Downward separation fails catastrophically for limited nondeterminism classes

Abstract

The /spl beta/ hierarchy consists of sets /spl betasub k/=NP[log/sup k/ n]/spl sube/NP. Unlike collapses in the polynomial hierarchy and the Boolean hierarchy, collapses in the /spl beta/ hierarchy do not seem to translate up, nor does closure under complement seem to cause the hierarchy to collapse. For any consistent set of collapses and separations of levels of the hierarchy that respects P=/spl betasub 1spl subespl betasub 2spl sube/.../spl sube/NP, we can construct an oracle relative to which those collapses and separations hold, yet any (or all) of the /spl betasub k/'s are closed under complement. We give a few relatively tame examples: first, for any k/spl ges/1, we construct an oracle relative to which P=/spl betasub kspl nespl betasub k+1spl nespl betasub k+2spl ne/..., and then another oracle relative to which P=/spl betasub kspl nespl betasub k+1/=PSPACE. We also construct an oracle relative to which /spl betasub 2k/=/spl betasub 2k+1spl nespl betasub 2k+2/ for all k. These results hold for more general nondeterminism hierarchies within NP, although they are in sharp contrast to the upward collapse results for Buss and Goldsmith's (1993) nondeterminism hierarchy in P.<>