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Properly enumeration degrees and the high/low hierarchy

Published online by Cambridge University Press:  12 March 2014

Matthew Giorgi ,
Andrea Sorbi  and
Yue Yang
Matthew Giorgi
Affiliation:
Dipartimento di Scienze Matematiche ed Informatiche, “Roberto Magari”, Pian dei Mantellini 44, 53100 Siena, Italy, E-mail: [email protected]
Andrea Sorbi
Affiliation:
Dipartimento di Scienze Matematiche ed Informatiche, “Roberto Magari”, Pian dei Mantellini 44, 53100 Siena, Italy, E-mail: [email protected]
Yue Yang
Affiliation:
Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, Singapore, 119260, E-mail: [email protected]
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Abstract

We show that there exist downwards properly (in fact noncuppable) e-degrees that are not high. We also show that every high e-degree bounds a noncuppable e-degree.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 4 , December 2006 , pp. 1125 - 1144
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Bereznyuk, S., Coles, R., and Sorbi, A., The distribution of properly enumeration degrees, this Journal, vol. 65 (2000), no. 1, pp. 19–32.Google Scholar
[2]Cooper, S. B., Enumeration reducibility, nondeterministic computations and relative computability of partial functions, Recursion Theory Week, Oberwolfach 1989 (Ambos-Spies, K., Müller, G., and Sacks, G. E., editors), Lecture Notes in Mathematics, vol. 1432, Springer-Verlag, Heidelberg, 1990, pp. 57–110.Google Scholar
[3]Cooper, S. B. and Copestake, C. S., Properly ∑2 enumeration degrees, Zeitschrift für Mathematische Logik and Grundlagen der Mathematik, vol. 34 (1988), pp. 491–522.CrossRefGoogle Scholar
[4]Cooper, S. B., Sorbi, A., and Yi, X., Cupping and noncupping in the enumeration degrees of sets, Annals of Pure and Applied Logic, vol. 82 (1997), pp. 317–342.CrossRefGoogle Scholar
[5]Dekker, J. C. E., A theorem on hypersimple sets, Proceedings of the American Mathematical Society, vol. 5 (1954), pp. 791–796.CrossRefGoogle Scholar
[6]Giorgi, M. B., A high noncuppable e-degree, to appear in the Archive for Mathematical Logic.Google Scholar
[7]Griffiths, E., Limit lemmas and jump inversion in the enumeration degrees, Archive for Mathematical Logic, vol. 42 (2003), no. 6.CrossRefGoogle Scholar
[8]Gutteridge, L., Some results on enumeration reducibility, Ph.D. thesis, Simon Fraser University, 1971.Google Scholar
[9]McEvoy, K., The structure of the enumeration degrees, Ph.D. thesis, School of Mathematics, University of Leeds, 1984.Google Scholar
[10]McEvoy, K. and Cooper, S. B., On minimal pairs of enumeration degrees, this Journal, vol. 50 (1985), pp. 983–1001.Google Scholar
[11]Nies, A. and Sorbi, A., Structural properties and enumeration degrees, this Journal, vol. 65 (2000), no. 1, pp. 285–292.Google Scholar
[12]Sacks, G. E., Recursive enumerability and the jump operator, Transactions of the American Mathematical Society, vol. 108 (1963), pp. 223–239.CrossRefGoogle Scholar
[13]Shore, R. and Sorbi, A., Jumps of high e-degrees and properly e-degrees, Recursion theory and complexity (Arslanov, M. and Lempp, S., editors), de Gruyter Series in Logic and its Applications, W. De Gruyter, Berlin, New York, 1999, pp. 157–172.CrossRefGoogle Scholar
[14]Soare, R. I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Heidelberg, 1987.CrossRefGoogle Scholar
[15]Sorbi, A., The enumeration degrees of thel sets, Complexity, logic and recursion theory (Sorbi, A., editor), Marcel Dekker, New York, 1997, pp. 303–330.Google Scholar