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arXiv · 2111.01472

Some Questions of Uniformity in Algorithmic Randomness

Laurent Bienvenu·Barbara Csima·Matthew Harrison-Trainor
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Abstract

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

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Laurent Bienvenu, Barbara Csima, Matthew Harrison-Trainor. 2021-11-02. Some Questions of Uniformity in Algorithmic Randomness. https://doi.org/10.1017/jsl.2021.58

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