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arXiv · 2610.39174

Martin-Löf randomness and rK-reducibility

George Barmpalias·Lu Liu
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Abstract

Can random real carry arbitrary information. In terms of (Turing) computability, Kučera and Gács addressed this question in affirmative. i.e., Every real $x\in 2^ω$ is computable from some Martin-Löf random real. This result motivated quantitative versions of coding into random reals. i.e., How many bits of the random oracle are needed to compute the first $n$ bits of $x$. Barmpalias and Lewis-Pye obtained a coding that uses $n+O(\log n)$ many bits. Is it possible for the coding to use $n$ bits exactly? This is equivalent to say, if every real is $rK$-reducible to a Martin-Löf random real. In this paper, we prove that such a coding does not exist. i.e., There exists some real that is not $rK$-reducible to any Martin-Löf random real.

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George Barmpalias, Lu Liu. 2026-10-29. Martin-Löf randomness and rK-reducibility. https://arxiv.org/abs/2610.39174

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