2012 IEEE 27th Conference on Computational Complexity (2011)

San Jose, California USA

June 8, 2011 to June 11, 2011

ISSN: 1093-0159

ISBN: 978-0-7695-4411-3

pp: 148-156

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/CCC.2011.21

ABSTRACT

We prove a strong Symmetry of Information relation for random strings (in the sense of Kolmogorov complexity) and establish tight bounds on the amount on nonuniformity that is necessary for extracting a string with randomness rate $1$ from a single source of randomness. More precisely, as instantiations of more general results, we show:\begin{itemize}\item For all $n$-bit random strings $x$ and $y$, $x$ is random conditioned by $y$ if and only if $y$ is random conditioned by $x$;\item While $O(1)$ amount of advice regarding the source is not enough for extracting a string with randomness rate $1$ from a source string with constant random rate, $\omega(1)$ amount of advice is.\end{itemize}The proofs use Kolmogorov extractors as the main technical device.

INDEX TERMS

Symmetry of information, random strings, randomness extraction, Kolmogorov extractors

CITATION

Marius Zimand,
"Symmetry of Information and Bounds on Nonuniform Randomness Extraction via Kolmogorov Extractors",

*2012 IEEE 27th Conference on Computational Complexity*, vol. 00, no. , pp. 148-156, 2011, doi:10.1109/CCC.2011.21