A Downward Collapse within the Polynomial Hierarchy
Edith Hemaspaandra, Lane A. Hemaspaandra, Harald Hempel⋆ · SIAM Journal on Computing · 1998
Downward collapse (also known as 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 ${\rm P}^{\Sigma^p_k[1]} = {\rm P}^{\Sigma^p_k[2]}$ then $\Sigma^p_k = \Pi^p_k = {\rm PH}$. We extend this to obtain a more general downward collapse result.