Abstract
The randomness rate of an infinite binary sequence is characterized by the sequence of ratios between the Kolmogorov complexity and the length of the initial segments of the sequence. It is known that there is no uniform effective procedure that transforms one input sequence into another sequence with higher randomness rate. By contrast, we display such a uniform effective procedure having as input two independent sequences with positive but arbitrarily small constant randomness rate. Moreover the transformation is a truth-table reduction and the output has randomness rate arbitrarily close to 1.
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References
Bienvenu, L., Doty, D., Stephan, F.: Constructive dimension and weak truth-table degrees. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) CiE 2007. LNCS, vol. 4497, Springer, Heidelberg (to appear, 2007); Available as Technical Report arXiv:cs/0701089 ar arxiv.org
Buhrman, H., Fortnow, L., Newman, I., Vereshchagin, N.: Increasing Kolmogorov complexity. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 412–421. Springer, Heidelberg (2005)
Barak, B., Impagliazzo, R., Wigderson, A.: Extracting randomness using few independent sources. In: Proceedings of the 36th ACM Symposium on Theory of Computing, pp. 384–393 (2004)
Calude, C., Zimand, M.: Algorithmically independent sequences, CORR Technical report arxiv:0802-0487 (2008)
Doty, D.: Dimension extractors and optimal decompression. Technical Report arXiv:cs/0606078, Computing Research Repository, arXiv.org, Theory of Computing Systems (to appear, May 2007)
Fortnow, L., Hitchcock, J., Pavan, A., Vinodchandran, N.V., Wang, F.: Extracting Kolmogorov complexity with applications to dimension zero-one laws. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 335–345. Springer, Heidelberg (2006)
Lutz, J.: The dimensions of individual strings and sequences. Information and Control 187, 49–79 (2003)
Mayordomo, E.: A Kolmogorov complexity characterization of constructive Hausdorff dimension. Information Processing Letters 84, 1–3 (2002)
Miller, J., Nies, A.: Randomness and computability. Open questions. Bull. Symb. Logic 12(3), 390–410 (2006)
Nies, A., Reimann, J.: A lower cone in the wtt degrees of non-integral effective dimension. In: Proceedings of IMS workshop on Computational Prospects of Infinity, Singapore (to appear, 2006)
Reimann, J.: Computability and fractal dimension. Technical report, Universität Heidelberg, Ph.D. thesis (2004)
Ryabko, B.: Coding of combinatorial sources and Hausdorff dimension. Doklady Akademii Nauk SSR 277, 1066–1070 (1984)
Shen, A.: Algorithmic information theory and Kolmogorov complexity. Technical Report 2000-034, Uppsala Universitet (December 2000)
Staiger, L.: Constructive dimension equals Kolmogorov complexity. Information Processing Letters 93, 149–153 (2005); Preliminary version: Research Report CDMTCS-210, Univ. of Auckland (January 2003)
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Zimand, M. (2008). Two Sources Are Better Than One for Increasing the Kolmogorov Complexity of Infinite Sequences. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds) Computer Science – Theory and Applications. CSR 2008. Lecture Notes in Computer Science, vol 5010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79709-8_33
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DOI: https://doi.org/10.1007/978-3-540-79709-8_33
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