Randomness for non-computable measures
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- by Adam R. Day and Joseph S. Miller PDF
- Trans. Amer. Math. Soc. 365 (2013), 3575-3591 Request permission
Abstract:
Different approaches have been taken to defining randomness for non-computable probability measures. We will explain the approach of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gács, Hoyrup and Rojas. We will show that these approaches are fundamentally equivalent.
Having clarified what it means to be random for a non-computable probability measure, we turn our attention to Levin’s neutral measures, for which all sequences are random. We show that every PA degree computes a neutral measure. We also show that a neutral measure has no least Turing degree representation and explain why the framework of the continuous degrees (a substructure of the enumeration degrees studied by Miller) can be used to determine the computational complexity of neutral measures. This allows us to show that the Turing ideals below neutral measures are exactly the Scott ideals. Since $X\in 2^\omega$ is an atom of a neutral measure $\mu$ if and only if it is computable from (every representation of) $\mu$, we have a complete understanding of the possible sets of atoms of a neutral measure. One simple consequence is that every neutral measure has a Martin-Löf random atom.
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Additional Information
- Adam R. Day
- Affiliation: School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, Wellington 6140, New Zealand
- Email: adam.day@msor.vuw.ac.nz
- Joseph S. Miller
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
- MR Author ID: 735102
- Email: jmiller@math.wisc.edu
- Received by editor(s): November 21, 2010
- Received by editor(s) in revised form: August 10, 2011
- Published electronically: January 17, 2013
- Additional Notes: The second author was supported by the National Science Foundation under grants DMS-0945187 and DMS-0946325, the latter being part of a Focused Research Group in Algorithmic Randomness.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3575-3591
- MSC (2010): Primary 03D32; Secondary 68Q30, 03D30
- DOI: https://doi.org/10.1090/S0002-9947-2013-05682-6
- MathSciNet review: 3042595