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On the Turing degrees of weakly computable real numbers
 JOURNAL OF LOGIC AND COMPUTATION
, 1986
"... The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semicomputable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others ..."
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Cited by 7 (4 self)
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The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semicomputable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others we show that, there are two real numbers of c.e. binary expansions such that their difference does not have an ω.c.e. Turing degree.
On BetaShifts Having Arithmetical Languages
"... Abstract. Let β be a real number with 1 < β < 2. We prove that the language of the βshift is ∆ 0 n iff β is a ∆nreal. The special case where n is 1 is the independently interesting result that the language of the βshift is decidable iff β is a computable real. The “if ” part of the proof is ..."
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Cited by 3 (2 self)
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Abstract. Let β be a real number with 1 < β < 2. We prove that the language of the βshift is ∆ 0 n iff β is a ∆nreal. The special case where n is 1 is the independently interesting result that the language of the βshift is decidable iff β is a computable real. The “if ” part of the proof is nonconstructive; we show that for Walters ’ version of the βshift, no constructive proof exists. 1