Symmetry of information and bounds on nonuniform randomness extraction via Kolmogorov extractors

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: (1) 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, and (2) 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. The proofs use Kolmogorov extractors as the main technical device.Comment: To appear in the proceedings of CCC 201

Nonuniform Kolmogorov extractors

We 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 with lower randomness rate. More precisely, as instantiations of more general results, we show that 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 subunitary random rate, \omega(1) amount of advice is.Comment: This is a part of the conference paper from CCC 2011. It corrects an erroneus result from there, namely Theorem 4.8 (the new version has weaker parameters

Space-Bounded Kolmogorov Extractors

An extractor is a function that receives some randomness and either "improves" it or produces "new" randomness. There are statistical and algorithmical specifications of this notion. We study an algorithmical one called Kolmogorov extractors and modify it to resource-bounded version of Kolmogorov complexity. Following Zimand we prove the existence of such objects with certain parameters. The utilized technique is "naive" derandomization: we replace random constructions employed by Zimand by pseudo-random ones obtained by Nisan-Wigderson generator.Comment: 12 pages, accepted to CSR201