Randomness and Recursive Enumerability

Abstract

Abstract. One recursively enumerable real α dominates another one β if there are nondecreasing recursive sequences of rational numbers (a[n] : n ∈ ω) approximating α and (b[n] : n ∈ ω) approximating β and a positive constant C such that for all n,

On Schnorr and computable randomness, martingales, and machines

“…A real is c. e. iff it is the measure of domain of a prefix-free machine and occupy the same distinguished place in algorithmic randomness that c. e. sets do in classical computability theory. Recent work of Kučera and Slaman ( [11]) has shown that, in some sense, Ω is essentially the only random c. e. real.…”

A Note on the Differences of Computably Enumerable Reals

“…reals has grown into a significant part of modern algorithmic randomness, and is best presented in [DH10, Chapters 5 and 9]. The present section is an original presentation of some facts regarding Martin-Löf random reals that stem from [Sol75,CHKW01,KS01] and are further elaborated on in [DHN02], which are essential for the proof of Theorem 1.1. Moreover, some of these facts are not given explicitly in the sources above, but can be recovered from the proofs.…”

“…Kučera and Slaman [KS01] proved that: if (α s ), (β s ) are left-c.e. approximations to α, β respectively and if α is MartinLöf random, then lim inf…”

There are 2^{ℵ₀} many 𝐻-degrees in the random reals

“…Downey, Hirschfeldt, Nies and Stephan ( [2] and [3]) have shown that for the c.e. reals, the H-degrees form a dense upper semi-lattice with arithmetic addition inducing a join operation, and following from a result Kučera and Slaman ( [5]) that the H-degree of Chaitin's Ω is the top degree. Recently Downey and Wu [4] have shown that there are minimal pairs in this structure.…”