Algorithmic Randomness and Turing Reducibility

Abstract

In this chapter, we look at the distribution of 1-random and n-random degrees among the Turing (and other) degrees. Among the major results we discuss are the Kučera-G´acs Theorem 8.3.2 [167, 215] that every set is computable from a 1-random set; Theorem 8.8.8, due to Barmpalias, Lewis, and Ng [24], that every PA degree is the join of two 1-random degrees; and Stillwell’s Theorem 8.15.1 [382] that the “almost all” theory of degrees is decidable. The latter uses Theorem 8.12.1, due to de Leeuw, Moore, Shannon, and Shapiro [92], that if a set is c.e. relative to positive measure many sets, then it is c.e. This result has as a corollary the fundamental result, first explicitly formulated by Sacks [341], that if a set is computable relative to positive measure many sets, then it is computable. Its proof will introduce a basic technique called the majority vote technique.