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

Authors: Scott Diehl and Dieter van Melkebeek

Published in: Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

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.

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Scott Diehl and Dieter van Melkebeek. Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@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/entities/document/10.4230/DagSemProc.06111.20},
  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}
}

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

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