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On the Turing degrees of weakly computable real numbers. (English) Zbl 1054.03040

A real \(\alpha\) is called computably enumerable or left computable iff there is a nondecreasing sequence of rationals \(q_0,q_1,\dots\) such that \(\alpha=\lim_s q_s\). There are natural objects of study in computable analysis. There are reals such that there is a computable function \(f(\, ,\,)\) with \(\alpha(i)= \lim_sf(i, s)\) (here \(\alpha(i)\) denotes position \(i\) in the dyadic expansion of \(\alpha\)) and such that \(f(i,s)= 1\) and \(f(i,s+ 1)= 0\) implies that for some \(j< i\), \(f(i, s)= 0\) and \(f(j, s+ 1)= 1\). Hence such reals are \(\omega\)-c.e. in a very strong way. K. Weihrauch and X. Zheng [Lect. Notes Comput. Sci. 1450, 798–806 (1998; Zbl 0920.03054)] looked at the related set of reals of the form \(\alpha-\beta\) with \(\alpha\), \(\beta\) c.e. These d.c.e. reals form a field. Here, Zheng proves that there exists a d.c.e. real which is not of \(\omega\)-c.e. Turing degree.

MSC:

03F60 Constructive and recursive analysis
03D80 Applications of computability and recursion theory
03D28 Other Turing degree structures

Citations:

Zbl 0920.03054
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