Randomness conservation inequalities; information and independence in mathematical theories
Information and Control
Kolmogorov complexity and Hausdorff dimension
Information and Computation
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
A Kolmogorov complexity characterization of constructive Hausdorff dimension
Information Processing Letters
The dimensions of individual strings and sequences
Information and Computation
Visualization 2001 Conference (Acm
Visualization 2001 Conference (Acm
Constructive Dimension and Weak Truth-Table Degrees
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
Constructive dimension equals Kolmogorov complexity
Information Processing Letters
Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
On Generating Independent Random Strings
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
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Two objects are independent if they do not affect each other. Independence is well-understood in classical information theory, but less in algorithmic information theory. Working in the framework of algorithmic information theory, the paper proposes two types of independence for arbitrary infinite binary sequences and studies their properties. Our two proposed notions of independence have some of the intuitive properties that one naturally expects. For example, for every sequence x, the set of sequences that are independent with xhas measure one. For both notions of independence we investigate to what extent pairs of independent sequences, can be effectively constructed via Turing reductions (from one or more input sequences). In this respect, we prove several impossibility results. For example, it is shown that there is no effective way of producing from an arbitrary sequence with positive constructive Hausdorff dimension two sequences that are independent (even in the weaker type of independence) and have super-logarithmic complexity. Finally, a few conjectures and open questions are discussed.