Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T04:02:32.873Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounding and nonbounding minimal pairs in the enumeration degrees

Published online by Cambridge University Press:  12 March 2014

S. Barry Cooper ,
Angsheng Li ,
Andrea Sorbi  and
Yue Yang
S. Barry Cooper
Affiliation:
Department of Pure Mathematics, School of Mathematics, University of Leeds, Leeds. LS2 9JT, UKE-mail:, s.b.cooper@leeds.ac.uk
Angsheng Li
Affiliation:
Institute of Software, Academia Sinica, Beijing, 100080., ChinaE-mail:, angsheng@gcl.iscas.ac.cn
Andrea Sorbi
Affiliation:
Dipartimento Di Scienze Matematiche, Ed Informatiche “Roberto Magari”, Siena. 53100, ItalyE-mail:, sorbi@unisi.it
Yue Yang
Affiliation:
Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, 119260, SingaporeE-mail:, matyangy@math.nus.edu.sg
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that every nonzero Δ20, e-degree bounds a minimal pair. On the other hand, there exist Σ20, e-degrees which bound no minimal pair.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 3 , September 2005 , pp. 741 - 766
Copyright
Copyright © Association for Symbolic Logic 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Arslanov, M.. Cooper, S. Barry, and Kalimullin, I. S., Splitting properties of total e-degrees. Algebra and Logic, vol. 42 (2003), pp. 1–13.CrossRefGoogle Scholar
[2]Cooper, S. B., Partial degrees and the density problem, this Journal, vol. 47 (1982). pp. 854–859.Google Scholar
[3]Cooper, S. B.. Enumeration reducibility using bounded information: counting minimal covers, Zeitschrift für Mathenatische Logik und Grundlagen der Mathematik, vol. 33 (1987), pp. 537–560.CrossRefGoogle Scholar
[4]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
[5]Copestake, K., l-genericity in the enumeration degrees, this Journal, (1988), no. 53, pp. 878–887.Google Scholar
[6]Friedberg, R. M. and Rogers, H. Jr, Reducibility and completeness for sets of integers, Zeitschrift für Mathenatische Logik und Grundlagen der Mathematik, vol. 5 (1959), pp. 117–125.CrossRefGoogle Scholar
[7]Gutteridge, L., Some results on enumeration reducibility. Ph.D. thesis. Simon Fraser University, 1971.Google Scholar
[8]Lachlan, A., Bounding minimal pairs, this Journal, vol. 44 (1979), pp. 626–642.Google Scholar
[9]McEvoy, K. and Cooper, S. B., On minimal pairs of enumeration degrees, this Journal, vol. 50 (1985), pp. 983–1001.Google Scholar
[10]Shore, R. and Sorbi, A., Jumps of Σ20 high e-degrees and properly Σ20 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
[11]Soare, R. I., Recursively Enumerable Sets and Degrees, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Heidelberg, 1987.CrossRefGoogle Scholar