Abstract
Symmetry of information states that C(x)+C(y|x)=C(x,y)+O(log(C(x))). In [Chernov, Shen, Vereshchagin, and Vovk, 2008] an online variant of Kolmogorov complexity is introduced and we show that a similar relation does not hold. Let the even (online Kolmogorov) complexity of an n-bitstring x_1 x_2...x_n be the length of a shortest program that computes x_2 on input x_1, computes x_4 on input x_1 x_2 x_3, etc; and similar for odd complexity. We show that for all n there exists an n-bit x such that both odd and even complexity are almost as large as the Kolmogorov complexity of the whole string. Moreover, flipping odd and even bits to obtain a sequence x_2 x_1 x_4 x_3..., decreases the sum of odd and even complexity to C(x). Our result is related to the problem of inferrence of causality in timeseries.
BibTeX - Entry
@InProceedings{bauwens:LIPIcs:2014:4452,
author = {Bruno Bauwens},
title = {{Asymmetry of the Kolmogorov complexity of online predicting odd and even bits}},
booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
pages = {125--136},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-65-1},
ISSN = {1868-8969},
year = {2014},
volume = {25},
editor = {Ernst W. Mayr and Natacha Portier},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2014/4452},
URN = {urn:nbn:de:0030-drops-44520},
doi = {10.4230/LIPIcs.STACS.2014.125},
annote = {Keywords: (On-line) Kolmogorov complexity, (On-line) Algorithmic Probability, Philosophy of Causality, Information Transfer}
}
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Keywords: |
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(On-line) Kolmogorov complexity, (On-line) Algorithmic Probability, Philosophy of Causality, Information Transfer |
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Collection: |
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31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014) |
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Issue Date: |
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2014 |
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Date of publication: |
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05.03.2014 |
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)