Hostname: page-component-76c49bb84f-hwk5l Total loading time: 0 Render date: 2025-07-06T18:21:50.291Z Has data issue: false hasContentIssue false
  • English
  • Français

Principal filters definable by parameters in 𝓔bT†

Published online by Cambridge University Press:  01 February 2009

ANGSHENG LI ,
WEILIN LI ,
YICHENG PAN  and
LINQING TANG
ANGSHENG LI
Affiliation:
State Key Laboratory of Computer Science, and Institute of Software, Chinese Academy of Sciences, P.O. Box 8718, Beijing, 100080, P.R. China Email: angsheng@ios.ac.cn
WEILIN LI
Affiliation:
State Key Laboratory of Computer Science, and Institute of Software, Chinese Academy of Sciences, P.O. Box 8718, Beijing, 100080, P.R. China Email: liweilin0919@gmail.com
YICHENG PAN
Affiliation:
State Key Laboratory of Computer Science, and Institute of Software, Chinese Academy of Sciences, P.O. Box 8718, Beijing, 100080, P.R. China Email: yicheng@ios.ac.cn
LINQING TANG
Affiliation:
State Key Laboratory of Computer Science, and Institute of Software, Chinese Academy of Sciences, P.O. Box 8718, Beijing, 100080, P.R. China Email: tanglinqing05@mails.gucas.ac.cn and lingqing2001@163.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that there exist c.e. bounded Turing degrees a, b such that 0 < a < 0′, and that for any c.e. bounded Turing degree x, we have b ∨ x = 0′ if and only if x ≥ a. The result gives an unexpected definability theorem in the structure of bounded Turing reducibility.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 19 , Special Issue 1: Theory and Applications of Models of Computation (TAMC) , February 2009 , pp. 153 - 167
Copyright
Copyright © Cambridge University Press 2009

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.)

Article purchase

Temporarily unavailable

References

Ambos-Spies, K., Fejer, P. A., Lempp, S. and Lerman, M. (1996) Decidability of the two-quantifier theory of the recursively enumerable weak truthtable degrees and other distributive upper semi-lattices. Journal of Symbolic Logic 61 880–905.CrossRefGoogle Scholar
Ambos-Spies, K., Lachlan, A. H. and Soare, R. I. (1993) The continuity of cupping to 0′. Annals of Pure and Applied Logic 64 195–209.CrossRefGoogle Scholar
Brodhead, P., Li, A. and Li, W. (2008) Continuity of capping in bT. Annals of Pure and Applied Logic 155 (1)1–15.CrossRefGoogle Scholar
Cooper, S. B. (2004) Computability Theory, Chapman and Hall/CRC Press.Google Scholar
Cooper, S. B. and Li, A. (2008) On Lachlan's major subdegree problem. Archive for Mathematical Logic 47 341–434.CrossRefGoogle Scholar
Harrington, L. and Soare, R. I. (1992) Games in recursion theory and continuity properties of capping degrees. In: Judah, H., Just, W. and Woodin, W. H. (eds.) Set Theory and the Continuum, Proceedings of Workshop on Set Theory and the Continuum. MSRI publications 26, Springer-Verlag 39–62.Google Scholar
Li, A. (to appear) A cuppable degree which is not low2 cuppable.Google Scholar
Li, A., Li, W., Pan, Y. and Tang, L. (2008) Definable filters in the structure of bounded Turing reductions (extended abstract). In: Agrawal, M., Du, D.-Z., Duan, Z. and Li, A. (eds.) Proceedings Theory and Applications of Models of Computation, TAMC 2008. Springer-Verlag Lecture Notes in Computer Science 4978 116–124.CrossRefGoogle Scholar
Lempp, S. and Nies, A. (1995) The undecidability of the Π4-theory for the r.e. wtt- and Turing degrees. Journal of Symbolic Logic 60 1118–1136.CrossRefGoogle Scholar
Lempp, S., Nies, A. and Slaman, T. A. (1998) The Π3-theory of the enumerable Turing degrees is undecidable. Transactions of the American Mathematical Society 350 2719–2736.CrossRefGoogle Scholar
Seetapun, D. (1991) Contributions to Recursion Theory, Ph.D. thesis, Trinity College, Cambridge University.Google Scholar
Soare, R. (1987) Recursively Enumerable Sets and Degrees, Springer-Verlag.CrossRefGoogle Scholar
Soare, R. I. (to appear) Computability Theory and Applications, Springer-Verlag.Google Scholar