Notre Dame Journal of Formal Logic

The K-Degrees, Low for K Degrees,and Weakly Low for K Sets

Joseph S. Miller

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


We call A weakly low for K if there is a c such that $K^A(\sigma)\geq K(\sigma)-c$ for infinitely many σ; in other words, there are infinitely many strings that A does not help compress. We prove that A is weakly low for K if and only if Chaitin's Ω is A-random. This has consequences in the K-degrees and the low for K (i.e., low for random) degrees. Furthermore, we prove that the initial segment prefix-free complexity of 2-random reals is infinitely often maximal. This had previously been proved for plain Kolmogorov complexity.

Article information

Notre Dame J. Formal Logic Volume 50, Number 4 (2009), 381-391.

First available: 11 February 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 68Q30
Secondary: 03D30 03D80

Martin-Lof randomness prefix-free Kolmogorov complexity


Miller , Joseph S. The K -Degrees, Low for K Degrees,and Weakly Low for K Sets. Notre Dame Journal of Formal Logic 50 (2009), no. 4, 381--391. doi:10.1215/00294527-2009-017.

Export citation


  • [1] Barmpalias, G., "Relative randomness and cardinality", forthcoming in Notre Dame Journal of Formal Logic, 2010.
  • [2] Barmpalias, G., A. E. M. Lewis, and M. Soskova, "Randomness, lowness and degrees", The Journal of Symbolic Logic, vol. 73 (2008), pp. 559--77.
  • [3] Bienvenu, L., and R. G. Downey, "Kolmogorov complexity and Solovay functions", pp. 147--58 in 26th Annual Symposium on Theoretical Aspects of Computer Science (STACS 2009), vol. 3 of Leibniz International Proceedings in Informatics, edited by S. Albers and J.-Y. Marion, Dagstuhl, Germany, 2009. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Germany.
  • [4] Calude, C. S., P. H. Hertling, B. Khoussainov, and Y. Wang, "Recursively enumerable reals and Chaitin $\Omega$ numbers", pp. 596--606 in STACS 98 (Paris, 1998), edited by M. Morvan, C. Meinel, and D. Krob, vol. 1373 of Lecture Notes in Computer Science, Springer, Berlin, 1998.
  • [5] Chaitin, G. J., "Incompleteness theorems for random reals", Advances in Applied Mathematics, vol. 8 (1987), pp. 119--46.
  • [6] Downey, R., and D. Hirschfeldt, Algorithmic Randomness and Complexity, Springer-Verlag, Berlin. forthcoming.
  • [7] Downey, R. G., D. R. Hirschfeldt, and G. LaForte, "Randomness and reducibility", pp. 316--27 in Mathematical Foundations of Computer Science, 2001 (Mariánské Lázn\u e), edited by J. Sgall, A. Pultr, and P. Kolman, vol. 2136 of Lecture Notes in Computer Science, Springer, Berlin, 2001.
  • [8] Downey, R. G., D. R. Hirschfeldt, and G. LaForte, "Randomness and reducibility", Journal of Computer and System Sciences, vol. 68 (2004), pp. 96--114.
  • [9] Downey, R. G., D. R. Hirschfeldt, J. S. Miller, and A. Nies, "Relativizing Chaitin's halting probability", Journal of Mathematical Logic, vol. 5 (2005), pp. 167--92.
  • [10] Hirschfeldt, D. R., A. Nies, and F. Stephan, "Using random sets as oracles", Journal of the London Mathematical Society. Second Series, vol. 75 (2007), pp. 610--22.
  • [11] Kautz, S., Degrees of Random Sets, Ph.D. thesis, Cornell University, Ithaca, 1991.
  • [12] Kjos-Hanssen, B., J. S. Miller, and R. Solomon, "Lowness notions, measure, and domination", in preparation.
  • [13] Kučera, A., and T. A. Slaman, "Randomness and recursive enumerability", SIAM Journal on Computing, vol. 31 (2001), pp. 199--211.
  • [14] Li, M., and P. Vitányi, An Introduction to Kolmogorov Complexity and Its Applications, Texts and Monographs in Computer Science. Springer-Verlag, New York, 1993.
  • [15] Martin-Löf, P., "Complexity oscillations in infinite binary sequences", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 19 (1971), pp. 225--30.
  • [16] Miller, J. S., "Every 2-random real is Kolmogorov random", The Journal of Symbolic Logic, vol. 69 (2004), pp. 907--13.
  • [17] Miller, J. S., and A. Nies, "Randomness and computability: Open questions", The Bulletin of Symbolic Logic, vol. 12 (2006), pp. 390--410.
  • [18] Miller, J. S., and L. Yu, "Oscillation in the initial segment complexity of random reals", forthcoming.
  • [19] 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--3210.
  • [20] Nies, A., Computability and Randomness, Oxford Logic Guides. Oxford University Press, Oxford, 2009.
  • [21] Nies, A., "Lowness properties and randomness", Advances in Mathematics, vol. 197 (2005), pp. 274--305.
  • [22] Nies, A., F. Stephan, and S. A. Terwijn, "Randomness, relativization and Turing degrees", The Journal of Symbolic Logic, vol. 70 (2005), pp. 515--35.
  • [23] Reimann, J., and T. A. Slaman, "Measures and their random reals", forthcoming.
  • [24] Schnorr, C. P., "A unified approach to the definition of random sequences", Mathematical Systems Theory, vol. 5 (1971), pp. 246--58.
  • [25] 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.
  • [26] Solovay, R. M., "Draft of paper (or series of papers) on Chaitin's work", (1975). unpublished notes, 215 pages.
  • [27] van Lambalgen, M., "The axiomatization of randomness", The Journal of Symbolic Logic, vol. 55 (1990), pp. 1143--67.
  • [28] Yu, L., D. Ding, and R. G. Downey, "The Kolmogorov complexity of random reals", Annals of Pure and Applied Logic, vol. 129 (2004), pp. 163--80.