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: • 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; • 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, ω(1) amount of advice is. The proofs use Kolmogorov extractors as the main technical device.

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Marius Zimand
• Keywords: CORR 2011, Education, Random, Random Strings, Randomness Rate