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)


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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© 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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