Downward separation fails catastrophically for limited nondeterminism classes

The hierarchy consists of sets k = NPPlog k n] NP. Unlike collapses in the polynomial hierarchy and the Boolean hierarchy, collapses in the 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 = 1 2 NP, we can construct an oracle relative to which those collapses and separations hold, yet any (or all) of the k 's are closed under complement. To give two relatively tame examples: For any k 1, we construct an oracle relative to which P = k 6 = k+1 6 = k+2 6 = and another oracle relative to which P = k 6 = k+1 = PSPACE: We also construct an oracle relative to which 2k = 2k+1 6 = 2k+2 for all k.