Schnorr dimension

Rodney Downey, Wolfgang Merkle, Jan Severin Reimann

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Following Lutz's approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr's concept(s) of randomness. In contrast to computable randomness and Schnorr randomness, the dimension concepts defined via computable martingales and Schnorr tests coincide, that is, the Schnorr Hausdorff dimension of a sequence always equals its computable Hausdorff dimension. Furthermore, we give a machine characterisation of the Schnorr dimension, based on prefix-free machines whose domain has computable measure. Finally, we show that there exist computably enumerable sets that are Schnorr (computably) irregular: while every c.e. set has Schnorr Hausdorff dimension 0, there are c.e. sets of computable packing dimension 1, which is, from Barzdiņš Theorem, an impossible property for the case of effective (constructive) dimension. In fact, we prove that every hyperimmune Turing degree contains a set of computable packing dimension 1.

Original languageEnglish (US)
Pages (from-to)789-811
Number of pages23
JournalMathematical Structures in Computer Science
Volume16
Issue number5
DOIs
StatePublished - Oct 1 2006

Fingerprint

Hausdorff Dimension
Randomness
Packing Dimension
Computably Enumerable Set
Prefix-free
Turing Degrees
Martingale
Irregular
Theorem
Concepts

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)
  • Computer Science Applications

Cite this

Downey, Rodney ; Merkle, Wolfgang ; Reimann, Jan Severin. / Schnorr dimension. In: Mathematical Structures in Computer Science. 2006 ; Vol. 16, No. 5. pp. 789-811.
@article{03112aa32ef24c6fbb9b8631f4337295,
title = "Schnorr dimension",
abstract = "Following Lutz's approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr's concept(s) of randomness. In contrast to computable randomness and Schnorr randomness, the dimension concepts defined via computable martingales and Schnorr tests coincide, that is, the Schnorr Hausdorff dimension of a sequence always equals its computable Hausdorff dimension. Furthermore, we give a machine characterisation of the Schnorr dimension, based on prefix-free machines whose domain has computable measure. Finally, we show that there exist computably enumerable sets that are Schnorr (computably) irregular: while every c.e. set has Schnorr Hausdorff dimension 0, there are c.e. sets of computable packing dimension 1, which is, from Barzdiņš Theorem, an impossible property for the case of effective (constructive) dimension. In fact, we prove that every hyperimmune Turing degree contains a set of computable packing dimension 1.",
author = "Rodney Downey and Wolfgang Merkle and Reimann, {Jan Severin}",
year = "2006",
month = "10",
day = "1",
doi = "10.1017/S0960129506005469",
language = "English (US)",
volume = "16",
pages = "789--811",
journal = "Mathematical Structures in Computer Science",
issn = "0960-1295",
publisher = "Cambridge University Press",
number = "5",

}

Schnorr dimension. / Downey, Rodney; Merkle, Wolfgang; Reimann, Jan Severin.

In: Mathematical Structures in Computer Science, Vol. 16, No. 5, 01.10.2006, p. 789-811.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Schnorr dimension

AU - Downey, Rodney

AU - Merkle, Wolfgang

AU - Reimann, Jan Severin

PY - 2006/10/1

Y1 - 2006/10/1

N2 - Following Lutz's approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr's concept(s) of randomness. In contrast to computable randomness and Schnorr randomness, the dimension concepts defined via computable martingales and Schnorr tests coincide, that is, the Schnorr Hausdorff dimension of a sequence always equals its computable Hausdorff dimension. Furthermore, we give a machine characterisation of the Schnorr dimension, based on prefix-free machines whose domain has computable measure. Finally, we show that there exist computably enumerable sets that are Schnorr (computably) irregular: while every c.e. set has Schnorr Hausdorff dimension 0, there are c.e. sets of computable packing dimension 1, which is, from Barzdiņš Theorem, an impossible property for the case of effective (constructive) dimension. In fact, we prove that every hyperimmune Turing degree contains a set of computable packing dimension 1.

AB - Following Lutz's approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr's concept(s) of randomness. In contrast to computable randomness and Schnorr randomness, the dimension concepts defined via computable martingales and Schnorr tests coincide, that is, the Schnorr Hausdorff dimension of a sequence always equals its computable Hausdorff dimension. Furthermore, we give a machine characterisation of the Schnorr dimension, based on prefix-free machines whose domain has computable measure. Finally, we show that there exist computably enumerable sets that are Schnorr (computably) irregular: while every c.e. set has Schnorr Hausdorff dimension 0, there are c.e. sets of computable packing dimension 1, which is, from Barzdiņš Theorem, an impossible property for the case of effective (constructive) dimension. In fact, we prove that every hyperimmune Turing degree contains a set of computable packing dimension 1.

UR - http://www.scopus.com/inward/record.url?scp=33750738235&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750738235&partnerID=8YFLogxK

U2 - 10.1017/S0960129506005469

DO - 10.1017/S0960129506005469

M3 - Article

VL - 16

SP - 789

EP - 811

JO - Mathematical Structures in Computer Science

JF - Mathematical Structures in Computer Science

SN - 0960-1295

IS - 5

ER -