Notre Dame Journal of Formal Logic

Superhighness

Bjørn Kjos-Hanssen and Andrée Nies

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

Abstract

We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2-random. We also study the class $superhigh^\diamond$ and show that it contains some, but not all, of the noncomputable K-trivial sets.

Article information

Source
Notre Dame J. Formal Logic Volume 50, Number 4 (2009), 445-452.

Dates
First available: 11 February 2010

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1265899124

Digital Object Identifier
doi:10.1215/00294527-2009-020

Zentralblatt MATH identifier
05778810

Mathematical Reviews number (MathSciNet)
MR2598873

Subjects
Primary: 03D28
Secondary: 03D32 68Q30

Keywords
Turing degrees highness and lowness notions algorithmic randomness truth-table degrees

Citation

Kjos-Hanssen , Bjørn; Nies , Andrée. Superhighness. Notre Dame Journal of Formal Logic 50 (2009), no. 4, 445--452. doi:10.1215/00294527-2009-020. http://projecteuclid.org/euclid.ndjfl/1265899124.


Export citation

References

  • [1] Barmpalias, G., "Tracing and domination in the Turing degrees". forthcoming.
  • [2] Binns, S., B. Kjos-Hanssen, M. Lerman, and R. Solomon, "On a conjecture of Dobrinen and Simpson concerning almost everywhere domination", The Journal of Symbolic Logic, vol. 71 (2006), pp. 119--36.
  • [3] Cholak, P., N. Greenberg, and J. S. Miller, "Uniform almost everywhere domination", The Journal of Symbolic Logic, vol. 71 (2006), pp. 1057--72.
  • [4] Cole, J. A., and S. G. Simpson, "Mass problems and hyperarithmeticity", Journal of Mathematical Logic, vol. 7 (2007), pp. 125--43.
  • [5] Cooper, S. B., "Minimal degrees and the jump operator", The Journal of Symbolic Logic, vol. 38 (1973), pp. 249--71.
  • [6] Dobrinen, N. L., and S. G. Simpson, "Almost everywhere domination", The Journal of Symbolic Logic, vol. 69 (2004), pp. 914--22.
  • [7] Friedberg, R., "A criterion for completeness of degrees of unsolvability", The Journal of Symbolic Logic, vol. 22 (1957), pp. 159--60.
  • [8] Jockusch, C. G., Jr., and R. I. Soare, "$\Pi \sp{0}\sb{1}$ classes and degrees of theories", Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33--56.
  • [9] Martin, D. A., "Classes of recursively enumerable sets and degrees of unsolvability", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295--310.
  • [10] Mohrherr, J., "Density of a final segment of the truth-table degrees", Pacific Journal of Mathematics, vol. 115 (1984), pp. 409--19.
  • [11] Nies, A., "Superhighness and strong jump traceability". forthcoming in Proceedings of the ICALP 2009.
  • [12] Nies, A., Computability and Randomness, Oxford University Press, Oxford, 2009.
  • [13] Schwarz, S., Index Sets of Computably Enumerable Sets, Quotient Lattices, and Computable Linear Orderings, Ph.D. thesis, University of Chicago, Chicago, 1982.
  • [14] Simpson, S. G., "Almost everywhere domination and superhighness", Mathematical Logic Quarterly, vol. 53 (2007), pp. 462--82.