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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2008.1335
URN: urn:nbn:de:0030-drops-13350
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2008/1335/
Bienvenu, Laurent ;
Muchnik, Andrej ;
Shen, Alexander ;
Veraschagin, Nikolay
Limit complexities revisited
Abstract
The main goal of this paper is to put some known results in a
common perspective and to simplify their proofs.
We start with a simple proof of a result from (Vereshchagin, 2002)
saying that $limsup_{nKS(x|n)$ (here $KS(x|n)$ is conditional
(plain) Kolmogorov complexity of $x$ when $n$ is known) equals
$KS^{mathbf{0'(x)$, the plain Kolmogorov complexity with
$mathbf{0'$-oracle.
Then we use the same argument to prove similar results for prefix
complexity (and also improve results of (Muchnik, 1987) about limit
frequencies), a priori probability on binary tree and measure of
effectively open sets. As a by-product, we get a criterion of
$mathbf{0'$ Martin-L"of randomness (called also $2$-randomness)
proved in (Miller, 2004): a sequence $omega$ is $2$-random if and
only if there exists $c$ such that any prefix $x$ of $omega$ is a
prefix of some string $y$ such that $KS(y)ge |y|-c$. (In the
1960ies this property was suggested in (Kolmogorov, 1968) as one of
possible randomness definitions; its equivalence to $2$-randomness
was shown in (Miller, 2004) while proving another $2$-randomness
criterion (see also (Nies et al. 2005)): $omega$ is $2$-random if
and only if $KS(x)ge |x|-c$ for some $c$ and infinitely many
prefixes $x$ of $omega$.
Finally, we show that the low-basis theorem can be used to get
alternative proofs for these results and to improve the result
about effectively open sets; this stronger version implies the
$2$-randomness criterion mentioned in the previous sentence.
BibTeX - Entry
@InProceedings{bienvenu_et_al:LIPIcs:2008:1335,
author = {Laurent Bienvenu and Andrej Muchnik and Alexander Shen and Nikolay Veraschagin},
title = {{Limit complexities revisited}},
booktitle = {25th International Symposium on Theoretical Aspects of Computer Science},
pages = {73--84},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-06-4},
ISSN = {1868-8969},
year = {2008},
volume = {1},
editor = {Susanne Albers and Pascal Weil},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2008/1335},
URN = {urn:nbn:de:0030-drops-13350},
doi = {10.4230/LIPIcs.STACS.2008.1335},
annote = {Keywords: Kolmogorov complexity, limit complexities, limit frequencies, 2-randomness, low basis}
}
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Keywords: |
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Kolmogorov complexity, limit complexities, limit frequencies, 2-randomness, low basis |
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Collection: |
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25th International Symposium on Theoretical Aspects of Computer Science |
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Issue Date: |
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2008 |
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Date of publication: |
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06.02.2008 |
