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CIE
2005
Springer

Schnorr Dimension

13 years 9 months ago
Schnorr Dimension
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, i.e. the Schnorr Hausdorff dimension of a sequence always equals its computable Hausdorff dimension. Furthermore, we give a machine characterization of Schnorr dimension, based on prefixfree machines whose domain has computable measure. Finally, we show that there exist computably enumerable sets which are Schnorr (computably) irregular: while every c.e. set has Schnorr Hausdorff dimension 0 there are c.e. sets of computable packing dimension 1, a property impossible in the case of effective (constructive) dimension, due to Barzdin¸ˇs’ Theorem. In fact, we prove that every hyperimmune Turing degree contains a set of com
Rodney G. Downey, Wolfgang Merkle, Jan Reimann
Added 26 Jun 2010
Updated 26 Jun 2010
Type Conference
Year 2005
Where CIE
Authors Rodney G. Downey, Wolfgang Merkle, Jan Reimann
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