License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2019.32
URN: urn:nbn:de:0030-drops-106087
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10608/
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Oliveira, Igor Carboni

Randomness and Intractability in Kolmogorov Complexity

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LIPIcs-ICALP-2019-32.pdf (0.5 MB)

Abstract

We introduce randomized time-bounded Kolmogorov complexity (rKt), a natural extension of Levin's notion [Leonid A. Levin, 1984] of Kolmogorov complexity. A string w of low rKt complexity can be decompressed from a short representation via a time-bounded algorithm that outputs w with high probability.
This complexity measure gives rise to a decision problem over strings: MrKtP (The Minimum rKt Problem). We explore ideas from pseudorandomness to prove that MrKtP and its variants cannot be solved in randomized quasi-polynomial time. This exhibits a natural string compression problem that is provably intractable, even for randomized computations. Our techniques also imply that there is no n^{1 - epsilon}-approximate algorithm for MrKtP running in randomized quasi-polynomial time.
Complementing this lower bound, we observe connections between rKt, the power of randomness in computing, and circuit complexity. In particular, we present the first hardness magnification theorem for a natural problem that is unconditionally hard against a strong model of computation.

BibTeX - Entry

@InProceedings{oliveira:LIPIcs:2019:10608,
  author =	{Igor Carboni Oliveira},
  title =	{{Randomness and Intractability in Kolmogorov Complexity}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{32:1--32:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10608},
  URN =		{urn:nbn:de:0030-drops-106087},
  doi =		{10.4230/LIPIcs.ICALP.2019.32},
  annote =	{Keywords: computational complexity, randomness, circuit lower bounds, Kolmogorov complexity}
}

Keywords: computational complexity, randomness, circuit lower bounds, Kolmogorov complexity
Collection: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 04.07.2019

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