Some Questions of Uniformity in Algorithmic Randomness
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The Ω numbers-the halting probabilities of universal prefix-free machines-are known to be exactly the Martin-Löf random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-Löf 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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