We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.
"Algorithmic randomness and measures of complexity." Bull. Symbolic Logic 19 (3) 318 - 350, September 2013. https://doi.org/10.2178/bsl.1903020