Coherence of Reducibilities with Randomness Notions

Abstract

Loosely speaking, when A is “more random” than B and B is “random”, then A should be random. The theory of algorithmic randomness has some formulations of “random” sets and “more random” sets. In this paper, we study which pairs (R, r) of randomness notions R and reducibilities r have the follwing property: if A is r-reducible to B and A is R-random, then B should be R-random. The answer depends on the notions R and r. The implications hold for most pairs, but not for some. We also give characterizations of n-randomness via complexity.