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From Bi-Immunity to Absolute Undecidability

  • Laurent Bienvenu (a1), Adam R. Day (a2) and Rupert Hölzl (a3)


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.



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[1] Arora, Sanjeev and Barak, Boaz, Computational complexity: A modern approach, Cambridge University Press, New York, USA, 2009.
[2] Downey, Rodney, Jockusch, Carl G., and Schupp, Paul, Asymptotic density and computably enumerable sets, in preparation.
[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. 472490.
[4] Jockusch, Carl G. Jr., The degrees of bi-immune sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 135140.
[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. 665694.
[6] Myasnikov, Alexei G. and Rybalov, Alexander N., Generic complexity of undecidable problems, this Journal, vol. 73 (2008), no. 2, pp. 656673.

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From Bi-Immunity to Absolute Undecidability

  • Laurent Bienvenu (a1), Adam R. Day (a2) and Rupert Hölzl (a3)


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