Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T11:12:51.243Z Has data issue: false hasContentIssue false
  • English
  • Français

From Bi-Immunity to Absolute Undecidability

Published online by Cambridge University Press:  12 March 2014

Laurent Bienvenu ,
Adam R. Day  and
Rupert Hölzl
Laurent Bienvenu
Affiliation:
Université Paris, 7 Denis Diderot, Case 7014, 75205 Paris Cedex 13, France, E-mail: laurent.bienvenu@liafa.univ-paris-diderot.fr
Adam R. Day
Affiliation:
Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, CA 94720-3840, USA, E-mail: adam.day@math.berkeley.edu
Rupert Hölzl
Affiliation:
Institut für Theoretische Informatik, Mathematik und Operations Research Fakultät für Informatik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany, E-mail: r@hoelzl.fr
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 4 , December 2013 , pp. 1218 - 1228
Copyright
Copyright © Association for Symbolic Logic 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Arora, Sanjeev and Barak, Boaz, Computational complexity: A modern approach, Cambridge University Press, New York, USA, 2009.Google Scholar
[2] Downey, Rodney, Jockusch, Carl G., and Schupp, Paul, Asymptotic density and computably enumerable sets, in preparation.Google Scholar
[3] Jockusch, Carl G. and Schupp, Paul, Generic computability, Turing degrees, and asymptotic density, Journal of the London Mathematical Society, vol. 85 (2012), no. 2, pp. 472–490.CrossRefGoogle Scholar
[4] Jockusch, Carl G. Jr., The degrees of bi-immune sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 135–140.CrossRefGoogle Scholar
[5] Kapovich, Ilya, Myasnikov, Alexei, Schupp, Paul, and Shpilrain, Vladimir, Generic-case complexity, decision problems in group theory, and random walks, Journal of Algebra, vol. 264 (2003), no. 2, pp. 665–694.Google Scholar
[6] Myasnikov, Alexei G. and Rybalov, Alexander N., Generic complexity of undecidable problems, this Journal, vol. 73 (2008), no. 2, pp. 656–673.Google Scholar