Randomness for non-computable measures

Authors:
Adam R. Day and Joseph S. Miller

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

Published electronically:
January 17, 2013

MathSciNet review:
3042595

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 is an atom of a neutral measure if and only if it is computable from (every representation of) , 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.

**1.**Patrick Billingsley,*Convergence of probability measures*, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR**1700749****2.**Rodney G. Downey and Denis R. Hirschfeldt,*Algorithmic randomness and complexity*, Theory and Applications of Computability, Springer, New York, 2010. MR**2732288****3.**Peter Gács,*Uniform test of algorithmic randomness over a general space*, Theoret. Comput. Sci.**341**(2005), no. 1-3, 91–137. MR**2159646**, https://doi.org/10.1016/j.tcs.2005.03.054**4.**Mathieu Hoyrup and Cristóbal Rojas,*Computability of probability measures and Martin-Löf randomness over metric spaces*, Inform. and Comput.**207**(2009), no. 7, 830–847. MR**2519075**, https://doi.org/10.1016/j.ic.2008.12.009**5.**Carl G. Jockusch Jr. and Robert I. Soare,*Π⁰₁ classes and degrees of theories*, Trans. Amer. Math. Soc.**173**(1972), 33–56. MR**0316227**, https://doi.org/10.1090/S0002-9947-1972-0316227-0**6.**Shizuo Kakutani,*A generalization of Brouwer’s fixed point theorem*, Duke Math. J.**8**(1941), 457–459. MR**0004776****7.**Daniel Lacombe,*Quelques procédés de définition en topologie recursive*, Constructivity in mathematics: Proceedings of the colloquium held at Amsterdam, 1957 (edited by A. Heyting), Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1959, pp. 129–158 (French). MR**0112838****8.**L. A. Levin,*The concept of a random sequence*, Dokl. Akad. Nauk SSSR**212**(1973), 548–550 (Russian). MR**0366096****9.**L. A. Levin,*Uniform tests for randomness*, Dokl. Akad. Nauk SSSR**227**(1976), no. 1, 33–35 (Russian). MR**0414222****10.**A. K. Zvonkin and L. A. Levin,*The complexity of finite objects and the basing of the concepts of information and randomness on the theory of algorithms*, Uspehi Mat. Nauk**25**(1970), no. 6(156), 85–127 (Russian). MR**0307889****11.**Per Martin-Löf,*The definition of random sequences*, Information and Control**9**(1966), 602–619. MR**0223179****12.**Joseph S. Miller,*Degrees of unsolvability of continuous functions*, J. Symbolic Logic**69**(2004), no. 2, 555–584. MR**2058189**, https://doi.org/10.2178/jsl/1082418543**13.**André Nies,*Computability and randomness*, Oxford Logic Guides, vol. 51, Oxford University Press, Oxford, 2009. MR**2548883****14.**Jan Reimann,*Effectively closed sets of measures and randomness*, Ann. Pure Appl. Logic**156**(2008), no. 1, 170–182. MR**2474448**, https://doi.org/10.1016/j.apal.2008.06.015**15.**Jan Reimann and Theodore A. Slaman.

Measures and their random reals.

To appear.**16.**M. Rozinas,*The semilattice of e-degrees*, Recursive functions (Russian), Ivanov. Gos. Univ., Ivanovo, 1978, pp. 71–84 (Russian). MR**604944****17.**Alan L. Selman,*Arithmetical reducibilities. I*, Z. Math. Logik Grundlagen Math.**17**(1971), 335–350. MR**0304150****18.**A. M. Turing.

On computable numbers, with an application to the Entscheidungsproblem. A correction.*Proceedings of the London Mathematical Society. Second Series*, 43:544-546, 1937.**19.**Klaus Weihrauch,*Computable analysis*, Texts in Theoretical Computer Science. An EATCS Series, Springer-Verlag, Berlin, 2000. An introduction. MR**1795407**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
03D32,
68Q30,
03D30

Retrieve articles in all journals with MSC (2010): 03D32, 68Q30, 03D30

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

Email:
jmiller@math.wisc.edu

DOI:
https://doi.org/10.1090/S0002-9947-2013-05682-6

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.

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.