Downward Collapse from a Weaker Hypothesis

Details Downward Collapse from a Weaker Hypothesis Downward Collapse from a Weaker Hypothesis

1998-08-24

arxiv.org/abs/cs/9808002v1

Computer Science

Computational Complexity

Scientific paper

Hemaspaandra et al. proved that, for $m > 0$ and $0 < i < k - 1$: if $\Sigma_i^p \BoldfaceDelta DIFF_m(\Sigma_k^p)$ is closed under complementation, then $DIFF_m(\Sigma_k^p) = coDIFF_m(\Sigma_k^p)$. 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 $DIFF_s(\Sigma_i^p) \BoldfaceDelta DIFF_m(\Sigma_k^p)$ is closed under complementation, then $DIFF_m(\Sigma_k^p) = coDIFF_m(\Sigma_k^p)$.

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