Effective Strong Dimension, Algorithmic Information, and Computational Complexity

TL;DR

This paper introduces a new gale-based characterization of packing dimension, extends it to effective strong dimensions, and explores their connections to randomness, complexity, and data compression.

Contribution

It provides the first gale characterization of packing dimension and develops effective strong dimensions linked to various computational and informational concepts.

Findings

Gale characterization of packing dimension is dual to that of Hausdorff dimension.

Effective strong dimensions relate to randomness, Kolmogorov complexity, and computational complexity.

Develops properties and applications of effective strong dimensions in complexity theory.

Abstract

The two most important notions of fractal dimension are {\it Hausdorff dimension}, developed by Hausdorff (1919), and {\it packing dimension}, developed by Tricot (1982). Lutz (2000) has recently proven a simple characterization of Hausdorff dimension in terms of {\it gales}, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual of -- and every bit as simple as -- the gale characterization of Hausdorff dimension. Effectivizing our gale characterization of packing dimension produces a variety of {\it effective strong dimensions}, which are exact duals of the effective dimensions mentioned above. We develop the basic properties of effective strong dimensions and prove a number of results relating them to fundamental aspects of randomness, Kolmogorov complexity, prediction, Boolean circuit-size complexity, polynomial-time degrees, and data compression.