Random semicomputable reals revisited

TL;DR

This paper provides an elementary exposition of the characterization of lower semicomputable random reals, connecting them to Chaitin Omega, Solovay reducibility, and busy beaver functions, simplifying previous complex proofs.

Contribution

It offers a simplified, accessible presentation of key results in the theory of lower semicomputable random reals, including new observations relating to busy beaver functions.

Findings

Characterization of lower semicomputable random reals as Chaitin Omega and maximal for Solovay reducibility

Elementary proofs requiring only basic algorithmic randomness knowledge

Connections established between random reals and busy beaver functions

Abstract

The aim of this expository paper is to present a nice series of results, obtained in the papers of Chaitin (1976), Solovay (1975), Calude et al. (1998), Kucera and Slaman (2001). This joint effort led to a full characterization of lower semicomputable random reals, both as those that can be expressed as a "Chaitin Omega" and those that are maximal for the Solovay reducibility. The original proofs were somewhat involved; in this paper, we present these results in an elementary way, in particular requiring only basic knowledge of algorithmic randomness. We add also several simple observations relating lower semicomputable random reals and busy beaver functions.