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arXiv · 1210.4937

From Bi-immunity to Absolute Undecidability

Laurent Bienvenu·Rupert Hölzl·Adam R. Day
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Abstract

An infinite binary sequence A is absolutely undecidable if it is impossible to compute A on a set of positions of positive upper density. Absolute undecidability is a weakening of bi-immunity. Downey, Jockusch and Schupp asked whether, unlike the case for bi-immunity, there is an absolutely undecidable set in every non-zero Turing degree. We provide a positive answer to this question by applying techniques from coding theory. We show how to use Walsh-Hadamard codes to build a truth-table functional which maps any sequence A to a sequence B, such that given any restriction of B to a set of positive upper density, one can recover A. This implies that if A is non-computable, then B is absolutely undecidable. Using a forcing construction, we show that this result cannot be strengthened in any significant fashion.

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Laurent Bienvenu, Rupert Hölzl, Adam R. Day. 2013-03-20. From Bi-immunity to Absolute Undecidability. https://arxiv.org/abs/1210.4937

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