From Bi-immunity to Absolute Undecidability

TL;DR

This paper demonstrates that for every non-zero Turing degree, there exists an absolutely undecidable sequence, using coding theory techniques to construct such sequences and establishing their properties through forcing methods.

Contribution

It proves the existence of absolutely undecidable sequences in every non-zero Turing degree, answering a longstanding open question.

Findings

Construction of sequences using Walsh-Hadamard codes

Existence of absolutely undecidable sequences in all non-zero Turing degrees

Limitations established via forcing construction

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.