Solovay Reducibility on D-c.e Real Numbers

  • Robert Rettinger
  • Xizhong Zheng
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3595)

Abstract

A c.e. real x is Solovay reducible to another c.e. real y if x can be approximated at least as efficiently as y by means of increasing computable sequences of rational numbers. The Solovay reducibility classifies elegantly the relative randomness of c.e. reals. Especially, the c.e. random reals are complete unter the Solovay reducibility for c.e. reals. In this paper we investigate an extension of the Solovay reducibility to the Δ\(^{\rm 0}_{\rm 2}\)-reals and show that the c.e. random reals are complete under (extended) Solovay reducibility for d-c.e. reals too. Actually we show that only the d-c.e. reals can be Solovay reducible to an c.e. random real. Furthermore, we show that this fails for the class of divergence bounded computable reals which extends the class of d-c.e. reals properly. In addition, we show also that any d-c.e. random reals are either c.e. or co-c.e.

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References

  1. 1.
    Ambos-Spies, K., Weihrauch, K., Zheng, X.: Weakly computable real numbers. Journal of Complexity 16(4), 676–690 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Calude, C.S., Hertling, P.H., Khoussainov, B., Wang, Y.: Recursively enumerable reals and Chaitin Ω numbers. Theoretical Computer Science 255, 125–149 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chaitin, G.: A theory of program size formally identical to information theory. J. of ACM 22, 329–340 (1975)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Downey, R.G.: Some recent progress in algorithmic randomness. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 42–83. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, Heidelberg (2000) (monograph to be published)Google Scholar
  6. 6.
    Downey, R.G., Hirschfeldt, D.R., 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
  7. 7.
    Downey, R.G., Hirschfeldt, D.R., Nies, A.: Randomness, computability, and density. SIAM J. Comput. 31(4), 1169–1183 (2002) (electronic)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kuçera, A., Slaman, T.A.: Randomness and recursive enumerability. SIAM J. Comput. 31(1), 199–211 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Levin, L.A.: The concept of a random sequence. Dokl. Akad. Nauk SSSR 212, 548–550 (1973); English translation: Soviet Math. Dokl. 212, 1413–1416 (1974) MathSciNetGoogle Scholar
  10. 10.
    Martin-Löf, P.: The definition of random sequences. Information and Control 9, 602–619 (1966)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Raichev, A.: D.c.e. reals, relative randomness, and real closed fields. In: CCA 2004, August 16-20, Lutherstadt Wittenberg, Germany (2004)Google Scholar
  12. 12.
    Solovay, R.M.: Draft of a paper (or a series of papers) on chaitin’s work.. IBM Thomas J. Watson Research Center, Yorktown Heights, NY, p. 215 (1975) (manuscript)Google Scholar
  13. 13.
    Zheng, X., Rettinger, R., Gengler, R.: Closure properties of real number classes under CBV functions. Theory of Computing Systems (2005) (to appear)Google Scholar
  14. 14.
    Zheng, X., Rettinger, R.: On the extensions of solovay-reducibility. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 360–369. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Robert Rettinger
    • 1
  • Xizhong Zheng
    • 2
    • 3
  1. 1.Theoretische Informatik IIFernUniversität HagenHagenGermany
  2. 2.Department of Computer ScienceJiangsu UniversityZhenjiangChina
  3. 3.Theoretische InformatikBTU CottbusCottbusGermany

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