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2011 26th Annual IEEE Conference on Computational Complexity
Symmetry of Information and Bounds on Nonuniform Randomness Extraction via Kolmogorov Extractors
San Jose, California USA
June 08-June 11
ISBN: 978-0-7695-4411-3
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," ccc, pp.148-156, 2011 26th Annual IEEE Conference on Computational Complexity, 2011
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