From Bi-immunity to Absolute Undecidability
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.
Keywords
Cite
@article{arxiv.1210.4937,
title = {From Bi-immunity to Absolute Undecidability},
author = {Laurent Bienvenu and Rupert Hölzl and Adam R. Day},
journal= {arXiv preprint arXiv:1210.4937},
year = {2013}
}