Conditional Dimension in Metric Spaces: A Natural Metric-Space Counterpart of Kolmogorov-Complexity-Based Mutual Dimension

Authors

Vladik Kreinovich, The University of Texas at El PasoFollow
Luc Longpre, The University of Texas at El PasoFollow
Olga Kosheleva, The University of Texas at El PasoFollow

Publication Date

9-2015

Comments

Technical Report: UTEP-CS-15-69

To appear in Mathematical Structures and Modeling, 2016, Vol. 37.

Abstract

It is known that dimension of a set in a metric space can be characterized in information-related terms -- in particular, in terms of Kolmogorov complexity of different points from this set. The notion of Kolmogorov complexity K(x) -- the shortest length of a program that generates a sequence x -- can be naturally generalized to conditionalKolmogorov complexity K(x:y) -- the shortest length of a program that generates x by using y as an input. It is therefore reasonable to use conditional Kolmogorov complexity to formulate a conditional analogue of dimension. Such a generalization has indeed been proposed, under the name of mutual dimension. However, somewhat surprisingly, this notion was formulated in pure Kolmogorov-complexity terms, without any analysis of possible metric-space meaning. In this paper, we describe the corresponding metric-space notion of conditional dimension -- a natural metric-space counterpart of the Kolmogorov-complexity-based mutual dimension.