Hausdorff dimension in exponential time

K. Ambos-Spies, W. Merkle, J. Reimann, F. Stephan
Proceedings 16th Annual IEEE Conference on Computational Complexity  
In this paper we investigate effective versions of Hausdorff dimension which have been recently introduced by Lutz. We focus on dimension in the class E of sets computable in linear exponential time. We determine the dimension of various classes related to fundamental structural properties including different types of autoreducibility and immunity. By a new general invariance theorem for resource-bounded dimension we show that the class of pm-complete sets for E has dimension 1 in E. Moreover,
more » ... e show that there are p-m-lower spans in E of dimension H(β) for any rational β between 0 and 1, where H(β) is the binary entropy function. This leads to a new general completeness notion for E that properly extends Lutz's concept of weak completeness. Finally we characterize resourcebounded dimension in terms of martingales with restricted betting ratios and in terms of prediction functions.
doi:10.1109/ccc.2001.933888 dblp:conf/coco/Ambos-SpiesMRS01 fatcat:evwb22hd3bca7b6tsds7rloxly