Random Semicomputable Reals Revisited

The aim of this paper is to present a nice series of results, obtained in the papers of Chaitin [2], Solovay [5], Calude et al. [1], Kucera and Slaman [3]. 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 Solovay reducibility. The original proofs were somewhat involved; in this paper, we present these results in an elementary way, in particular requiring no prior knowledge of algorithmic randomness. We add also several simple observations relating lower semicomputable random reals and busy beaver functions. 1 Lower semicomputable reals and the 1-relation A real number α is lower semicomputable if it is a limit of a computable increasing sequence of rational numbers. (Equivalent definition: if the set of all rational numbers less than α is enumerable). There exist lower semicomputable but not computable reals. Corresponding sequences of rational numbers have non-computable convergence (there is no algorithm that produces N(ε) given ε). We want to classify computable sequences according to their convergence speed and formalize the intuitive idea “one sequence converges better (not worse) than the other one”. Definition 1 Let ai→ α and b j→ β be two computable strictly increasing sequences. We say that (ai) [resp. (bi)] is a computable approximation from below of α [resp. of β ]. We say that the approximation an→ α converges “better” (not worse) than the approximation bn→ β if there exists a total computable function h such that