Downward Collapse from a Weaker Hypothesis

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 $DIFF_m(\Sigma_k^p) = coDIFF_m(\Sigma_k^p)$

What's Up with Downward Collapse: Using the Easy-Hard Technique to Link Boolean and Polynomial Hierarchy Collapses

During the past decade, nine papers have obtained increasingly strong consequences from the assumption that boolean or bounded-query hierarchies collapse. The final four papers of this nine-paper progression actually achieve downward collapse---that is, they show that high-level collapses induce collapses at (what beforehand were thought to be) lower complexity levels. For example, for each $k\geq 2$ it is now known that if \psigkone=\psigktwo then \ph=\sigmak. This article surveys the history, the results, and the technique---the so-called easy-hard method---of these nine papers.Comment: 37 pages. an extended abstract appeared in SIGACT News, 29, 10-22, 199

Query Order and the Polynomial Hierarchy

Hemaspaandra, Hempel, and Wechsung [cs.CC/9909020] initiated the field of query order, which studies the ways in which computational power is affected by the order in which information sources are accessed. The present paper studies, for the first time, query order as it applies to the levels of the polynomial hierarchy. We prove that the levels of the polynomial hierarchy are order-oblivious. Yet, we also show that these ordered query classes form new levels in the polynomial hierarchy unless the polynomial hierarchy collapses. We prove that all leaf language classes - and thus essentially all standard complexity classes - inherit all order-obliviousness results that hold for P.Comment: 14 page