Extending Downward Collapse from 1-versus-2 Queries to j-versus-j+1 Queries

The above figure shows some classes from the boolean and (truth-table) bounded-query hierarchies. It is well-known that if either collapses at a given level, then all higher levels collapse to that same level. This is a standard "upward translation of equality" that has been known for over a decade. The issue of whether these hierarchies can translate equality downwards has proven vastly more challenging. In particular, with regard to the figure above, consider the following claim: Pm-ttΣkp = Pm+1-ttΣkp ⇒ DIFFm(Sigma;kp) = coDIFFm(Sigma;kp) = BH(Sigma;kp). This claim, if true, says that equality translates downwards between levels of the bounded-query hierarchy and the boolean hierarchy levels that (before the fact) are immediately below them. Until recently, it was not known whether (**) ever held, except in the trivial m = 0 case. Then Hemaspaandra et al. [15] proved that (**) holds for all m, whenever k > 2. For the case k = 2, Buhrman and Fortnow [5] then showed that (**) holds when m = 1. In this paper, we prove that for the case k = 2, (**) holds for all values of m. As Buhrman and Fortnow showed that no relativizable technique can prove "for k = 1, (**) holds for all m," our achievement of the k = 2 case is unlikely to be strengthened to k = 1 any time in the foreseeable future. The new downward translation we obtain tightens the collapse in the polynomial hierarchy implied by a collapse in the bounded-query hierarchy of the second level of the polynomial hierarchy.