Translator Disclaimer
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

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 [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.

Citation

Download Citation

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

Information

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

JOURNAL ARTICLE
11 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.78 • No. 4 • December 2013
Back to Top