Notre Dame Journal of Formal Logic

Two More Characterizations of K-Triviality

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

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

K-triviality Kolmogorov complexity Martin-Löf randomness


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.

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  • [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
  • [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.