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DOI: 10.4230/LIPIcs.STACS.2010.2462

Collapsing and Separating Completeness Notions under Average-Case and Worst-Case Hypotheses

Authors: Xiaoyang Gu, John M. Hitchcock, and Aduri Pavan

Published in: LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Abstract

This paper presents the following results on sets that are complete for $\NP$. \begin{enumerate} \item If there is a problem in $\NP$ that requires $\twonO$ time at almost all lengths, then every many-one NP-complete set is complete under length-increasing reductions that are computed by polynomial-size circuits. \item If there is a problem in $\CoNP$ that cannot be solved by polynomial-size nondeterministic circuits, then every many-one complete set is complete under length-increasing reductions that are computed by polynomial-size circuits. \item If there exist a one-way permutation that is secure against subexponential-size circuits and there is a hard tally language in $\NP \cap \CoNP$, then there is a Turing complete language for $\NP$ that is not many-one complete. \end{enumerate} Our first two results use worst-case hardness hypotheses whereas earlier work that showed similar results relied on average-case or almost-everywhere hardness assumptions. The use of average-case and worst-case hypotheses in the last result is unique as previous results obtaining the same consequence relied on almost-everywhere hardness results.

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Xiaoyang Gu, John M. Hitchcock, and Aduri Pavan. Collapsing and Separating Completeness Notions under Average-Case and Worst-Case Hypotheses. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 429-440, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{gu_et_al:LIPIcs.STACS.2010.2462,
  author =	{Gu, Xiaoyang and Hitchcock, John M. and Pavan, Aduri},
  title =	{{Collapsing and Separating Completeness Notions under Average-Case and Worst-Case Hypotheses}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{429--440},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Marion, Jean-Yves and Schwentick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2462},
  URN =		{urn:nbn:de:0030-drops-24627},
  doi =		{10.4230/LIPIcs.STACS.2010.2462},
  annote =	{Keywords: Computational complexity, NP-completeness}
}

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DOI: 10.4230/LIPIcs.FSTTCS.2009.2320

Kolmogorov Complexity in Randomness Extraction

Authors: John M. Hitchcock, Aduri Pavan, and N. V. Vinodchandran

Published in: LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

Abstract

We clarify the role of Kolmogorov complexity in the area of randomness extraction. We show that a computable function is an almost randomness extractor if and only if it is a Kolmogorov complexity extractor, thus establishing a fundamental equivalence between two forms of extraction studied in the literature: Kolmogorov extraction and randomness extraction. We present a distribution ${\cal M}_k$ based on Kolmogorov complexity that is complete for randomness extraction in the sense that a computable function is an almost randomness extractor if and only if it extracts randomness from ${\cal M}_k$.

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John M. Hitchcock, Aduri Pavan, and N. V. Vinodchandran. Kolmogorov Complexity in Randomness Extraction. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 215-226, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{hitchcock_et_al:LIPIcs.FSTTCS.2009.2320,
  author =	{Hitchcock, John M. and Pavan, Aduri and Vinodchandran, N. V.},
  title =	{{Kolmogorov Complexity in Randomness Extraction}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{215--226},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-13-2},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{4},
  editor =	{Kannan, Ravi and Narayan Kumar, K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2320},
  URN =		{urn:nbn:de:0030-drops-23201},
  doi =		{10.4230/LIPIcs.FSTTCS.2009.2320},
  annote =	{Keywords: Extractors, Kolmogorov extractors, randomness}
}

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Documents authored by Pavan, Aduri

Collapsing and Separating Completeness Notions under Average-Case and Worst-Case Hypotheses

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Kolmogorov Complexity in Randomness Extraction

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