License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/DagSemProc.06111.20
URN: urn:nbn:de:0030-drops-6054
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2006/605/
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Diehl, Scott ;
van Melkebeek, Dieter
Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines
Abstract
In this talk, we establish lower bounds for the running time of randomized
machines with two-sided error which use a small amount of workspace to
solve complete problems in the polynomial-time hierarchy. In particular,
we show that for integers $l > 1$, a randomized machine with two-sided error
using subpolynomial space requires time $n^{l - o(1)}$ to solve QSATl, where
QSATl denotes the problem of deciding the validity of a Boolean first-order
formula with at most $l-1$ quantifier alternations. This represents the first
time-space lower bounds for complete problems in the polynomial-time
hierarchy on randomized machines with two-sided error.
Corresponding to $l = 1$, we show that a randomized machine with one-sided
error using subpolynomial space requires time $n^{1.759}$ to decide the set
of Boolean tautologies. As a corollary, this gives the same lower bound for
satisfiability on deterministic machines, improving on the previously best
known such result.
BibTeX - Entry
@InProceedings{diehl_et_al:DagSemProc.06111.20,
author = {Diehl, Scott and van Melkebeek, Dieter},
title = {{Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines}},
booktitle = {Complexity of Boolean Functions},
pages = {1--33},
series = {Dagstuhl Seminar Proceedings (DagSemProc)},
ISSN = {1862-4405},
year = {2006},
volume = {6111},
editor = {Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2006/605},
URN = {urn:nbn:de:0030-drops-6054},
doi = {10.4230/DagSemProc.06111.20},
annote = {Keywords: Time-space lower bounds, lower bounds, randomness, polynomial-time hierarchy}
}
Keywords: |
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Time-space lower bounds, lower bounds, randomness, polynomial-time hierarchy |
Collection: |
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06111 - Complexity of Boolean Functions |
Issue Date: |
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2006 |
Date of publication: |
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09.10.2006 |