Notre Dame Journal of Formal Logic

Subclasses of the Weakly Random Reals

Johanna N. Y. Franklin

Abstract

The weakly random reals contain not only the Schnorr random reals as a subclass but also the weakly 1-generic reals and therefore the n-generic reals for every n. While the class of Schnorr random reals does not overlap with any of these classes of generic reals, their degrees may. In this paper, we describe the extent to which this is possible for the Turing, weak truth-table, and truth-table degrees and then extend our analysis to the Schnorr random and hyperimmune reals.

Article information

Source
Notre Dame J. Formal Logic, Volume 51, Number 4 (2010), 417-426.

Dates
First available in Project Euclid: 29 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1285765796

Digital Object Identifier
doi:10.1215/00294527-2010-026

Mathematical Reviews number (MathSciNet)
MR2741834

Zentralblatt MATH identifier
1229.03037

Subjects
Primary: 03D32: Algorithmic randomness and dimension [See also 68Q30]

Keywords
weak randomness Kurtz randomness Schnorr randomness recursive randomness genericity hyperimmune

Citation

Franklin, Johanna N. Y. Subclasses of the Weakly Random Reals. Notre Dame J. Formal Logic 51 (2010), no. 4, 417--426. doi:10.1215/00294527-2010-026. https://projecteuclid.org/euclid.ndjfl/1285765796


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References

  • [1] Demuth, O., and A. Kučera, "Remarks on $1$-genericity, semigenericity and related concepts", Commentationes Mathematicae Universitatis Carolinae, vol. 28 (1987), pp. 85--94.
  • [2] Downey, R. G., and E. J. Griffiths, "Schnorr randomness", The Journal of Symbolic Logic, vol. 69 (2004), pp. 533--54.
  • [3] Downey, R. G., and D. R. Hirschfeldt, Algorithmic Randomness and Complexity, Springer-Verlag, New York, 2010.
  • [4] Franklin, J. N. Y., "Schnorr triviality and genericity", The Journal of Symbolic Logic, vol. 75 (2010), pp. 191--207.
  • [5] Gács, P., "Every sequence is reducible to a random one", Information and Control, vol. 70 (1986), pp. 186--92.
  • [6] Jockusch, C. G., Jr., "Degrees of generic sets", pp. 110--39 in Recursion Theory: Its Generalisation and Applications (Proceedings of the Logic Colloquium, Leeds, 1979), vol. 45 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1980.
  • [7] Kautz, S. M., Degrees of Random Sets, Ph.D. thesis, Cornell University, Ithaca, 1991.
  • [8] Kjos-Hanssen, B., W. Merkle, and F. Stephan, "Kolmogorov complexity and the recursion theorem", pp. 149--61 in STACS 2006, vol. 3884 of Lecture Notes in Computer Science, Springer, Berlin, 2006.
  • [9] Kučera, A., "Measure, $\Pi^0_1$-classes and complete extensions of ${\rm PA}$", pp. 245--59 in Recursion Theory Week (Oberwolfach, 1984), vol. 1141 of Lecture Notes in Mathematics, Springer, Berlin, 1985.
  • [10] Kučera, A., "Randomness and generalizations of fixed point free functions", pp. 245--54 in Recursion Theory Week (Oberwolfach, 1989), vol. 1432 of Lecture Notes in Mathematics, Springer, Berlin, 1990.
  • [11] Kurtz, S. A., Randomness and Genericity in the Degrees of Unsolvability, Ph.D. thesis, University of Illinois, Urbana-Champaign, 1981.
  • [12] Martin-Löf, P., "The definition of random sequences", Information and Computation, vol. 9 (1966), pp. 602--19.
  • [13] Nies, A., F. Stephan, and S. A. Terwijn, "Randomness, relativization and Turing degrees", The Journal of Symbolic Logic, vol. 70 (2005), pp. 515--35.
  • [14] Odifreddi, P. G., Classical Recursion Theory. Vol. II, vol. 143 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1999.
  • [15] Odifreddi, P., Classical Recursion Theory. The Theory of Functions and Sets of Natural Numbers, vol. 125 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1989.
  • [16] Schnorr, C.-P., Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung der Wahrscheinlichkeitstheorie, vol. 218 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1971.
  • [17] Soare, R. I., Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987.
  • [18] Solovay, R. M., "A model of set-theory in which every set of reals is Lebesgue measurable", Annals of Mathematics. Second Series, vol. 92 (1970), pp. 1--56.
  • [19] Wang, Y., Randomness and Complexity, Ph.D. thesis, University of Heidelberg, Heidelberg, 1996.