Notre Dame Journal of Formal Logic

Two More Characterizations of K-Triviality

Noam Greenberg, Joseph S. Miller, Benoit Monin, and Daniel Turetsky

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 give two new characterizations of K-triviality. We show that if for all Y such that Ω is Y-random, Ω is (YA)-random, then A is K-trivial. The other direction was proved by Stephan and Yu, giving us the first titular characterization of K-triviality and answering a question of Yu. We also prove that if A is K-trivial, then for all Y such that Ω is Y-random, (YA)LRY. This answers a question of Merkle and Yu. The other direction is immediate, so we have the second characterization of K-triviality.

The proof of the first characterization uses a new cupping result. We prove that if ALRB, then for every set X there is a B-random set Y such that X is computable from YA.

Article information

Source
Notre Dame J. Formal Logic, Volume 59, Number 2 (2018), 189-195.

Dates
Received: 22 March 2015
Accepted: 9 May 2015
First available in Project Euclid: 2 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1517540521

Digital Object Identifier
doi:10.1215/00294527-2017-0021

Mathematical Reviews number (MathSciNet)
MR3778306

Zentralblatt MATH identifier
06870287

Subjects
Primary: 03D32: Algorithmic randomness and dimension [See also 68Q30]
Secondary: 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.) [See also 03D32]

Keywords
K-triviality Kolmogorov complexity Martin-Löf randomness

Citation

Greenberg, Noam; Miller, Joseph S.; Monin, Benoit; Turetsky, Daniel. Two More Characterizations of K -Triviality. Notre Dame J. Formal Logic 59 (2018), no. 2, 189--195. doi:10.1215/00294527-2017-0021. https://projecteuclid.org/euclid.ndjfl/1517540521


Export citation

References

  • [1] Chaitin, G. J., “Algorithmic information theory,” IBM Journal of Research and Development, vol. 21 (1977), pp. 350–59.
  • [2] Day, A. R., and J. S. Miller, “Cupping with random sets,” Proceedings of the American Mathematical Society, vol. 142 (2014), pp. 2871–79.
  • [3] Downey, R., and D. R. Hirschfeldt, Algorithmic Randomness and Complexity, Theory and Applications of Computability, Springer, New York, 2010.
  • [4] Downey, R., D. R. Hirschfeldt, J. S. Miller, and A. Nies, “Relativizing Chaitin’s halting probability,” Journal of Mathematical Logic, vol. 5 (2005), pp. 167–92.
  • [5] Gács, P., “Every sequence is reducible to a random one,” Information and Control, vol. 70 (1986), pp. 186–92.
  • [6] Hirschfeldt, D. R., A. Nies, and F. Stephan, “Using random sets as oracles,” Journal of the London Mathematical Society (2), vol. 75 (2007), pp. 610–22.
  • [7] Hölzl, R., and A. Nies, “CCR 2014: Open questions,” in Logic Blog, 2014, Part 1, Section 1, edited by A. Nies, available at http://arxiv.org/abs/1504.08163.
  • [8] Kjos-Hanssen, B., “Low for random reals and positive-measure domination,” Proceedings of the American Mathematical Society, vol. 135 (2007), pp. 3703–9.
  • [9] Kučera, A., “Measure, $\Pi^{0}_{1}$-classes and complete extensions of $\mathrm{PA}$,” pp. 245–59 in Recursion Theory Week (Oberwolfach, 1984), vol. 1141 of Lecture Notes in Mathematics, Springer, Berlin, 1985.
  • [10] Miller, J. S., and L. Yu, “On initial segment complexity and degrees of randomness,” Transactions of the American Mathematical Society, vol. 360 (2008), pp. 3193–210.
  • [11] Nies, A., “Lowness properties and randomness,” Advances in Mathematics, vol. 197 (2005), pp. 274–305.
  • [12] Nies, A., Computability and Randomness, vol. 51 of Oxford Logic Guides, Oxford University Press, Oxford, 2009.
  • [13] Posner, D. B., and R. W. Robinson, “Degrees joining to $\mathbf{0}^{\prime}$,” Journal of Symbolic Logic, vol. 46 (1981), pp. 714–22.
  • [14] Reimann, J., and T. A. Slaman, “Measures and their random reals,” Transactions of the American Mathematical Society, vol. 367 (2015), pp. 5081–97.
  • [15] Simpson, S. G., and F. Stephan, “Cone avoidance and randomness preservation,” Annals of Pure and Applied Logic, vol. 166 (2015), pp. 713–28.
  • [16] Solovay, R. M., unpublished notes, May 1975.
  • [17] van Lambalgen, M., “The axiomatization of randomness,” Journal of Symbolic Logic, vol. 55 (1990), pp. 1143–67.
  • [18] Yu, L., “Characterizing strong randomness via Martin-Löf randomness,” Annals of Pure and Applied Logic, vol. 163 (2012), pp. 214–24.