L1-Computability, Layerwise Computability and Solovay Reducibility

We propose a hierarchy of classes of functions that corresponds to the hierarchy of randomness notions. Each class of functions converges at the corresponding random points. We give various characterizations of the classes, that is, characterizations via integral tests, L-computability and layerwise computability. Furthermore, the relation among these classes is formulated using a version of Solovay reducibility for lower semicomputable functions.