(poly(log log n), poly(log log n))—Restricted verifiers are unlikely to exist for languages in $\mathcal{N}\mathcal{P}$*

Abstract

The aim of this paper is to present a proof of the equivalence of the equalities $\mathcal{N}\mathcal{P} = \mathcal{P}\mathcal{C}\mathcal{P}$(log log n,1) and $\mathcal{P} = \mathcal{N}\mathcal{P}$. The proof is based on producing long pseudo-random bit strings through random walks on expander graphs. This technique also implies that for any language in $\mathcal{N}\mathcal{P}$ there exists a restricted verifier using log n + c, c is a constant, random bits. Furthermore, we prove that the equality of classes $\mathcal{N}\mathcal{P}$ and $\mathcal{P}\mathcal{C}\mathcal{P}$ (poly(log log n), poly(log log n)) implies the inclusion of $\mathcal{N}\mathcal{P}$ in $\mathcal{D}\mathcal{T}\mathcal{I}\mathcal{M}\mathcal{E}$(n ^{poly(log log n)}). Also, some technical details of the proof of $\mathcal{N}\mathcal{P} = \mathcal{P}\mathcal{C}\mathcal{P}$ (log n, 1) are used for showing that a certain class of (poly(log log n), poly(log log n))-restricted verifiers does not exist for languages in $\mathcal{N}\mathcal{P}$ unless $\mathcal{P} = \mathcal{N}\mathcal{P}$.

This work was supported by the Information Technology Programme of EU under the project ALCOM-IT.