Advertisement

Computably Enumerable Sets in the Solovay and the Strong Weak Truth Table Degrees

  • George Barmpalias
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3526)

Abstract

The strong weak truth table reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky and Weinberger on applications of computability to differential geometry. Yu and Ding showed that the relevant degree structure restricted to the c.e. reals has no greatest element, and asked for maximal elements. We answer this question for the case of c.e. sets. Using a doubly non-uniform argument we show that there are no maximal elements in the sw degrees of the c.e. sets. We note that the same holds for the Solovay degrees of c.e. sets.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Downey, R., Hirschfeldt, D., LaForte, G.: Randomness and reducibility. Journal of Computer and System Sciences 68, 96–114 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Downey, R., Hirschfeldt, D., LaForte, G.: Randomness and reducibility. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 316–327. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  3. 3.
    Downey, R., Hirschfeldt, D., Nies, A.: Randomness, computability, and density. SIAM Journal of Computation 31(4), 1169–1183 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Downey, R.: Some Recent Progress in Algorithmic Randomness (2004) (preprint)Google Scholar
  5. 5.
    Odifreddi, P.: Classical recursion theory. Amsterdam Oxford, North-Holand (1989)zbMATHGoogle Scholar
  6. 6.
    Yu, L., Ding, D.: There is no SW-complete c.e. real. To appear in J. Symb. Logic Google Scholar
  7. 7.
    Soare, R.: Computability Theory and Differential Geometry. To appear in the Bulletin of Symbolic LogicGoogle Scholar
  8. 8.
    Soare, R.: Recursively enumerable sets and degrees. Springer, Berlin London (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • George Barmpalias
    • 1
  1. 1.School of MathematicsUniversity of LeedsLeedsU.K

Personalised recommendations