Schnorr dimension

Rodney Downey, Wolfgang Merkle, Jan Reimann

Research output: Contribution to journalConference articlepeer-review

4 Scopus citations

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 denned 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.

Original languageEnglish (US)
Pages (from-to)96-105
Number of pages10
JournalLecture Notes in Computer Science
Volume3526
DOIs
StatePublished - Jan 1 2005
EventFirst Conference on Computability in Europe, CiE 2005: New Computational Paradigms - Amsterdam, Netherlands
Duration: Jun 8 2005Jun 12 2005

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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