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Every 2-random real is Kolmogorov random

  • Joseph S. Miller (a1)
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.

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[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. 329340.
[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. 113122.
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[10]Miller, Joseph S. and Yu, Liang, On initial segment complexity and degrees of randomness, in preparation.
[11]Nies, André, Stephan, Frank, and Terwijn, Sebastiaan A., Randomness, relativization and Turing degrees, submitted to this Journal.
[12]Schnorr, C. P., A unified approach to the definition of random sequences, Mathematical Systems Theory, vol. 5 (1971), pp. 246258.
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[14]Solovay, Robert M., Draft of paper (or series of papers) on Chaitin's work, unpublished notes. 215 pages, 05 1975.
[15]Yu, Liang, Decheng, Ding, and Downey, Rod G., The Kolmogorov complexity of the random reals, submitted.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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