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Every 2-random real is Kolmogorov random

Published online by Cambridge University Press:  12 March 2014

Joseph S. Miller
Joseph S. Miller*
Affiliation:
School of Mathematical and Computing Sciences, Victoria University, P.O. Box 600 Wellington, New Zealand, E-mail: [email protected]
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Abstract.

We study reals with infinitely many incompressible prefixes. Call A ∈ 2ωKolmogorov random if . where C denotes plain Kolmogorov complexity. This property was suggested by Loveland and studied by Martin-Löf. Schnorr and Solovay. We prove that 2-random reals are Kolmogorov random. Together with the converse—proved by Nies. Stephan and Terwijn [11]—this provides a natural characterization of 2-randomness in terms of plain complexity. We finish with a related characterization of 2-randomness.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 3 , September 2004 , pp. 907 - 913
Copyright
Copyright © Association for Symbolic Logic 2004

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References

REFERENCES

[1]Chaitin, Gregory J., A theory of program size formally identical to information theory, Journal of the Association for Computing Machinery, vol. 22 (1975), pp. 329–340.CrossRefGoogle Scholar
[2]Daley, Robert P., Complexity and randomness, Computational complexity (Courant Computer Science Symposium 7, New York University, New York, 1971), Algorithmics Press, New York, 1973, pp. 113–122.Google Scholar
[3]Downey, R. and Hirschfeldt, D., Algorithmic randomness and complexity, Springer-Verlag, Berlin, to appear.Google Scholar
[4]Kolmogorov, A. N., Three approaches to the definition of the concept “quantity of information”, Problemy Peredači Informacii, vol. 1 (1965), no. vyp. 1, pp. 3–11.Google Scholar
[5]Levin, L. A., The concept of a random sequence, Doklady Akademii Nauk SSSR, vol. 212 (1973), pp. 548–550.Google Scholar
[6]Li, M. and Vitányi, P., An introduction to Kolmogorov complexity and its applications, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1993.CrossRefGoogle Scholar
[7]Loveland, Donald W., On minimal-program complexity measures, ACM Symposium on the Theory of Computing (STOC), ACM Press, New York, 05 1969, pp. 61–78.Google Scholar
[8]Martin-Löf, Per, The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602–619.CrossRefGoogle Scholar
[9]Martin-Löf, Per, Complexity oscillations in infinite binary sequences, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 19 (1971), pp. 225–230.CrossRefGoogle Scholar
[10]Miller, Joseph S. and Yu, Liang, On initial segment complexity and degrees of randomness, in preparation.Google Scholar
[11]Nies, André, Stephan, Frank, and Terwijn, Sebastiaan A., Randomness, relativization and Turing degrees, submitted to this Journal.Google Scholar
[12]Schnorr, C. P., A unified approach to the definition of random sequences, Mathematical Systems Theory, vol. 5 (1971), pp. 246–258.CrossRefGoogle Scholar
[13]Solomonoff, R. J., A formal theory of inductive inference I and II, Information and Control, vol. 7 (1964). pp. 1–22, 224–254.CrossRefGoogle Scholar
[14]Solovay, Robert M., Draft of paper (or series of papers) on Chaitin's work, unpublished notes. 215 pages, 05 1975.Google Scholar
[15]Yu, Liang, Decheng, Ding, and Downey, Rod G., The Kolmogorov complexity of the random reals, submitted.Google Scholar