Schnorr Dimension

  • Rodney Downey
  • Wolfgang Merkle
  • Jan Reimann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3526)


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. Furthermore, we give a machine characterization of Schnorr dimension, based on prefix free machines whose domain has computable measure. Finally, we show that there exist computably enumerable sets which are Schnorr irregular: while every c.e. set has Schnorr Hausdorff dimension 0 there are c.e. sets of Schnorr packing dimension 1, a property impossible in the case of effective (constructive) dimension, due to Barzdin’s Theorem.


Kolmogorov Complexity Packing Dimension Constructive Dimension Algorithmic Randomness Free Machine 
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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Rodney Downey
    • 1
  • Wolfgang Merkle
    • 2
  • Jan Reimann
    • 2
  1. 1.School of Mathematics, Statistics and Computer ScienceVictoria UniversityWellingtonNew Zealand
  2. 2.Arbeitsgruppe Mathematische Logik und Theoretische Informatik, Institut für Informatik, Fakultät für Mathematik und InformatikRuprecht-Karls-Universität HeidelbergHeidelbergGermany

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