Advertisement

High Minimal Pairs in the Enumeration Degrees

  • Andrea Sorbi
  • Guohua Wu
  • Yue Yang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5532)

Abstract

The natural embedding of the Turing degrees into the enumeration degrees preserves the jump operation, and maps isomorphically the computably enumerable Turing degrees onto the \({\it \Pi}^0_1\) enumeration degrees. The embedding does not preserve minimal pairs, though, unless one of the two sides is low. In particular it is known that there exist high minimal pairs of c.e. Turing degrees that do not embed to minimal pairs of e-degrees. We show however that high minimal pairs of \({\it \Pi}^0_1\) e-degrees do exist.

Keywords

Natural Embedding Minimal Pair Computable Enumeration True Stage True Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cooper, S.B., Copestake, C.S.: Properly Σ 2 enumeration degrees. Z. Math. Logik Grundlag. Math. 34, 491–522 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Lachlan, A.H.: Lower bounds for pairs of recursively enumerable degrees. Proc. London Math. Soc. 16, 537–569 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    McEvoy, K.: The Structure of the Enumeration Degrees. PhD thesis, School of Mathematics, University of Leeds (1984)Google Scholar
  4. 4.
    McEvoy, K., Cooper, S.B.: On minimal pairs of enumeration degrees. J. Symbolic Logic 50, 983–1001 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Shore, R., Sorbi, A.: Jumps of \(\Sigma^0_2\) high e-degrees and properly \(\Sigma^0_2\) e-degrees. In: Arslanov, M., Lempp, S. (eds.) Recursion Theory and Complexity. De Gruyter Series in Logic and Its Applications, pp. 157–172. W. De Gruyter, Berlin (1999)Google Scholar
  6. 6.
    Yates, C.E.M.: A minimal pair of recursively enumerable degrees. J. Symbolic Logic 31, 159–168 (1966)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Andrea Sorbi
    • 1
  • Guohua Wu
    • 2
  • Yue Yang
    • 3
  1. 1.University of SienaSienaItaly
  2. 2.Nanyang Technological UniversitySingapore
  3. 3.National University of SingaporeSingapore

Personalised recommendations