Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-14T16:12:45.615Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get accessRights & Permissions [Opens in a new window]

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.

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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