Notre Dame Journal of Formal Logic

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

Joseph S. Miller


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.

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

First available in Project Euclid: 11 February 2010

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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.

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  • [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.