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Randomness, lowness and degrees

Published online by Cambridge University Press:  12 March 2014

George Barmpalias ,
Andrew E. M. Lewis  and
Mariya Soskova
George Barmpalias
Affiliation:
University of Leeds, School of Mathematics, Leeds. LS2 9JT, UK, E-mail: [email protected]
Andrew E. M. Lewis
Affiliation:
Universita Degli Studi Di Siena, Dipartimento Di Scienze Matematiche Ed Informatiche, Via Del Capitano 15, 53100 Siena, Italy, E-mail: [email protected]
Mariya Soskova
Affiliation:
University of Leeds, School of Mathematics, Leeds, LS2 9JT, UK, E-mail: [email protected]
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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 ∝ is not GL2 the LR degree of ∝ bounds 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 2 , June 2008 , pp. 559 - 577
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[1]Barmpalias, George and Montalban, Antonio, A cappable almost everywhere dominating computably enumerable degree. Electronic Notes in Theoretical Computer Science, vol. 167 (2007), pp. 17–31.CrossRefGoogle Scholar
[2]Binns, Stephen, Kjos-Hanssen, Bjørn, Miller, Joseph S., and Solomon, Reed, Lowness notions, measure and domination, in preparation.Google Scholar
[3]Cooper, S. Barry, Computability Theory, Chapman & Hall/ CRC Press, Boca Raton, FL, New York, London, 2004.Google Scholar
[4]Dobrinen, Natasha L. and Simpson, Stephen G., Almost everywhere domination, this Journal, vol. 69 (2004), no. 3, pp. 914–922.Google Scholar
[5]Downey, Rodney and Hirschfeldt, Denis, Algorithmic Randomness and Complexity, Springer, 2008, in print. Current draft available at http://www.mcs.vuw.ac.nz/~downey/.Google Scholar
[6]Ershov, Yu. L., A hierarchy of sets. Algebra i Logika, vol. 7 (1968), pp. 47–74, translation: Algebra and Logic, vol. 7 (1968), pp. 25–43.Google Scholar
[7]Hay, L. and Lerman, M., On the degrees of boolean combinations of r.e. sets, Recursive Function Theory Newsletter, 1976.Google Scholar
[8]Jockusch, Carl G., Simple proofs of some theorems on high degrees of unsolvahility, Canadian Journal of Mathematics, vol. 29 (1977), no. 5, pp. 1072–1080.CrossRefGoogle Scholar
[9]Kjos-Hanssen, Bjorn, Low for random reals and positive-measure domination. Proceedings of the American Mathematical Society, vol. 135 (2007), pp. 3703–3709.CrossRefGoogle Scholar
[10]Kurtz, Stuart A., Randomness and genericity in the degrees of unsolvability, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1981.Google Scholar
[11] Antonín Kučera, Measure, classes and complete extensions of PA, Recursion Theory Week (Oberwolfach, 1984), Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, 1985, pp. 245–259.CrossRefGoogle Scholar
[12]Kučera, Antonín and Terwijn, Sebastiaan A., Lowness for the class of random sets, this Journal, vol. 64 (1999), pp. 1396–1402.Google Scholar
[13]Lerman, Manuel, Degrees of Unsolvahility: Local and Global Theory, Springer-Verlag, 1983.CrossRefGoogle Scholar
[14]Nies, André, Computability and Randomness, monograph to appear. Current draft available at http://www.cs.auckland.ac.nz/~nies/.Google Scholar
[15]Nies, André, Low for random sets: The story, unpublished draft, which is available at the author's webpage http://www.cs.auckland.ac.nz/~nies/.Google Scholar
[16]Nies, André, Lowness properties and randomness, Advances in Mathematics, vol. 197 (2005), pp. 274–305.CrossRefGoogle Scholar
[17]Odifreddi, Piergiorgio, Classical Recursion Theory, vol. I and II, North-Holland, Amsterdam, Oxford, 1989 and 1999.Google Scholar
[18]Sacks, Gerald, Degrees of Unsolvability, Princeton University Press, 1963.Google Scholar
[19]Shore, Richard and Slaman, Theodore, Working below a high recursively enumerable degree, this Journal, vol. 58 (1993), pp. 824–859.Google Scholar
[20]Simpson, Stephen G., Almost everywhere domination and superhighness, Mathematical Logic Quarterly, vol. 53 (2007), pp. 462–482.CrossRefGoogle Scholar
[21]Soare, Robert I., Recursively Eumerable Sets and Degrees, Springer-Verlag, Berlin, London, 1987.CrossRefGoogle Scholar
[22]Trakhtenbrot, B. A., On autoreducibility, Rossiskaya Akademiya Nauk, vol. 192 (1970), pp. 1224–1227, translation: Soviet Mathematics Doklady, vol. 11 (1970), pp. 814–817.Google Scholar