Downward Collapse from a Weaker Hypothesis

Abstract

Hemaspaandra et al. proved that, for $m>0$ and $0<i<k-1$: if $Σ_i^p \BoldfaceDelta DIFF_m(Σ_k^p)$ is closed under complementation, then $DIF{F}_m({\Sigma }_k^p)=coDIF{F}_m({\Sigma }_k^p)$. This sharply asymmetric result fails to apply to the case in which the hypothesis is weakened by allowing the ${\Sigma }_i^p$ to be replaced by any class in its difference hierarchy. We so extend the result by proving that, for $s,m>0$ and $0<i<k-1$: if $DIFF_s(Σ_i^p) \BoldfaceDelta DIFF_m(Σ_k^p)$ is closed under complementation, then $DIF{F}_m({\Sigma }_k^p)=coDIF{F}_m({\Sigma }_k^p)$.