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CUPPING AND JUMP CLASSES IN THE COMPUTABLY ENUMERABLE DEGREES

Part of: Computability and recursion theory

Published online by Cambridge University Press:  30 October 2020

NOAM GREENBERG ,
KENG MENG NG  and
GUOHUA WU
NOAM GREENBERG
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY OF WELLINGTONWELLINGTON, NEW ZEALANDE-mail: greenberg@msor.vuw.ac.nz
KENG MENG NG
Affiliation:
DIVISION OF MATHEMATICAL SCIENCES SCHOOL OF PHYSICAL AND MATHEMATICAL SCIENCES NANYANG TECHNOLOGICAL UNIVERSITY SINGAPORE637371, SINGAPOREE-mail: kmng@ntu.edu.sgE-mail: guohua@ntu.edu.sg
GUOHUA WU
Affiliation:
DIVISION OF MATHEMATICAL SCIENCES SCHOOL OF PHYSICAL AND MATHEMATICAL SCIENCES NANYANG TECHNOLOGICAL UNIVERSITY SINGAPORE637371, SINGAPOREE-mail: kmng@ntu.edu.sgE-mail: guohua@ntu.edu.sg
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Abstract

We show that there is a cuppable c.e. degree, all of whose cupping partners are high. In particular, not all cuppable degrees are ${\operatorname {\mathrm {low}}}_3$-cuppable, or indeed ${\operatorname {\mathrm {low}}}_n$ cuppable for any n, refuting a conjecture by Li. On the other hand, we show that one cannot improve highness to superhighness. We also show that the ${\operatorname {\mathrm {low}}}_2$-cuppable degrees coincide with the array computable-cuppable degrees, giving a full understanding of the latter class.

Keywords

computably enumerable degreesjoinjump classesarray computable

MSC classification

Primary: 03D25: Recursively (computably) enumerable sets and degrees
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 4 , December 2020 , pp. 1499 - 1545
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Copyright
© The Association for Symbolic Logic 2020

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References

REFERENCES

Ambos-Spies, K., Jockusch, C. G. Jr., Shore, R. A., and Soare, R. I., An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees. Transactions of the American Mathematical Society, vol. 281 (1984), no. 1, pp. 109–128.CrossRefGoogle Scholar
Ambos-Spies, K., Lachlan, A. H., and Soare, R. I., The continuity of cupping to 0′. Annals of Pure and Applied Logic, vol. 64 (1993), no. 3, pp. 195–209.CrossRefGoogle Scholar
Bickford, M. and Mills, C. F., Lowness properties of r.e. sets, UW Madison. Manuscript, 1982.Google Scholar
Cholak, P., Groszek, M., and Slaman, T., An almost deep degree, this Journal, vol.66 (2001), no. 2, pp. 881–901.Google Scholar
Diamondstone, D., Promptness does not imply superlow cuppability, this Journal, vol. 74 (2009), no. 4, pp. 1264–1272.Google Scholar
Downey, R., Greenberg, N., Miller, J. S., and Weber, R., Prompt simplicity, array computability and cupping, Computational Prospects of Infinity. Part II. Presented Talks, Lecture Notes Series, vol. 15, Institute for Mathematical Sciences, National University of Singapore, World Scientific Publishing, Hackensack, 2008, pp. 59–77.CrossRefGoogle Scholar
Downey, R. G., Jockusch, C. G. Jr, and Stob, M., Array nonrecursive sets and multiple permitting arguments, Recursion Theory Week (Oberwolfach, 1989), Lecture Notes in Mathematics, vol. 1432, Springer, Berlin, 1990, pp. 141–173.CrossRefGoogle Scholar
Downey, R. G., Array nonrecursive degrees and genericity, Computability, Enumerability, Unsolvability, London Mathematical Society Lecture Note Series, vol. 224, Cambridge University Press, Cambridge, 1996, pp. 93–104.CrossRefGoogle Scholar
Fejer, P. A. and Soare, R. I., The plus-cupping theorem for the recursively enumerable degrees, Logic Year 1979–80 (Proc. Seminars and Conf. Math. Logic, Univ. Connecticut, Storrs, Conn., 1979/80), Lecture Notes in Mathematics, vol. 859, Springer, Berlin, 1981, pp. 49–62.Google Scholar
Harrington, L. A. and Shelah, S., The undecidability of the recursively enumerable degrees. Bulletin of the American Mathematical Society, vol. 6 (1982), pp. 79–80.CrossRefGoogle Scholar
Harrington, L. and Soare, R. I., Games in recursion theory and continuity properties of capping degrees, Set Theory of the Continuum (Berkeley, CA, 1989), volume 26 of Mathematical Sciences Research Institute Publications, Springer, New York, 1992, pp. 39–62.CrossRefGoogle Scholar
Ishmukhametov, S., Weak recursive degrees and a problem of spector, Recursion Theory and Complexity (Kazan, 1997), de Gruyter Series in Logic and its Applications, vol. 2, de Gruyter, Berlin, 1999, pp. 81–87.Google Scholar
Li, A., A hierarchy characterisation of cuppable degrees. University of Leeds, Department of Pure Mathematics, Preprint 2001, series, No. 1, 21 pp.Google Scholar
Li, A., Wu, G., and Zhang, Z., A hierarchy for cuppable degrees. Illinois Journal of Mathematics, vol. 44 (2000), no. 3, pp. 619–632.CrossRefGoogle Scholar
Miller, D. P., High recursively enumerable degrees and the anticupping property, Logic Year 1979–80 (Proc. Seminars and Conf. Math. Logic, Univ. Connecticut, Storrs, Conn., 1979/80) , Lecture Notes in Mathematics, vol. 859, Springer, Berlin, 1981, pp. 230–245.Google Scholar
Nies, A., Shore, R. A., and Slaman, T. A., Interpretability and definability in the recursively enumerable degrees. Proceedings of the London Mathematical Society, vol. 77 (1998), no. 2, 241–291.CrossRefGoogle Scholar