Computer Science – Computational Complexity
Scientific paper
2002-11-21
Computer Science
Computational Complexity
35 pages
Scientific paper
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.
Athreya Krishna B.
Hitchcock John M.
Lutz Jack H.
Mayordomo Elvira
No associations
LandOfFree
Effective Strong Dimension, Algorithmic Information, and Computational Complexity does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Effective Strong Dimension, Algorithmic Information, and Computational Complexity, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Effective Strong Dimension, Algorithmic Information, and Computational Complexity will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-189900