Propagation of partial randomness

Abstract

Let f be a computable function from finite sequences of 0's and 1's to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-L"of random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f-randomness relative to a PA-degree implies strong f-randomness, hence f-randomness does not imply f-randomness relative to a PA-degree.