A computable analysis of majorizing martingales

Abstract

We give upper bounds for several highness properties in computability randomness theory. First, we prove that a certain discrete covering property (which requires to almost cover all sets in a certain class) does not imply the ability to compute a 1‐random real, answering a question of Greenberg, Miller and Nies. This also implies that an infinite set of incompressible strings does not necessarily compute a 1‐random real. Second, we prove that given a homogeneous binary tree that does not admit an infinite computable path, a sequence of bounded martingales whose initial capitals tend to zero (where a martingale is bounded if its range is a bounded set of reals), there exists a martingale S majorizing infinitely many of them such that S does not compute an infinite path of the tree. This implies that: (1) a certain highness notion (which degenerates non 1‐random into noncomputably random) does not imply PA‐completeness, answering a question of Miller; (2) the computably random reducibility is not equivalent to Turing reducibility, answering a question of Nies. The proof of the second result suggests that the coding power of the universal computably enumerable martingale lies in its infinite variance.