Hausdorff dimension in exponential time

K. Ambos-Spies, W. Merkle, Jan Severin Reimann, F. Stephan

Research output: Contribution to journalConference articlepeer-review

25 Scopus citations

Abstract

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 p-m-complete sets for E has dimension 1 in E. Moreover, we show that there are p-m-lower spans in E of dimension Η(β) for any rational β between 0 and 1, where Η(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 resource-bounded dimension in terms of martingales with restricted betting ratios and in terms of prediction functions.

Original languageEnglish (US)
Pages (from-to)210-217
Number of pages8
JournalProceedings of the Annual IEEE Conference on Computational Complexity
StatePublished - Jan 1 2001

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Computational Mathematics

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