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Minimal Pair Constructions and Iterated Trees of Strategies

  • S. Lempp
  • M. Lerman
  • F. Weber
Chapter
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 12)

Abstract

We use the iterated trees of strategies approach developed in [LL1], [LL2] to prove some theorems about minimal pairs. In Section 1-3, we show how to use these methods to prove the Minimal Pair Theorem of Lachlan [L] and Yates [Y]:

Theorem 3.4 (Minimal Pair). There exist nonrecursive r.e. degreesaandbsuch thata λ b = 0.

Keywords

Basic Module Minimal Pair True Path Finite Tree Finite Outcome 
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.

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References

  1. [A]
    Ambos-Spies, K. [1984], An extension of the non-diamond theorem in classical and α-recursion theory. J. Symbolic Logic, 49, 586–607.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [C]
    Cooper, S.B. [1974], Minimal pairs and high recursively enumerable degrees. J. Symbolic Logic, 39, 665–660.MathSciNetCrossRefGoogle Scholar
  3. [L]
    Lachlan, A.H. [1966], Lower bounds for pairs of recursively enumerable degrees. Proc. London Math. Soc, 16, 537–569.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [LL1]
    Lempp, S and M. Lerman [1990], Priority arguments using iterated trees of strategies. In: Recursion Theory Week, 1989, Lecture Notes in Mathematics #1482, 277–296. Springer-Verlag, Berlin, Heidelberg, New York.CrossRefGoogle Scholar
  5. [LL2]
    Lempp, S. and M. Lerman, The decidability of the existential theory of the poset of recursively enumerable degrees with jump relations. In preparation.Google Scholar
  6. [P]
    Posner, D. [1977], High degrees. Ph.D. Dissertation, University of California, Berkeley.Google Scholar
  7. [S]
    Shoenfield, J.R. [1959], On degrees of unsolvability. Ann. Math, 69, 644–653.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [SS]
    Shore, R.A. and T.A. Slaman, Working below a high recursively enumerable degree. To appear.Google Scholar
  9. [So]
    Soare, R.I. [1987], Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic, Springer-Verlag, Berlin, Heidelberg, New York.Google Scholar
  10. [Y]
    Yates, C.E.M. [1966], A minimal pair of recursively enumerable degrees. J. Symbolic Logic, 31, 159–168.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • S. Lempp
    • 1
  • M. Lerman
    • 2
  • F. Weber
    • 3
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA
  2. 2.Department of MathematicsUniversity of ConnecticutStorrsUSA
  3. 3.Department of MathematicsConcordia University, WisconsinMequonUSA

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