Hausdorff dimension in exponential time

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

Research output: Contribution to journalConference article

25 Citations (Scopus)

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

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Exponential time
Hausdorff Dimension
Invariance
Structural properties
Entropy
Completeness
Binary entropy
Entropy Function
Resources
Immunity
Martingale
Structural Properties
Prediction
Theorem
Class

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Computational Mathematics

Cite this

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Hausdorff dimension in exponential time. / Ambos-Spies, K.; Merkle, W.; Reimann, Jan Severin; Stephan, F.

In: Proceedings of the Annual IEEE Conference on Computational Complexity, 01.01.2001, p. 210-217.

Research output: Contribution to journalConference article

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