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 Z_$n$_Z-bit random strings Z_$x$_Z and Z_$y$_Z, Z_$x$_Z is random conditioned by Z_$y$_Z if and only if Z_$y$_Z is random conditioned by Z_$x$_Z;\item While Z_$O(1)$_Z amount of advice regarding the source is not enough for extracting a string with randomness rate Z_$1$_Z from a source string with constant random rate, Z_$\omega(1)$_Z amount of advice is.\end{itemize}The proofs use Kolmogorov extractors as the main technical device.