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Monotonically computable real numbers
 Math. Log. Quart
, 2002
"... Key words hmonotone computable real, ωmonotone computable real. ..."
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Key words hmonotone computable real, ωmonotone computable real.
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 6 (3 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. 1
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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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.
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 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.
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 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 nonc ..."
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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