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Limits on jump inversion for strong reducibilities

Published online by Cambridge University Press:  12 March 2014

Barbara F. Csima ,
Rod Downey  and
Keng Meng Ng
Barbara F. Csima
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada, E-mail: [email protected], URL: www.math.uwaterloo.ca/~csima
Rod Downey
Affiliation:
School of Mathematics, Statistics and Computer Science, Victoria University of Wellington, PO BOX 600, Wellington, New Zealand, E-mail: [email protected]
Keng Meng Ng
Affiliation:
University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706, USA, E-mail: [email protected]
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Abstract

We show that Sacks' and Shoenfield's analogs of jump inversion fail for both tt- and wtt-reducibilities in a strong way. In particular we show that there is a δ20 set B >tt ∅′ such that there is no c.e. set A with A′ ≡wttB. We also show that there is a Σ20 set C >tt ∅′ such that there is no δ20 set D with D′ ≡wttC.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 4 , December 2011 , pp. 1287 - 1296
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[And08]Anderson, B., Automorhisms of the truth-table degrees are fixed on some cone, preprint, 2008.Google Scholar
[Coo04]Cooper, S. Barry, Computability theory, Chapman & Hall/CRC, Boca Raton, FL, 2004.Google Scholar
[Dem88]Demuth, O., Remarks on the structure of tt-degrees based on constructive measure theory, Commentationes Mathematicae Universitatis Carolinae, vol. 29 (1988), no. 2, pp. 233–247.Google Scholar
[DR89]Downey, R. and Remmel, J., Classification of degree classes associated with r.e. subspaces, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 105–124.CrossRefGoogle Scholar
[Fri58]Friedberg, R., A criterion for completeness of degrees of unsolvability, this Journal, vol. 22 (1958), pp. 159–160.Google Scholar
[Moh84]Mohrherr, J., Density of a final segment of the truth-table degrees, Pacific Journal of Mathematics, vol. 115 (1984), pp. 409–419.CrossRefGoogle Scholar
[Pos44]Post, E., Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 284–316.CrossRefGoogle Scholar
[RS08a]Reimann, J. and Slaman, T., Measures and their random reals, preprint, 2008.Google Scholar
[RS08b]Reimann, J. and Slaman, T., Probability measures and effective randomness, preprint, 2008.Google Scholar
[Rob71]Robinson, R., Jump restricted interpolation in the recursively enumerable degrees, Annals of Mathematics, vol. 93 (1971), no. 2, pp. 586–596.CrossRefGoogle Scholar
[Sac63]Sacks, G., Recursive enumerability and the jump operator, 1963, vol. 108.Google Scholar
[Sho59]Shoenfield, J., On degrees of unsolvability, Annals of Mathematics, vol. 69 (1959), pp. 644–653.CrossRefGoogle Scholar
[Soa87]Soare, R., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, 1987.CrossRefGoogle Scholar