### 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 language | English (US) |
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Pages (from-to) | 210-217 |

Number of pages | 8 |

Journal | Proceedings of the Annual IEEE Conference on Computational Complexity |

State | Published - Jan 1 2001 |

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### All Science Journal Classification (ASJC) codes

- Software
- Theoretical Computer Science
- Computational Mathematics

### Cite this

*Proceedings of the Annual IEEE Conference on Computational Complexity*, 210-217.

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*Proceedings of the Annual IEEE Conference on Computational Complexity*, pp. 210-217.

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

Research output: Contribution to journal › Conference article

TY - JOUR

T1 - Hausdorff dimension in exponential time

AU - Ambos-Spies, K.

AU - Merkle, W.

AU - Reimann, Jan Severin

AU - Stephan, F.

PY - 2001/1/1

Y1 - 2001/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0034871624&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034871624&partnerID=8YFLogxK

M3 - Conference article

AN - SCOPUS:0034871624

SP - 210

EP - 217

JO - Proceedings of the Annual IEEE Conference on Computational Complexity

JF - Proceedings of the Annual IEEE Conference on Computational Complexity

SN - 1093-0159

ER -