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Extracting information is hard: a Turing degree of non-integral effective Hausdorff dimension. (English) Zbl 1214.03030
The major result in the paper under review is that there is a $$\Delta^0_2$$ real with effective Hausdorff dimension $$\frac{1}{2}$$ that does not compute a real of higher dimension.
Reviewer: Liang Yu (Nanjing)

##### MSC:
 03D32 Algorithmic randomness and dimension 68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
##### Keywords:
effective Hausdorff dimension; Turing degrees
Full Text:
##### References:
 [1] Athreya, Krishna B.; Hitchcock, John M.; Lutz, Jack H.; Mayordomo, Elvira, Effective strong dimension in algorithmic information and computational complexity, (), 632-643 · Zbl 1122.68068 [2] Bienvenu, Laurent; Doty, David; Stephan, Frank, Constructive dimension and Turing degrees, Theory comput. syst., 45, 4, 740-755, (2009) · Zbl 1183.68281 [3] Chaitin, Gregory J., A theory of program size formally identical to information theory, J. assoc. comput. Mach., 22, 329-340, (1975) · Zbl 0309.68045 [4] R. Downey, D. Hirschfeldt, Algorithmic Randomness and Complexity, Springer-Verlag, Berlin, in press. · Zbl 1221.68005 [5] Downey, Rod; Greenberg, Noam, Turing degrees of reals of positive effective packing dimension, Inform. process. lett., 108, 5, 298-303, (2008) · Zbl 1191.68304 [6] Downey, Rod; Hirschfeldt, Denis R.; Nies, André; Terwijn, Sebastiaan A., Calibrating randomness, Bull. symbolic logic, 12, 3, 411-491, (2006) · Zbl 1113.03037 [7] Fortnow, Lance; Hitchcock, John M.; Pavan, A.; Vinodchandran, N.V.; Wang, Fengming, Extracting Kolmogorov complexity with applications to dimension zero-one laws, (), 335-345 · Zbl 1223.68060 [8] Noam Greenberg, Joseph S. Miller, Diagonally non-recursive functions and effective Hausdorff dimension, Bull. London Math. Soc., in press. · Zbl 1225.03055 [9] Kjos-Hanssen, Bjørn; Merkle, Wolfgang; Stephan, Frank, Kolmogorov complexity and the recursion theorem, (), 149-161 · Zbl 1137.03026 [10] Levin, L.A., Laws of information conservation (non-growth) and aspects of the foundation of probability theory, Probl. inf. transm., 10, 3, 206-210, (1974) [11] Li, M.; Vitányi, P., An introduction to Kolmogorov complexity and its applications, Texts monogr. comput. sci., (1993), Springer-Verlag New York · Zbl 0805.68063 [12] Lutz, Jack H., Gales and the constructive dimension of individual sequences, (), 902-913 · Zbl 0973.68087 [13] Lutz, Jack H., Effective fractal dimensions, Math. log. Q., 51, 1, 62-72, (2005) · Zbl 1058.03044 [14] Martin-Löf, Per, The definition of random sequences, Inform. control, 9, 602-619, (1966) · Zbl 0244.62008 [15] Mayordomo, Elvira, A Kolmogorov complexity characterization of constructive Hausdorff dimension, Inform. process. lett., 84, 1, 1-3, (2002) · Zbl 1045.68570 [16] Miller, Joseph S.; Nies, André, Randomness and computability: open questions, Bull. symbolic logic, 12, 3, 390-410, (2006) · Zbl 1169.03033 [17] Nies, André, Computability and randomness, Oxford logic guides, vol. 51, (2009), Oxford University Press Oxford · Zbl 1169.03034 [18] Nies, André; Reimann, Jan, A lower cone in the wtt degrees of non-integral effective dimension, (), 249-260 · Zbl 1158.03026 [19] J. Reimann, Computability and fractal dimension, Doctoral dissertation, Universität Heidelberg, 2004, available at http://math.uni-heidelberg.de/logic/reimann/publications/phdthesis.pdf. · Zbl 1080.03031 [20] Reimann, Jan, Effectively closed sets of measures and randomness, Ann. pure appl. logic, 156, 1, 170-182, (2008) · Zbl 1153.03021 [21] Ryabko, B.Ya., Coding of combinatorial sources and Hausdorff dimension, Dokl. akad. nauk SSSR, 277, 5, 1066-1070, (1984) · Zbl 0581.94007 [22] Sántha, Miklós; Vazirani, Umesh V., Generating quasirandom sequences from semirandom sources, Twenty-fifth annual symposium on foundations of computer science, Singer Island, Fla., 1984, J. comput. system sci., 33, 1, 75-87, (1986) · Zbl 0612.94004 [23] Shaltiel, Ronen, Recent developments in extractors, () · Zbl 1082.68124 [24] Vazirani, Umesh V., Strong communication complexity or generating quasirandom sequences from two communicating semirandom sources, Combinatorica, 7, 4, 375-392, (1987) · Zbl 0643.94002 [25] Vereshchagin, Nikolai K.; Vyugin, Michael V., Independent minimum length programs to translate between given strings, Theoret. comput. sci., 271, 1-2, 131-143, (2002), Kolmogorov complexity · Zbl 0992.68083 [26] von Neumann, John, Various techniques for use in connection with random digits, (), 768-770 [27] Zimand, Marius, Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences, corr, (2007), Informal publication · Zbl 1143.68020
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