From bi-immunity to absolute undecidability

Laurent Bienvenu (1) , Rupert Hölzl (1) , Adam R. Day (2)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
2 Department of Mathematics [Berkeley]

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.

Mots clés: Computability theory; Bi-immunity; Absolute Undecidability

Domaines: Logique [math.LO]

Laurent Bienvenu, Rupert Hölzl, Adam R. Day. From bi-immunity to absolute undecidability. 2012. ⟨hal-00743008⟩