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Schnorr triviality and genericity

Published online by Cambridge University Press:  12 March 2014

Johanna N.Y. Franklin*
Affiliation:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada, E-mail: jfranklin@math.uwaterloo.ca

Abstract

We study the connection between Schnorr triviality and genericity. We show that while no 2-generic is Turing equivalent to a Schnorr trivial and no 1-generic is tt-equivalent to a Schnorr trivial, there is a 1-generic that is Turing equivalent to a Schnorr trivial. However, every such 1-generic must be high. As a corollary, we prove that not all K-trivials are Schnorr trivial. We also use these techniques to extend a previous result and show that the bases of cones of Schnorr trivial Turing degrees are precisely those whose jumps are at least 0″.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2010

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References

[1] Chaitin, Gregory J., A theory of program size formally identical to information theory, Journal of the Association for Computing Machinery, vol. 22 (1975), pp. 329340.CrossRefGoogle Scholar
[2] Downey, Rod, Griffiths, Evan, and Laforte, Geoffrey, On Schnorr and computable randomness, martingales, and machines, Mathematical Logic Quarterly, vol. 50 (2004), no. 6, pp. 613627.CrossRefGoogle Scholar
[3] Downey, Rod, Hirschfeldt, Denis R., Nies, Andre, and Terwijn, Sebastiaan A., Calibrating randomness, The Bulletin of Symbolic Logic, vol. 12 (2006), no. 3, pp. 411491.CrossRefGoogle Scholar
[4] Downey, Rod G., Hirschfeldt, Denis R., Nies, Andre, and Stephan, Frank, Trivial reals, Proceedings of the 7th and 8th Asian Logic Conferences, Singapore University Press, Singapore, 2003, pp. 103131.CrossRefGoogle Scholar
[5] Downey, Rodney G. and Griffiths, Evan J., Schnorr randomness, this Journal, vol. 69 (2004), no. 2, pp. 533554.Google Scholar
[6] Franklin, Johanna N. Y., Schnorr trivial reals: A construction, Archive for Mathematical Logic, vol. 46 (2008), no. 7–8, pp. 665678.CrossRefGoogle Scholar
[7] Franklin, Johanna N. Y. and Stephan, Frank, Schnorr trivial sets and truth-table reducibility, Technical Report TRA3/08, School of Computing, National University of Singapore, 2008.Google Scholar
[8] Kummer, Martin, A proof of Beigel's cardinality conjecture, this Journal, vol. 57 (1992), no. 2, pp. 677681.Google Scholar
[9] Martin, Donald A., Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295310.CrossRefGoogle Scholar
[10] Nies, André, Lowness properties and randomness, Advances in Mathematics, vol. 197 (2005), no. 1, pp. 274305.CrossRefGoogle Scholar
[11] Odifreddi, P. G., Classical recursion theory. Volume II, Studies in Logic and the Foundations of Mathematics, vol. 143, North-Holland Publishing Co., Amsterdam, 1999.Google Scholar
[12] Schnorr, C.-P., Zufälligkeit und Wahrscheintichkeit, Lecture Notes in Mathematics, vol. 218, Springer-Verlag, Heidelberg, 1971.CrossRefGoogle Scholar
[13] Soare, Robert I., Recursively enumerable sets and degrees. Perspectives in Mathematical Logic, Springer-Verlag, 1987.CrossRefGoogle Scholar
[14] Zambella, Domenico, On sequences with simple initial segments, Technical Report ILLC ML-1990-05, University of Amsterdam, 1990.Google Scholar
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