Downward Collapse from a Weaker Hypothesis

Hemaspaandra et al. proved that, for $m>0$ and $0<i<k-1$: if ${\mathrm{\Sigma }}_i^p\text{BoldfaceDelta}DIF{F}_m({\mathrm{\Sigma }}_k^p)$ is closed under complementation, then $DIF{F}_m({\mathrm{\Sigma }}_k^p)=coDIF{F}_m({\mathrm{\Sigma }}_k^p)$. This sharply asymmetric result fails to apply to the case in which the hypothesis is weakened by allowing the ${\mathrm{\Sigma }}_i^p$ 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$DIFF_s(\Sigma_i^p) \BoldfaceDelta DIFF_m(\Sigma_k^p)$ is closed under complementation, then $DIF{F}_m({\mathrm{\Sigma }}_k^p)=coDIF{F}_m({\mathrm{\Sigma }}_k^p)$.