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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: [email protected]
Adam R. Day
Affiliation:
Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, CA 94720-3840, USA, E-mail: [email protected]
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: [email protected]
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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

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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