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URN: urn:nbn:de:0030-drops-6054
URL: http://drops.dagstuhl.de/opus/volltexte/2006/605/

Diehl, Scott ; van Melkebeek, Dieter

Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines

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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:DSP:2006:605,
  author =	{Scott Diehl and Dieter van Melkebeek},
  title =	{Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines},
  booktitle =	{Complexity of Boolean Functions},
  year =	{2006},
  editor =	{Matthias Krause and Pavel Pudl{\'a}k and R{\"u}diger Reischuk and Dieter van Melkebeek},
  number =	{06111},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2006/605},
  annote =	{Keywords: Time-space lower bounds, lower bounds, randomness, polynomial-time hierarchy}
}

Keywords: Time-space lower bounds, lower bounds, randomness, polynomial-time hierarchy
Seminar: 06111 - Complexity of Boolean Functions
Issue date: 2006
Date of publication: 09.10.2006


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