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Some ComputabilityTheoretical Aspects of Reals and Randomness
 the Lect. Notes Log. 18, Assoc. for Symbol. Logic
, 2001
"... We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. ..."
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Cited by 25 (7 self)
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We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.
Randomness, computability, and density
 SIAM Journal of Computation
, 2002
"... 1 Introduction In this paper we are concerned with effectively generated reals in the interval (0; 1] and their relative randomness. In what follows, real and rational will mean positive real and positive rational, respectively. It will be convenient to work modulo 1, that is, identifying n + ff and ..."
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Cited by 13 (6 self)
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1 Introduction In this paper we are concerned with effectively generated reals in the interval (0; 1] and their relative randomness. In what follows, real and rational will mean positive real and positive rational, respectively. It will be convenient to work modulo 1, that is, identifying n + ff and ff for any n 2! and ff 2 (0; 1], and we do this below without further comment.
Presentations of computably enumerable reals
 Theoretical Computer Science
, 2002
"... Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefixfree set of binary strings. ..."
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Cited by 10 (5 self)
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Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefixfree set of binary strings.
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
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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Cited by 3 (1 self)
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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.
A Finite Hierarchy of the Recursively Enumerable Real Numbers
 In Proceedings of MFCS’98
, 1998
"... . For any set A of natural numbers, denote by x A the corresponding real number such that A is just the set of "1" positions in its binary expansion. In this paper we characterize the number x A for some classes of recursively enumerable sets A. Applying finite injury priority methods we show that t ..."
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. For any set A of natural numbers, denote by x A the corresponding real number such that A is just the set of "1" positions in its binary expansion. In this paper we characterize the number x A for some classes of recursively enumerable sets A. Applying finite injury priority methods we show that there is a dr.e. set A such that x A is not a semicomputable real number (which corresponds to the limit of computable monotonic sequence of rational numbers) and that there is an #r.e. set A such that x A can't be represented as a sum of two semicomputable real numbers. 1 Introduction Arealnumberx is computable, if there is a computable Cauchy sequence (r n ) n#IN of rational numbers which converges e#ectively to x (see e.g. [3,5,6].) Where a sequence (r n ) n#IN of rational numbers is computable means that there are recursive functions a, b, c :IN# IN s u c h t h a t r n =(a(n)  b(n))/(c(n)+1)foralln # IN and the sequence (r n ) n#IN converges e#ectively means that r n+m  ...