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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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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.
Approximation representations for ∆2 reals
"... Abstract. We study ∆2 reals x in terms of how they can be approximated symmetrically by a computable sequence of rationals. We deal with a natural notion of ‘approximation representation ’ and study how these are related computationally for a fixed x. This is a continuation of earlier work; it aims ..."
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Abstract. We study ∆2 reals x in terms of how they can be approximated symmetrically by a computable sequence of rationals. We deal with a natural notion of ‘approximation representation ’ and study how these are related computationally for a fixed x. This is a continuation of earlier work; it aims at a classification of ∆2 reals based on approximation and it turns out to be quite different than the existing ones (based on information content etc.) 1.
On the Hierarchy of ∆ 0 2Real Numbers
, 2004
"... A real number x is called ∆0 2 if its binary expansion corresponds to a ∆02set of natural numbers. Such reals are just the limits of computable sequences of rational numbers and hence also called computably approximable. Depending on how fast the sequences converge, ∆0 2reals have different levels ..."
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A real number x is called ∆0 2 if its binary expansion corresponds to a ∆02set of natural numbers. Such reals are just the limits of computable sequences of rational numbers and hence also called computably approximable. Depending on how fast the sequences converge, ∆0 2reals have different levels of effectiveness. This leads to various hierarchies of ∆0 2 reals. In this paper we summarize several recent developments related to such kind of hierarchies.
On the Extensions of SolovayReducibility
"... Abstract. A c.e. real x is Solovay reducible (Sreducible) to another c.e. real y if y is at least as difficult to be approximated as x. In this case, y is at least as random as x. Thus, the Sreducibility classifies relative randomness of c.e. reals such that the c.e. random reals are complete in t ..."
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Abstract. A c.e. real x is Solovay reducible (Sreducible) to another c.e. real y if y is at least as difficult to be approximated as x. In this case, y is at least as random as x. Thus, the Sreducibility classifies relative randomness of c.e. reals such that the c.e. random reals are complete in the class of c.e. reals under the Sreducibility. In this paper we investigate extensions of the Sreducibility outside the c.e. reals. We show that the straightforward extension does not behave satisfactorily. Then we introduce two new extensions which coincide with the Sreducibility on the c.e. reals and behave reasonably outside the c.e. reals. Both of these extensions imply the rHreducibility of Downey, Hirschfeldt and LaForte [6]. At last we show that even under the rHreducibility the computably approximable random reals cannot be characterized as complete elements of this reduction. 1
On the Monotonic Computability of SemiComputable Real Numbers
"... Let h: N → Q be a computable function. A real number x is hmonotonically computable (hmc, for short) if there is a computable sequence (xs) of rational numbers which converges to x in such a way that the ratios of the approximation errors are bounded by h. In this paper we discuss the hmonotonic ..."
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Let h: N → Q be a computable function. A real number x is hmonotonically computable (hmc, for short) if there is a computable sequence (xs) of rational numbers which converges to x in such a way that the ratios of the approximation errors are bounded by h. In this paper we discuss the hmonotonic computability of semicomputable real numbers which are limits of monotone computable sequences of rational numbers. Especially, we show a sufficient and necessary condition for the function h such that the hmonotonic computability is simply equivalent to the normal computability.