Downward Collapse from a Weaker Hypothesis

Hemaspaandra et al. [1] proved that, for m > 0 and 0 < i < k 1: if i DIFFm( p k) is closed under complementation, then DIFFm( p k) = coDIFFm( p k). This sharply asymmetric result fails to apply to the case in which the hypothesis is weakened by allowing the i to be replaced by any class in its di erence hierarchy. We so extend the result by proving that, for s;m > 0 and 0 < i < k 1: if DIFFs( p i ) DIFFm( p k) is closed under complementation, then DIFFm( p k) = coDIFFm( p k).