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Schnorr dimension

Published online by Cambridge University Press:  11 October 2006

RODNEY DOWNEY ,
WOLFGANG MERKLE  and
JAN REIMANN
RODNEY DOWNEY
Affiliation:
School of Mathematics, Statistics, and Computer Science, Victoria University of Wellington, Wellington, New Zealand Email: Rod.Downey@mcs.vuw.ac.nz
WOLFGANG MERKLE
Affiliation:
Institut für Informatik, Ruprecht-Karls-Uninversität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany Email: merkle@math.uni-heidelberg.dereimann@math.uni-heidelberg.de
JAN REIMANN
Affiliation:
Institut für Informatik, Ruprecht-Karls-Uninversität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany Email: merkle@math.uni-heidelberg.dereimann@math.uni-heidelberg.de
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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.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 16 , Issue 5 , October 2006 , pp. 789 - 811
Copyright
2006 Cambridge University Press

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