Skip to main content Accessibility help
×
Home
Hostname: page-component-747cfc64b6-7hjq6 Total loading time: 0.229 Render date: 2021-06-14T06:33:22.665Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

CUPPING AND JUMP CLASSES IN THE COMPUTABLY ENUMERABLE DEGREES

Published online by Cambridge University Press:  30 October 2020

NOAM GREENBERG
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY OF WELLINGTON WELLINGTON, NEW ZEALAND E-mail: greenberg@msor.vuw.ac.nz
KENG MENG NG
Affiliation:
DIVISION OF MATHEMATICAL SCIENCES SCHOOL OF PHYSICAL AND MATHEMATICAL SCIENCES NANYANG TECHNOLOGICAL UNIVERSITY SINGAPORE 637371, SINGAPORE E-mail: kmng@ntu.edu.sg E-mail: guohua@ntu.edu.sg
GUOHUA WU
Affiliation:
DIVISION OF MATHEMATICAL SCIENCES SCHOOL OF PHYSICAL AND MATHEMATICAL SCIENCES NANYANG TECHNOLOGICAL UNIVERSITY SINGAPORE 637371, SINGAPORE E-mail: kmng@ntu.edu.sg E-mail: guohua@ntu.edu.sg
Corresponding

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.

Type
Articles
Copyright
© The Association for Symbolic Logic 2020

Access options

Get access to the full version of this content by using one of the access options below.

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. 109128.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. 195209.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. 881901.Google Scholar
Diamondstone, D., Promptness does not imply superlow cuppability, this Journal, vol. 74 (2009), no. 4, pp. 12641272.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. 5977.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. 141173.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. 93104.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. 4962.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. 7980.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. 3962.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. 8187.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. 619632.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. 230245.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, 241291.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

CUPPING AND JUMP CLASSES IN THE COMPUTABLY ENUMERABLE DEGREES
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

CUPPING AND JUMP CLASSES IN THE COMPUTABLY ENUMERABLE DEGREES
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

CUPPING AND JUMP CLASSES IN THE COMPUTABLY ENUMERABLE DEGREES
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *