Random semicomputable reals revisited

Authors:
Laurent Bienvenu;Alexander Shen
Affiliations:
LIAFA, CNRS & Université Paris Diderot, Paris Cedex 13, France;LIRMM, CNRS & Université Montpellier 2, Montpellier Cedex 5, France
Venue:
WTCS'12 Proceedings of the 2012 international conference on Theoretical Computer Science: computation, physics and beyond
Year:
2012

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Abstract

The aim of this expository paper is to present a nice series of results, obtained in the papers of Chaitin [3], Solovay [8], Calude et al. [2], Ku $\mathrm{\check{c}}$ era and Slaman [5]. 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.