Some Questions of Uniformity in Algorithmic Randomness

TL;DR

The paper investigates limitations in uniformly constructing universal machines and related reals from Martin-Löf random left-c.e. reals, revealing inherent non-uniformities in algorithmic randomness.

Contribution

It proves that one cannot uniformly generate a universal prefix-free machine with a given halting probability from a Martin-Löf random left-c.e. real, and addresses a question on the uniform generation of certain left-c.e. reals.

Findings

Cannot uniformly produce a universal machine with a given halting probability from a Martin-Löf random left-c.e. real.

Cannot uniformly produce a left-c.e. real  such that  is neither left-c.e. nor right-c.e.

Addresses a question by Barmpalias and Lewis-Pye on uniform real construction.

Abstract

The $\Omega$ numbers-the halting probabilities of universal prefix-free machines-are known to be exactly the Martin-L{\"o}f random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-L{\"o}f random left-c.e. real $\alpha$, a universal prefix-free machine U whose halting probability is $\alpha$. We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real $\alpha$, one cannot uniformly produce a left-c.e. real $\beta$ such that $\alpha$ -- $\beta$ is neither left-c.e. nor right-c.e.