Journal of Symbolic Logic

Lowness and Π₂⁰ nullsets

Rod Downey, Andre Nies, Rebecca Weber, and Liang Yu

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We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Löf randomness.

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J. Symbolic Logic, Volume 71, Issue 3 (2006), 1044-1052.

First available in Project Euclid: 4 August 2006

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Downey, Rod; Nies, Andre; Weber, Rebecca; Yu, Liang. Lowness and Π₂⁰ nullsets. J. Symbolic Logic 71 (2006), no. 3, 1044--1052. doi:10.2178/jsl/1154698590.

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  • R. Downey, E. Griffiths, and S. Reid, On Kurtz randomness, Theoretical Computer Science, vol. 321 (2004), pp. 249--270.
  • R. Downey and D. Hirschfeldt Algorithmic Randomness and Complexity, Springer-Verlag, to appear. Current version available at,
  • S. Figueira, A. Nies, and F. Stephan, Lowness properties and approximations of the jump, Proceedings of the Workshop of Logic, Language, Information and Computation (WoLLIC), 2005, Electronic Lecture Notes in Theoretical Computer Science, vol. 143(2006), pp. 45--57.
  • P. Gács, Every set is reducible to a random one, Information and Control, vol. 70 (1986), pp. 186--192.
  • H. Gaifman and M. Snir, Probabilities over rich languages, testing and randomness, Journal of Symbolic Logic, vol. 47 (1982), pp. 495--548.
  • S. Kautz Degrees of random sets, Ph.D. thesis, Cornell University, 1991.,
  • S. Kurtz Randomness and genericity in the degrees of unsolvability, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1981.,
  • A. Kučera, Measure, $\Pi^0_1$ classes, and complete extensions of PA, Lecture Notes in Mathematics, vol. 1141, Springer, 1985, pp. 245--259.
  • A. Kučera and S. Terwijn, Lowness for the class of random sets, Journal of Symbolic Logic, vol. 64 (1999), no. 4, pp. 1396--1402.
  • P. Martin-Löf, The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602--619.
  • A. Nies Computability and Randomness, to appear.,
  • --------, Low for random sets: The story, preprint, available at nies.
  • --------, Non-cupping and randomness, Proceedings of the American Mathematical Society, to appear.
  • G. Sacks Degrees of Unsolvability, Princeton University Press, 1963.,
  • F. Stephan personal communication.,
  • Y. Wang Randomness and Complexity, Ph.D. thesis, University of Heidelberg, 1996.,