Lowness and Π₂⁰ nullsets
Rod Downey, Andre Nies, Rebecca Weber, and Liang Yu
Source: J. Symbolic Logic Volume 71, Issue 3
(2006), 1044-1052.
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.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1154698590
Digital Object Identifier: doi:10.2178/jsl/1154698590
Mathematical Reviews number (MathSciNet): MR2251554
Zentralblatt MATH identifier: 1112.03040
References
R. Downey, E. Griffiths, and S. Reid, On Kurtz randomness, Theoretical Computer Science, vol. 321 (2004), pp. 249--270.
Mathematical Reviews (MathSciNet): MR2076147
Digital Object Identifier: doi:10.1016/j.tcs.2004.03.055
Zentralblatt MATH: 1070.68054
R. Downey and D. Hirschfeldt Algorithmic Randomness and Complexity, Springer-Verlag, to appear. Current version available at http://www.mcs.vuw.ac.nz/~downey.,
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.
Mathematical Reviews (MathSciNet): MR2270232
P. Gács, Every set is reducible to a random one, Information and Control, vol. 70 (1986), pp. 186--192.
Mathematical Reviews (MathSciNet): MR859105
Digital Object Identifier: doi:10.1016/S0019-9958(86)80004-3
Zentralblatt MATH: 0628.03024
H. Gaifman and M. Snir, Probabilities over rich languages, testing and randomness, Journal of Symbolic Logic, vol. 47 (1982), pp. 495--548.
Mathematical Reviews (MathSciNet): MR666815
Digital Object Identifier: doi:10.2307/2273587
Project Euclid: euclid.jsl/1183741085
Zentralblatt MATH: 0501.60006
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.
Mathematical Reviews (MathSciNet): MR820784
A. Kučera and S. Terwijn, Lowness for the class of random sets, Journal of Symbolic Logic, vol. 64 (1999), no. 4, pp. 1396--1402.
Mathematical Reviews (MathSciNet): MR1780059
Digital Object Identifier: doi:10.2307/2586785
Project Euclid: euclid.jsl/1183745926
Zentralblatt MATH: 0954.68080
P. Martin-Löf, The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602--619.
Mathematical Reviews (MathSciNet): MR223179
A. Nies Computability and Randomness, to appear.,
Mathematical Reviews (MathSciNet): MR2548883
--------, Low for random sets: The story, preprint, available at http://www.cs.auckland.ac.nz/% nies.
--------, Non-cupping and randomness, Proceedings of the American Mathematical Society, to appear.
G. Sacks Degrees of Unsolvability, Princeton University Press, 1963.,
Mathematical Reviews (MathSciNet): MR186554
Zentralblatt MATH: 0143.25302
F. Stephan personal communication.,
Y. Wang Randomness and Complexity, Ph.D. thesis, University of Heidelberg, 1996.,