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

Hemaspaandra, Edith; Hemaspaandra, Lane A.; Hempel, Harald

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

Authors

Hemaspaandra, Edith

Hemaspaandra, Lane A.

Hempel, Harald

Description

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.
37 pages. an extended abstract appeared in SIGACT News, 29, 10-22, 1998