Schnorr dimension

Authors:
Rodney Downey;Wolfgang Merkle;Jan Reimann
Affiliations:
School of Mathematics, Statistics and Computer Science, Victoria University, Wellington, New Zealand;Arbeitsgruppe Mathematische Logik und Theoretische Informatik, Institut für Informatik, Fakultät für Mathematik und Informatik, Ruprecht-Karls-Universität Heidelberg, Heidelber ...;Arbeitsgruppe Mathematische Logik und Theoretische Informatik, Institut für Informatik, Fakultät für Mathematik und Informatik, Ruprecht-Karls-Universität Heidelberg, Heidelber ...
Venue:
CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
Year:
2005

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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. 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.