Notre Dame Journal of Formal Logic

Compressibility and Kolmogorov Complexity

Stephen Binns and Marie Nicholson


This paper continues the study of the metric topology on 2 N that was introduced by S. Binns. This topology is induced by a directional metric where the distance from Y 2 N to X 2 N is given by

lim sup n C ( X n Y n ) n .

This definition is closely related to the notions of effective Hausdorff and packing dimensions. Here we establish that this is a path-connected topology on 2 N and that under it the functions X dim H X and X dim p X are continuous.

We also investigate the scalar multiplication operation that was introduced by Binns. The multiplication of a real X 2 N by an element α [ 0 , 1 ] represents a dilution of the information in X by a factor of α .

Our main result is to show that every regular real is the dilution of a real of Hausdorff dimension 1. That is, that the information in every regular real can be maximally compressed.

Article information

Notre Dame J. Formal Logic, Volume 54, Number 1 (2013), 105-123.

First available in Project Euclid: 14 December 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D32: Algorithmic randomness and dimension [See also 68Q30] 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.) [See also 03D32]

effective Hausdorff dimension effective packing dimension Kolmogorov complexity computability theory effective metric


Binns, Stephen; Nicholson, Marie. Compressibility and Kolmogorov Complexity. Notre Dame J. Formal Logic 54 (2013), no. 1, 105--123. doi:10.1215/00294527-1731416.

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