On partial randomness

If x = x1x2 ··· xn ··· is ar andom sequence, then the sequence y = 0x10x2 ··· 0xn ··· is clearly not random; however, y seems to be “about half random”. L. Staiger [Kolmogorov complexity and Hausdorff dimension, Inform. and Comput. 103 (1993) 159–194 and A tight upper bound on Kolmogorov complexity and uniformly optimal prediction, Theory Comput. Syst. 31 (1998) 215–229] and K. Tadaki [A generalisation of Chaitin’s halting probability Ω an dh alting self-similar sets, Hokkaido Math. J. 31 (2002) 219–253] have studied the degree of randomness of sequences or reals by measuring their “degree o fc ompression”. This line of study leads to various definitions of partial randomness. In this paper we explore some relations between these definitions. Among other results we obtain a characterisation of Σ1-dimension (as defined by Schnorr and Lutz in terms of martingales) in terms of strong Martin-Lof e-tests (a variant of Martin-Lof tests), and we show that e-randomness for e ∈ (0, 1) is different (and more difficult to study) than the classical 1-randomness. © 2005 Elsevier B.V. All rights reserved.

[1]  F. Hausdorff.  Dimension und äußeres Maß , 1918 .

[2]  Cristian S. Calude.  Information and Randomness , 2002, Texts in Theoretical Computer Science An EATCS Series.

[3]  Alexander Shen,et al.  Relations between varieties of kolmogorov complexities , 1996 .

[4]  Ludwig Staiger,et al.  Kolmogorov Complexity and Hausdorff Dimension , 1989, FCT.

[5]  C. Schnorr.  Zufälligkeit und Wahrscheinlichkeit , 1971 .

[6]  Robert P. Daley.  The Extent and Density of Sequences within the Minimal-Program Complexity Hierarchies , 1974, J. Comput. Syst. Sci..

[7]  Kohtaro Tadaki,et al.  A Generalization of Chaitin's Halting Probability \Omega and Halting Self-Similar Sets , 2002, ArXiv.

[8]  Jack H. Lutz,et al.  The dimensions of individual strings and sequences , 2002, Inf. Comput..

[9]  Ludwig Staiger,et al.  A Tight Upper Bound on Kolmogorov Complexity and Uniformly Optimal Prediction , 1998, Theory of Computing Systems.

[10]  Sebastiaan A. Terwijn,et al.  Complexity and Randomness , 2003 .

[11]  Gregory J. Chaitin,et al.  Algorithmic Information Theory , 1987, IBM J. Res. Dev..

[12]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[13]  André Nies,et al.  Randomness, relativization and Turing degrees , 2005, J. Symb. Log..

[14]  Cristian S. Calude.  Information and Randomness: An Algorithmic Perspective , 1994 .

[15]  Cristian S. Calude.  Information and Randomness , 1994, Monographs in Theoretical Computer Science An EATCS Series.

[16]  Gregory J. Chaitin,et al.  Algorithmic Information Theory , 1987, IBM J. Res. Dev..

[17]  Frank Stephan,et al.  Hausdorff dimension in exponential time , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[18]  Alexander Shen,et al.  Relations between varieties of kolmogorov complexities , 1996, Mathematical systems theory.

[19]  Jean-Luc Ville.  Étude critique de la notion de collectif , 1939 .

[20]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .