Schnorr Randomness

Journal of Symbolic Logic 69 (2):533 - 554 (2004)
Abstract
Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. After its initial development in the 1970s the notion received considerably less attention than Martin-Löf randomness, but recently interest has increased in a range of randomness concepts. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of "Schnorr reducibility" which allows us to calibrate the Schnorr complexity of reals. We define the class of "Schnorr trivial" reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2178/jsl/1082418542
Options
 Save to my reading list
Follow the author(s)
Edit this record
My bibliography
Export citation
Find it on Scholar
Mark as duplicate
Request removal from index
Revision history
Download options
Our Archive


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 30,349
Through your library
References found in this work BETA

No references found.

Add more references

Citations of this work BETA
Calibrating Randomness.Rod Downey, Denis R. Hirschfeldt, André Nies & Sebastiaan A. Terwijn - 2006 - Bulletin of Symbolic Logic 12 (3):411-491.
Randomness and Lowness Notions Via Open Covers.Laurent Bienvenu & Joseph S. Miller - 2012 - Annals of Pure and Applied Logic 163 (5):506-518.
Truth-Table Schnorr Randomness and Truth-Table Reducible Randomness.Kenshi Miyabe - 2011 - Mathematical Logic Quarterly 57 (3):323-338.

Add more citations

Similar books and articles
Subclasses of the Weakly Random Reals.Johanna N. Y. Franklin - 2010 - Notre Dame Journal of Formal Logic 51 (4):417-426.
Every 2-Random Real is Kolmogorov Random.Joseph S. Miller - 2004 - Journal of Symbolic Logic 69 (3):907-913.
Computational Randomness and Lowness.Sebastiaan A. Terwijn & Domenico Zambella - 2001 - Journal of Symbolic Logic 66 (3):1199-1205.
Van Lambalgen's Theorem and High Degrees.Johanna N. Y. Franklin & Frank Stephan - 2011 - Notre Dame Journal of Formal Logic 52 (2):173-185.
Relative Randomness and Cardinality.George Barmpalias - 2010 - Notre Dame Journal of Formal Logic 51 (2):195-205.
Relative Randomness and Real Closed Fields.Alexander Raichev - 2005 - Journal of Symbolic Logic 70 (1):319 - 330.
Degrees of Monotone Complexity.William C. Calhoun - 2006 - Journal of Symbolic Logic 71 (4):1327 - 1341.
General Random Sequences and Learnable Sequences.C. P. Schnorr & P. Fuchs - 1977 - Journal of Symbolic Logic 42 (3):329-340.
Lowness and $\Pi _{2}^{0}$ Nullsets.Rod Downey, Andre Nies, Rebecca Weber & Liang Yu - 2006 - Journal of Symbolic Logic 71 (3):1044 - 1052.
Added to PP index
2010-08-24

Total downloads
6 ( #570,144 of 2,193,597 )

Recent downloads (6 months)
1 ( #290,647 of 2,193,597 )

How can I increase my downloads?

Monthly downloads
My notes
Sign in to use this feature