The Boolean hierarchy I: structural properties
SIAM Journal on Computing
The polynomial time hierarchy collapses if the Boolean hierarchy collapses
SIAM Journal on Computing
The Boolean hierarchy II: applications
SIAM Journal on Computing
Downward translations of equality
Theoretical Computer Science
Bounded queries to SAT and the Boolean hierarchy
Theoretical Computer Science
Upward separation for FewP and related classes
Information Processing Letters
Fine hierarchies and Boolean terms
Journal of Symbolic Logic
Saving queries with randomness
Journal of Computer and System Sciences
Defying upward and downward separation
Information and Computation
A note on parallel queries and the symmetric-difference hierarchy
Information Processing Letters
A Downward Collapse within the Polynomial Hierarchy
SIAM Journal on Computing
The Boolean Hierarchy and the Polynomial Hierarchy: A Closer Connection
SIAM Journal on Computing
Two Refinements of the Polynomial Hierarcht
STACS '94 Proceedings of the 11th Annual Symposium on Theoretical Aspects of Computer Science
FCT '85 Fundamentals of Computation Theory
COCO '98 Proceedings of the Thirteenth Annual IEEE Conference on Computational Complexity
Translating Equality Downwards
Translating Equality Downwards
The equivalence problem for regular expressions with squaring requires exponential space
SWAT '72 Proceedings of the 13th Annual Symposium on Switching and Automata Theory (swat 1972)
Some connections between bounded query classes and non-uniform complexity
Information and Computation
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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.