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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: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: 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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