1,424 research outputs found
Query Order and the Polynomial Hierarchy
Hemaspaandra, Hempel, and Wechsung [cs.CC/9909020] initiated the field of
query order, which studies the ways in which computational power is affected by
the order in which information sources are accessed. The present paper studies,
for the first time, query order as it applies to the levels of the polynomial
hierarchy. We prove that the levels of the polynomial hierarchy are
order-oblivious. Yet, we also show that these ordered query classes form new
levels in the polynomial hierarchy unless the polynomial hierarchy collapses.
We prove that all leaf language classes - and thus essentially all standard
complexity classes - inherit all order-obliviousness results that hold for P.Comment: 14 page
What's Up with Downward Collapse: Using the Easy-Hard Technique to Link Boolean and Polynomial Hierarchy Collapses
During the past decade, nine papers have obtained increasingly strong
consequences from the assumption that boolean or bounded-query hierarchies
collapse. The final four papers of this nine-paper progression actually achieve
downward collapse---that is, they show that high-level collapses induce
collapses at (what beforehand were thought to be) lower complexity levels. For
example, for each it is now known that if \psigkone=\psigktwo then
\ph=\sigmak. This article surveys the history, the results, and the
technique---the so-called easy-hard method---of these nine papers.Comment: 37 pages. an extended abstract appeared in SIGACT News, 29, 10-22,
199
Downward Collapse from a Weaker Hypothesis
Hemaspaandra et al. proved that, for and : if
\Sigma_i^p \BoldfaceDelta DIFF_m(\Sigma_k^p) is closed under complementation,
then . This sharply asymmetric
result fails to apply to the case in which the hypothesis is weakened by
allowing the to be replaced by any class in its difference
hierarchy. We so extend the result by proving that, for and : if DIFF_s(\Sigma_i^p) \BoldfaceDelta DIFF_m(\Sigma_k^p) is closed
under complementation, then
A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Cook and Krajíček [9] have obtained the following Karp-Lipton result in bounded arithmetic: if the theory proves , then collapses to , and this collapse is provable in . Here we show the converse implication, thus answering an open question from [9]. We obtain this result by formalizing in a hard/easy argument of Buhrman, Chang, and Fortnow [3]. In addition, we continue the investigation of propositional proof systems using advice, initiated by Cook and Krajíček [9]. In particular, we obtain several optimal and even p-optimal proof systems using advice. We further show that these p-optimal systems are equivalent to natural extensions of Frege systems
Quantum Bounded Query Complexity
We combine the classical notions and techniques for bounded query classes
with those developed in quantum computing. We give strong evidence that quantum
queries to an oracle in the class NP does indeed reduce the query complexity of
decision problems. Under traditional complexity assumptions, we obtain an
exponential speedup between the quantum and the classical query complexity of
function classes.
For decision problems and function classes we obtain the following results: o
P_||^NP[2k] is included in EQP_||^NP[k] o P_||^NP[2^(k+1)-2] is included in
EQP^NP[k] o FP_||^NP[2^(k+1)-2] is included in FEQP^NP[2k] o FP_||^NP is
included in FEQP^NP[O(log n)] For sets A that are many-one complete for PSPACE
or EXP we show that FP^A is included in FEQP^A[1]. Sets A that are many-one
complete for PP have the property that FP_||^A is included in FEQP^A[1]. In
general we prove that for any set A there is a set X such that FP^A is included
in FEQP^X[1], establishing that no set is superterse in the quantum setting.Comment: 11 pages LaTeX2e, no figures, accepted for CoCo'9
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