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2013 Compressibility and Kolmogorov Complexity
Stephen Binns, Marie Nicholson
Notre Dame J. Formal Logic 54(1): 105-123 (2013). DOI: 10.1215/00294527-1731416


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.


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Stephen Binns. Marie Nicholson. "Compressibility and Kolmogorov Complexity." Notre Dame J. Formal Logic 54 (1) 105 - 123, 2013.


Published: 2013
First available in Project Euclid: 14 December 2012

zbMATH: 1271.03058
MathSciNet: MR3007965
Digital Object Identifier: 10.1215/00294527-1731416

Primary: 03D32 , 68Q30

Keywords: computability theory , effective Hausdorff dimension , effective metric , effective packing dimension , Kolmogorov complexity

Rights: Copyright © 2013 University of Notre Dame


Vol.54 • No. 1 • 2013
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