A Downward Collapse within the Polynomial Hierarchy

TL;DR

This paper proves that within the polynomial hierarchy, the equality of certain classes implies the collapse of the hierarchy to a lower level, revealing new relationships between complexity classes.

Contribution

It establishes a new downward collapse result within the polynomial hierarchy, showing specific class equalities imply the hierarchy's collapse.

Findings

If ^{k+1} = ^{k+1} for k > 2, then ^{k} = ^{k} = ext{PH}

Provides a more general downward collapse result within the polynomial hierarchy

Enhances understanding of class relationships and hierarchy collapse conditions

Abstract

Downward collapse (a.k.a. upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial hierarchy. In particular, we prove that, for k > 2, if $\psigkone = \psigktwo$ then $\sigmak = \pik = \ph$. We extend this to obtain a more general downward collapse result.