On the computational power of random strings

Abstract

There are two fundamental computably enumerable sets associated with any Kolmogorov complexity measure. These are the set of non-random strings and the overgraph. This paper investigates the computational power of these sets. It follows work done by Kummer, Muchnik and Positselsky, and Allender and co-authors. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not $tt$-complete. This paper answers this question in the negative by proving that the overgraph of any optimal monotone machine, or any optimal process machine, is $tt$-complete. The monotone results are shown for both descriptional complexity $K_m$ and $KM$, the complexity measure derived from algorithmic probability. A distinction is drawn between two definitions of process machines that exist in the literature. For one class of process machines, designated strict process machines, it is shown that there is a universal machine whose set of non-random strings is not $tt$-complete.