Equivalent characterizations of partial randomness for a recursively enumerable real

TL;DR

This paper generalizes multiple characterizations of randomness for recursively enumerable real numbers by introducing a parameter T, providing a unified framework for partial randomness.

Contribution

It introduces a parameterized approach to characterize partial randomness, extending existing notions to a broader class of recursively enumerable reals.

Findings

Multiple equivalent characterizations of partial randomness for recursively enumerable reals.

Unified framework connecting program-size complexity, Martin-Löf tests, and other notions.

Extension of randomness concepts to a continuum parameter T in (0,1].

Abstract

A real number \alpha is called recursively enumerable if there exists a computable, increasing sequence of rational numbers which converges to \alpha. The randomness of a recursively enumerable real \alpha can be characterized in various ways using each of the notions; program-size complexity, Martin-L\"{o}f test, Chaitin's \Omega number, the domination and \Omega-likeness of \alpha, the universality of a computable, increasing sequence of rational numbers which converges to \alpha, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each of the notions above by a real number T\in(0,1]. We thus present several equivalent characterizations of partial randomness for a recursively enumerable real number.