December 2013 From bi-immunity to absolute undecidability
Laurent Bienvenu, Adam R. Day, Rupert Hölzl
J. Symbolic Logic 78(4): 1218-1228 (December 2013). DOI: 10.2178/jsl.7804120


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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Laurent Bienvenu. Adam R. Day. Rupert Hölzl. "From bi-immunity to absolute undecidability." J. Symbolic Logic 78 (4) 1218 - 1228, December 2013.


Published: December 2013
First available in Project Euclid: 5 January 2014

zbMATH: 1349.03044
MathSciNet: MR3156521
Digital Object Identifier: 10.2178/jsl.7804120

Rights: Copyright © 2013 Association for Symbolic Logic


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Vol.78 • No. 4 • December 2013
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