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Lowness and nullsets

Published online by Cambridge University Press:  12 March 2014

Rod Downey ,
Andre Nies ,
Rebecca Weber  and
Liang Yu
Rod Downey
Affiliation:
Victoria University, School of Mathematics, Statistics and Computer Science, P.O. Box 600, Wellington, New Zealand.E-mail:[email protected]
Andre Nies
Affiliation:
The University of Auckland, Department of Computer Science, Private Bag 92019, Auckland, New Zealand.E-mail:[email protected]
Rebecca Weber
Affiliation:
Dartmouth College, Department of Mathematics, 6188 Bradley Hall, Hanover, NH 03755, USA.E-mail:[email protected]
Liang Yu
Affiliation:
National University of Singapore, Department of Mathematics, Lower Kent Ridge Road, Singapore 117543, Singapore.E-mail:[email protected]
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Abstract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 3 , September 2006 , pp. 1044 - 1052
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Downey, R., Griffiths, E., and Reid, S., On Kurtz randomness, Theoretical Computer Science, vol. 321 (2004), pp. 249–270.CrossRefGoogle Scholar
[2]Downey, R. and Hirschfeldt, D., Algorithmic Randomness and Complexity, Springer-Verlag, to appear. Current version available at http://www.mcs.vuw.ac.nz/~downey.Google Scholar
[3]Figueira, S., Nies, A., and Stephan, F., 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.Google Scholar
[4]GÁcs, P., Every set is reducible to a random one, Information and Control, vol. 70 (1986), pp. 186–192.CrossRefGoogle Scholar
[5]Gaifman, H. and Snir, M., Probabilities over rich languages, testing and randomness, this Journal, vol. 47 (1982), pp. 495–548.Google Scholar
[6]Kautz, S., Degrees of random sets, Ph.D. thesis, Cornell University, 1991.Google Scholar
[7]Kurtz, S., Randomness and genericity in the degrees of unsolvability, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1981.Google Scholar
[8]Kučera, A., Measure, classes, and complete extensions of PA, Lecture Notes in Mathematics, vol. 1141, Springer, 1985, pp. 245–259.Google Scholar
[9]Kučera, A. and Terwijn, S., Lowness for the class of random sets, this Journal, vol. 64 (1999), no. 4, pp. 1396–1402.Google Scholar
[10]Martin-Löf, P., The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602–619.CrossRefGoogle Scholar
[11]Nies, A., Computability and Randomness, to appear.Google Scholar
[12]Nies, A., Low for random sets: The story, preprint, available at http://www.cs.auckland.ac.nz/~nies.Google Scholar
[13]Nies, A., Non-cupping and randomness, Proceedings of the American Mathematical Society, to appear.Google Scholar
[14]Sacks, G., Degrees of Unsolvability, Princeton University Press, 1963.Google Scholar
[15]Stephan, F., personal communication.Google Scholar
[16]Wang, Y., Randomness and Complexity, Ph.D. thesis, University of Heidelberg, 1996.Google Scholar