What's up with downward collapse

Abstract

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 before-the-fact seemed to be) lower complexity levels. For example, for each k ≥ 2 it. is now known that if one query to σ p k is as powerful as two queries to σ p k (i.e., P Σ[1] = P Σ[2] ), then PH = σ p k . This article surveys the history, the results, and the method---the so-called easy-hard technique---of the just-mentioned nine-paper progression:1. J. Kadin. The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal on Computing , 17(6):1263--1282, 1988. Erratum appears in the same journal. 20(2):404.2. K. Wagner. Number-of-query hierarchies. Technical Report 158, Institut für Mathematik. Universität Augsburg, Augsburg, Germany, October 1987.3. K. Wagner. Number-of-query hierarchies. Technical Report 4. Institut für Informatik. Universität Würzburg, Würzburg, Germany, February 1989.4. R. Chang and J. Kadin. The boolean hierarchy and the polynomial hierarchy: A closer connection. SIAM Journal on Computing . 25(2):340--354. 1996.5. R. Beigel. R. Chang. and M. Ogiwara. A relationship between difference hierarchies and relativized polynomial hierarchies. Mathematical Systems Theory . 26(3):293--310, 1993.6. E. Hemaspaandra, L. Hemaspaandra. and H. Hempel. An upward separation in the polynomial hierarchy. Technical Report Math/Inf/96/15, Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany, June 1996.7. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. A downward collapse within the polynomial hierarchy. SIAM Journal on Computing , 28(2):383--393, 1999.8. H. Buhrman and L. Fortnow. Two queries. In Proceedings of the 13th Annual IEEE Conference on Computational Complexity . 13--19. IEEE Computer Society Press, June 1998.9. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. Translating equality downwards. Technical Report TR-657. Department of Computer Science, University of Rochester, Rochester, NY. April 1997.