### Randomness, lowness and degrees

George Barmpalias, Andrew E. M. Lewis, and Mariya Soskova
Source: J. Symbolic Logic Volume 73, Issue 2 (2008), 559-577.

#### Abstract

We say that A≤LRB if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e., restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α is not GL2 the LR degree of α bounds 20 degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees.

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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1208359060
Digital Object Identifier: doi:10.2178/jsl/1208359060
Mathematical Reviews number (MathSciNet): MR2414465
Zentralblatt MATH identifier: 1145.03020

### References

George Barmpalias and Antonio Montalban, A cappable almost everywhere dominating computably enumerable degree, Electronic Notes in Theoretical Computer Science, vol. 167 (2007), pp. 17--31.
Mathematical Reviews (MathSciNet): MR2321777
Stephen Binns, Bjørn Kjos-Hanssen, Joseph S. Miller, and Reed Solomon, Lowness notions, measure and domination, in preparation.
S. Barry Cooper, Computability Theory, Chapman & Hall/ CRC Press, Boca Raton, FL, New York, London, 2004.
Mathematical Reviews (MathSciNet): MR2017461
Zentralblatt MATH: 1041.03001
Natasha L. Dobrinen and Stephen G. Simpson, Almost everywhere domination, Journal of Symbolic Logic, vol. 69 (2004), no. 3, pp. 914--922.
Mathematical Reviews (MathSciNet): MR2078930
Digital Object Identifier: doi:10.2178/jsl/1096901775
Project Euclid: euclid.jsl/1096901775
Zentralblatt MATH: 1075.03021
Rodney Downey and Denis Hirschfeldt, Algorithmic Randomness and Complexity, Springer, 2008, in print. Current draft available at http://www.mcs.vuw.ac.nz/$\sim$downey/.
Yu. L. Ershov, A hierarchy of sets, Algebra i Logika, vol. 7 (1968), pp. 47--74, translation: Algebra and Logic, vol. 7 (1968), pp. 25--43.
Mathematical Reviews (MathSciNet): MR270911
L. Hay and M. Lerman, On the degrees of boolean combinations of r.e. sets, Recursive Function Theory Newsletter, 1976.
Carl G. Jockusch, Simple proofs of some theorems on high degrees of unsolvability, Canadian Journal of Mathematics, vol. 29 (1977), no. 5, pp. 1072--1080.
Mathematical Reviews (MathSciNet): MR476460
Bjørn Kjos-Hanssen, Low for random reals and positive-measure domination, Proceedings of the American Mathematical Society, vol. 135 (2007), pp. 3703--3709.
Mathematical Reviews (MathSciNet): MR2336587
Digital Object Identifier: doi:10.1090/S0002-9939-07-08648-0
Zentralblatt MATH: 1128.03031
Stuart A. Kurtz, Randomness and genericity in the degrees of unsolvability, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1981.
Antonín Kučera, Measure, $\Pi^0_1$ classes and complete extensions of $PA$, Recursion Theory Week (Oberwolfach, 1984), Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, 1985, pp. 245--259.
Mathematical Reviews (MathSciNet): MR820784
Antonín Kučera and Sebastiaan A. Terwijn, Lowness for the class of random sets, Journal of Symbolic Logic, vol. 64 (1999), pp. 1396--1402.
Mathematical Reviews (MathSciNet): MR1780059
Digital Object Identifier: doi:10.2307/2586785
Project Euclid: euclid.jsl/1183745926
Zentralblatt MATH: 0954.68080
Manuel Lerman, Degrees of Unsolvability: Local and Global Theory, Springer-Verlag, 1983.
Mathematical Reviews (MathSciNet): MR708718
Zentralblatt MATH: 0542.03023
André Nies, Computability and Randomness, monograph to appear. Current draft available at http://www.cs.auckland.ac.nz/$\sim$nies/.
--------, Low for random sets: The story, unpublished draft, which is available at the author's webpage http://www.cs.auckland.ac.nz/$\sim$nies/.
--------, Lowness properties and randomness, Advances in Mathematics, vol. 197 (2005), pp. 274--305.
Mathematical Reviews (MathSciNet): MR2166184
Digital Object Identifier: doi:10.1016/j.aim.2004.10.006
Zentralblatt MATH: 1141.03017
Piergiorgio Odifreddi, Classical Recursion Theory, vol. I and II, North-Holland, Amsterdam, Oxford, 1989 and 1999.
Mathematical Reviews (MathSciNet): MR982269
Zentralblatt MATH: 0661.03029
Gerald Sacks, Degrees of Unsolvability, Princeton University Press, 1963.
Mathematical Reviews (MathSciNet): MR186554
Zentralblatt MATH: 0143.25302
Richard Shore and Theodore Slaman, Working below a high recursively enumerable degree, Journal of Symbolic Logic, vol. 58 (1993), pp. 824--859.
Mathematical Reviews (MathSciNet): MR1242041
Digital Object Identifier: doi:10.2307/2275099
Project Euclid: euclid.jsl/1183744300
Zentralblatt MATH: 0797.03043
Stephen G. Simpson, Almost everywhere domination and superhighness, Mathematical Logic Quarterly, vol. 53 (2007), pp. 462--482.
Mathematical Reviews (MathSciNet): MR2351944
Digital Object Identifier: doi:10.1002/malq.200710012
Zentralblatt MATH: 1123.03040
Robert I. Soare, Recursively Eumerable Sets and Degrees, Springer-Verlag, Berlin, London, 1987.
Mathematical Reviews (MathSciNet): MR882921
B. A. Trakhtenbrot, On autoreducibility, Rossiskaya Akademiya Nauk, vol. 192 (1970), pp. 1224--1227, translation: Soviet Mathematics Doklady, vol. 11 (1970), pp. 814--817.
Mathematical Reviews (MathSciNet): MR274287