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

• Type: Conference
• Year: 2005
• Where: CIE
• Authors: Rodney G. Downey, Wolfgang Merkle, Jan Reimann
• Keywords: CIE 2005, Computable, Computable Hausdorff Dimension, Schnorr Hausdorff Dimension