on the class NP and on the NP hierarchy of Meyer and Stockmeyer [7] to further investigate these ideas. Assuming that the NP hierarchy exists (specifically, assuming that $P \stackrel{\subset}{+} NP = \sum^{P}_{1} \stackrel{\subset}{+} \sum^{P}_{2}$ we show that, while every complete set in $\sum^{P}_{2}$ is functionally self-reducible, there exist sets in $\sum^{P}_{2}$ which are not functionally self-reducible.