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Every 1-generic computes a properly 1-generic

Published online by Cambridge University Press:  12 March 2014

Barbara F. Csima ,
Rod Downey ,
Noam Greenberg ,
Denis R. Hirschfeldt  and
Joseph S. Miller
Barbara F. Csima
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON. N2L 3G1., Canada, E-mail: csima@math.uwaterloo.ca
Rod Downey
Affiliation:
School of Mathematics, Statistics and Computer Science, Victoria University, P.O. Box 600 Wellington. New Zealand, E-mail: Rod.Downey@vuw.ac.nz
Noam Greenberg
Affiliation:
School of Mathematics, Statistics and Computer Science, Victoria University, P.O. Box 600 Wellington. New Zealand, E-mail: greenberg@mcs.vuw.ac.nz
Denis R. Hirschfeldt
Affiliation:
Mathematics Department, University of Chicago, 5734 S. University Ave. Chicago. IL 60637., USA, E-mail: drh@math.uchicago.edu
Joseph S. Miller
Affiliation:
Mathematics Department, University of Connecticut, Storrs, CT 06269., USA, E-mail: joseph.miller@math.uconn.edu
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Abstract

A real is called properly n-generic if it is n-generic but not n + 1-generic. We show that every 1-generic real computes a properly 1-generic real. On the other hand, if m > n ≥ 2 then an m-generic real cannot compute a properly n-generic real.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 4 , December 2006 , pp. 1385 - 1393
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REEERENCES

[1]Downey, Rod and Hirschfeldt, Denis R., Algorithmic randomness and complexity, Monographs in Computer Science, Springer-Verlag, to appear, preliminary version www.mcs.vuw.ac.nz/~downey.Google Scholar
[2]Downey, Rod, Hirschfeldt, Denis R., Nies, André, and Terwijn, Sebastiaan A., Calibrating randomness, The Bulletin of Symbolic Logic, vol. 12 (2006), no. 3, pp. 411–491.CrossRefGoogle Scholar
[3]Greenberg, Noam and Montalbán, Antonio, Embedding and coding below a l-generic degree, Notre Dame Journal of Formal Logic, vol. 44 (2003), no. 4, pp. 200–216.CrossRefGoogle Scholar
[4]Haught, Christine Ann. The degrees below a l-generic degree < 0′, this Journal, vol. 51 (1986). no. 3. pp. 770–777.Google Scholar
[5]Jockusch, Carl G. Jr., Degrees of generic sets, Recursion theory: its generalisation and applications. London Mathematical Society Lecture Note Series, vol. 45, Cambridge University Press, Cambridge, 1980, pp. 110–139.CrossRefGoogle Scholar
[6]Jockusch, Carl G. Jr. and Posner, David B., Double jumps of minimal degrees, this Journal, vol. 43 (1978). no. 4, pp. 715–724.Google Scholar
[7]Kurtz, Stewart. Randomness and genericity in the degrees of unsolvability, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1981.Google Scholar
[8]Martin-Löf, Per, The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602–619.CrossRefGoogle Scholar
[9]Miller, Joseph S. and Yu, Liang, On initial segment complexity and degrees of randomness, Transactions of the American Mathematical Society, to appear.Google Scholar
[10]Nies, André, Computability and randomness, Monograph in preparation.Google Scholar
[11]Lambalgen, Michiel Van, Random sequences, Ph.D. thesis, University of Amsterdam, 1987.Google Scholar
[12]Yu, Liang, Lowness for genericity. Archive for Mathematical Logic, to appear.Google Scholar