Journal of Symbolic Logic

From bi-immunity to absolute undecidability

Laurent Bienvenu, Adam R. Day, and Rupert Hölzl

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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 [2] 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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J. Symbolic Logic, Volume 78, Issue 4 (2013), 1218-1228.

First available in Project Euclid: 5 January 2014

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Bienvenu, Laurent; Day, Adam R.; Hölzl, Rupert. From bi-immunity to absolute undecidability. J. Symbolic Logic 78 (2013), no. 4, 1218--1228. doi:10.2178/jsl.7804120.

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