On the turing degrees of divergence bounded computable reals

Authors:
Robert Rettinger;Xizhong Zheng
Affiliations:
Theoretische Informatik II, FernUniversität Hagen, Hagen, Germany;Department of Computer Science, Jiangsu University, Zhenjiang, China
Venue:
CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
Year:
2005

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A hierarchy of Turing degrees of divergence bounded computable real numbers

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Abstract

The d-c.e. (difference of c.e.) and dbc (divergence bounded computable) reals are two important subclasses of Δ$_{\rm 2}^{\rm 0}$-reals which have very interesting computability-theoretical as well as very nice analytical properties. Recently, Downey, Wu and Zheng [2] have shown by a double witness technique that not every Δ$_{\rm 2}^{\rm 0}$-Turing degree contains a d-c.e. real. In this paper we show that the classes of Turing degrees of d-c.e., dbc and Δ$_{\rm 2}^{\rm 0}$ reals are all different.