Dimension Is Compression

Abstract

Effective fractal dimension was defined by Lutz (2003) in order to quantitatively analyze the structure of complexity classes. Interesting connections of effective dimension with information theory were also found, in fact the cases of polynomial-space and constructive dimension can be precisely characterized in terms of Kolmogorov complexity, while analogous results for polynomial-time dimension haven't been found. In this paper we remedy the situation by using the natural concept of reversible time-bounded compression for finite strings. We completely characterize polynomial-time dimension in terms of polynomial-time compressors.

Pushdown Compression

“…This leaves us with the natural open question of whether each pushdown compressor can be transformed into a pushdown prediction algorithm, for which the log-loss unpredictability coincides with the compression ratio of the initial compressor, that is, whether the natural concept of pushdown dimension defined in [3] coincides with pushdown compressibility. A positive answer would get pushdown computation closer to finite-state devices, and a negative one would make it closer to polynomial-time algorithms, for which the answer is likely to be negative [12].…”

Algorithmic Fractal Dimensions in Geometric Measure Theory

“…The development of resource-bounded dimension was based on a characterization of Hausdorff dimension in terms of betting strategies, imposing different complexity constraints on those strategies to obtain the different resource-bounded dimensions. Contrary to the case of computability constraints introduced in section 3, many important resource-bounds such as polynomial time dimension do not have corresponding algorithmic information characterizations (although more elaborated compression algorithms characterizations have been obtained in [50,38]).…”

SIGACT news complexity theory column 81

“…Hitchcock [25,35] obtained a characterization of pspace-dimension in terms of space-bounded Kolmogorov complexity that is closely analogous to Mayordomo's Kolmogorov complexity characterization of constructive dimension [59]. Obtaining a data compression characterization of pdimension was more problematic, but López-Valdés and Mayordomo [45] have recently achieved this.…”