Randomness, Computability, and Density

We study effectively given positive reals (more specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [32] and studied by Calude, Hertling, Khoussainov, and Wang [6], Calude [2], Kuˇcera and Slaman [20], and Downey, Hirschfeldt, and LaForte [15], among others. This measure is called domination or Solovay reducibility, and is defined by saying that α dominates β if there are a constant c and a partial computable function ϕ such that for all positive rationals q < α we have ϕ(q) ↓< β and β − ϕ(q) c(α − q). The intuition is that an approximating sequence for α generates one for β whose rate of convergence is not much slower than that of the original sequence. It is not hard to show that if α dominates β then the initial segment complexity of α is at least that of β. In this paper we are concerned with structural properties of the degree structure generated by Solovay reducibility. We answer a natural questio...

• Type: Conference
• Year: 2001
• Where: STACS
• Authors: Rodney G. Downey, Denis R. Hirschfeldt, André Nies
• Keywords: Solovay, Solovay Degree, Solovay Reducibility, STACS 2001, Theoretical Computer Science