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15
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.
Monotonically Computable Real Numbers
, 2001
"... A real number x is called kmonotonically computable (kmc), for constant k> 0, if there is a computable sequence (xn)n∈N of rational numbers which converges to x such that the convergence is kmonotonic in the sense that k · x−xn  ≥ x−xm  for any m> n and x is monotonically computable (m ..."
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Cited by 6 (4 self)
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A real number x is called kmonotonically computable (kmc), for constant k> 0, if there is a computable sequence (xn)n∈N of rational numbers which converges to x such that the convergence is kmonotonic in the sense that k · x−xn  ≥ x−xm  for any m> n and x is monotonically computable (mc) if it is kmc for some k> 0. x is weakly computable if there is a computable sequence (xs)s∈N of rational numbers converging to x such that the sum s∈N xs − xs+1  is finite. In this paper we show that all mc real numbers are weakly computable but the converse fails. Furthermore, we show also an infinite hierarchy of mc real numbers.
ON THE COMPUTABILITY OF CONDITIONAL PROBABILITY
"... Abstract. We study the problem of computing conditional probabilities, a fundamental operation in statistics and machine learning. In the elementary discrete setting, conditional probability is defined axiomatically and the search for more constructive definitions is the subject of a rich literature ..."
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Cited by 5 (4 self)
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Abstract. We study the problem of computing conditional probabilities, a fundamental operation in statistics and machine learning. In the elementary discrete setting, conditional probability is defined axiomatically and the search for more constructive definitions is the subject of a rich literature in probability theory and statistics. In the discrete or dominated setting, under suitable computability hypotheses, conditional probabilities are computable. However, we show that in general one cannot compute conditional probabilities. We do this by constructing a pair of computable random variables in the unit interval whose conditional distribution encodes the halting problem at almost every point. We show that this result is tight, in the sense that given an oracle for the halting problem, one can compute this conditional distribution. On the other hand, we show that conditioning in abstract settings is computable in the presence of certain additional structure, such as independent absolutely continuous noise. 1.
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
A note on the Turing degree of divergence bounded computable real numbers
 CCA 2004, August 1620, Lutherstadt
, 2004
"... The Turing degree of a real number is defined as the Turing degree of its binary expansion. In this note we apply the double witnesses technique recently developed by Downey, Wu and Zheng [2] and show that there exists a ∆0 2Turing degree which contains no divergence bounded computable real numbers ..."
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The Turing degree of a real number is defined as the Turing degree of its binary expansion. In this note we apply the double witnesses technique recently developed by Downey, Wu and Zheng [2] and show that there exists a ∆0 2Turing degree which contains no divergence bounded computable real numbers. This extends the result of [2] that not every ∆0 2Turing degree contains a dc.e. real.
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.
Approximation Representations for Reals and their wttDegrees
 Mathematical Logic Quarterly
, 2004
"... Abstract. We study the approximation properties of computably enumerable reals. We deal with a natural notion of approximation representation and study their wttdegrees. Also, we show that a single representation may correspond to a quite diverse variety of reals. 1. ..."
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Abstract. We study the approximation properties of computably enumerable reals. We deal with a natural notion of approximation representation and study their wttdegrees. Also, we show that a single representation may correspond to a quite diverse variety of reals. 1.
Computable real functions of bounded variation and semicomputable real numbers
 In Proceedings of COCOON 2002
"... Abstract. In this paper we discuss some basic properties of computable real functions of bounded variation (CBVfunctions for short). Especially, it is shown that the image set of semicomputable real numbers under CBVfunctions is a proper subset of the class of weakly computable real numbers; Two ..."
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Abstract. In this paper we discuss some basic properties of computable real functions of bounded variation (CBVfunctions for short). Especially, it is shown that the image set of semicomputable real numbers under CBVfunctions is a proper subset of the class of weakly computable real numbers; Two applications of CBVfunctions to semicomputable real numbers produce the whole closure of semicomputable real numbers under total computable real functions, and the image sets of semicomputable real numbers under monotone computable functions and CBVfunctions are different. 1
Variants of computability
"... The definition is far from constructive. How do we effectivize? 6 ..."
Computable Functions of Bounded Variation and SemiComputable Real Numbers
, 2002
"... In this paper we discuss some basic properties of computable real functions which have bounded variations (CBVfunctions for short). Especially, it is shown that the image set of semicomputable real numbers under CBVfunctions is a proper subset of weakly computable real number class; Two applicati ..."
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In this paper we discuss some basic properties of computable real functions which have bounded variations (CBVfunctions for short). Especially, it is shown that the image set of semicomputable real numbers under CBVfunctions is a proper subset of weakly computable real number class; Two applications of CBVfunctions to semicomputable real numbers produce the whole closure of semicomputable real numbers under total computable real functions, and the image sets of semicomputable real numbers under monotone computable functions and CBVfunctions are different.