Kolmogorov complexity and Hausdorff dimension
Information and Computation
An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
Information and Randomness: An Algorithmic Perspective
Information and Randomness: An Algorithmic Perspective
Proceedings on Mathematical Foundations of Computer Science
The dimensions of individual strings and sequences
Information and Computation
Constructive dimension equals Kolmogorov complexity
Information Processing Letters
The extent and density of sequences within the minimal-program complexity hierarchies
Journal of Computer and System Sciences
Constructive dimension and Hausdorff dimension: the case of exact dimension
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
On oscillation-free chaitin h-random sequences
WTCS'12 Proceedings of the 2012 international conference on Theoretical Computer Science: computation, physics and beyond
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In this paper we discuss three notions of partial randomness or @e-randomness. @e-randomness should display all features of randomness in a scaled down manner. However, as Reimann and Stephan [J. Reimann, and F. Stephan, On hierarchies of randomness tests, in: Mathematical Logic in Asia, Proceedings of the 9th Asian Logic Conference, Novosibirsk, World Scientific, Singapore 2006] proved, Tadaki [K. Tadaki, A generalization of Chaitin's halting probability @Wand halting self-similar sets, Hokkaido Math. J. 31 (2002), 219-253] and Calude et al. [C.S. Calude, L. Staiger, and S.A. Terwijn. On partial randomness, Annals of Applied and Pure Logic, 138 (2006), 20-30] proposed at least three different concepts of partial randomness. We show that all of them satisfy the natural requirement that any @e-non-null set contains an @e-random infinite word. This allows us to focus our investigations on the strongest one which is based on a priori complexity. We investigate this concept of partial randomness and show that it allows-similar to the random infinite words-oscillation-free (w.r.t. to a priori complexity) @e-random infinite words if only @e is a computable number. The proof uses the dilution principle. Alternatively, for certain sets of infinite words (@w-languages) we show that their most complex infinite words are oscillation-free @e-random. Here the parameter @e is also computable and depends on the set chosen.