2 research outputs found
Using the No-Search Easy-Hard Technique for Downward Collapse
The top part of the preceding figure [figure appears in actual paper] shows
some classes from the (truth-table) bounded-query and boolean hierarchies. It
is well-known that if either of these hierarchies collapses at a given level,
then all higher levels of that hierarchy 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 {\em
downwards\/} has proven vastly more challenging. In particular, with regard to
the figure above, consider the following claim:
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 (*) {\em ever\/} held, except for
the degenerate cases and . Then Hemaspaandra, Hemaspaandra, and
Hempel \cite{hem-hem-hem:j:downward-translation} proved that (*) holds for all
, for . Buhrman and Fortnow~\cite{buh-for:j:two-queries} then showed
that, when , (*) holds for the case . In this paper, we prove that
for the case , (*) holds for all values of . Since there is an oracle
relative to which ``for , (*) holds for all '' fails
\cite{buh-for:j:two-queries}, our achievement of the case cannot to be
strengthened to by any relativizable proof technique. The new downward
translation we obtain also tightens the collapse in the polynomial hierarchy
implied by a collapse in the bounded-query hierarchy of the second level of the
polynomial hierarchy.Comment: 22 pages. Also appears as URCS technical repor
Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets
It is known that for any class C closed under union and intersection, the
Boolean closure of C, the Boolean hierarchy over C, and the symmetric
difference hierarchy over C all are equal. We prove that these equalities hold
for any complexity class closed under intersection; in particular, they thus
hold for unambiguous polynomial time (UP). In contrast to the NP case, we prove
that the Hausdorff hierarchy and the nested difference hierarchy over UP both
fail to capture the Boolean closure of UP in some relativized worlds.
Karp and Lipton proved that if nondeterministic polynomial time has sparse
Turing-complete sets, then the polynomial hierarchy collapses. We establish the
first consequences from the assumption that unambiguous polynomial time has
sparse Turing-complete sets: (a) UP is in Low_2, where Low_2 is the second
level of the low hierarchy, and (b) each level of the unambiguous polynomial
hierarchy is contained one level lower in the promise unambiguous polynomial
hierarchy than is otherwise known to be the case.Comment: 27 page