Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 54, Number 1 (2013), 105-123.
Compressibility and Kolmogorov Complexity
This paper continues the study of the metric topology on that was introduced by S. Binns. This topology is induced by a directional metric where the distance from to is given by
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 and that under it the functions and are continuous.
We also investigate the scalar multiplication operation that was introduced by Binns. The multiplication of a real by an element represents a dilution of the information in by a factor of .
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.
Notre Dame J. Formal Logic, Volume 54, Number 1 (2013), 105-123.
First available in Project Euclid: 14 December 2012
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Binns, Stephen; Nicholson, Marie. Compressibility and Kolmogorov Complexity. Notre Dame J. Formal Logic 54 (2013), no. 1, 105--123. doi:10.1215/00294527-1731416. https://projecteuclid.org/euclid.ndjfl/1355494526