Results 1 
8 of
8
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. ..."
Abstract

Cited by 25 (7 self)
 Add to MetaCart
We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.
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. ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
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.
Degrees of d.c.e. reals
 Mathematical Logic Quartely
, 2004
"... A real α is called a c. e. real if it is the halting probability of a prefix free Turing machine. Equivalently, α is c. e. if it is left computable in the sense that L(α) ={q ∈ Q: q ≤ α} is a computably enumerable set. The natural field formed by the c. e. reals turns out to be the field formed by t ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
A real α is called a c. e. real if it is the halting probability of a prefix free Turing machine. Equivalently, α is c. e. if it is left computable in the sense that L(α) ={q ∈ Q: q ≤ α} is a computably enumerable set. The natural field formed by the c. e. reals turns out to be the field formed by the collection of the d. c. e. reals, which are of the form α − β,whereα and β are c. e. reals. While c. e. reals can only be found in the c. e. degrees, Zheng has proven that there are ∆ 0 2 degrees that are not even nc. e. for any n and yet contain d. c. e. reals, where a degree is nc. e. if it contains an nc. e. set. In this paper we will prove that every ωc. e. degree contains a d. c. e. real, but there are ω +1c. e. degrees and, hence ∆ 0 2 degrees, containing no d. c. e. real. 1
Computability, Definability and Algebraic Structures
, 1999
"... In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set
Singular Coverings and NonUniform Notions of Closed Set Computability
, 2007
"... Abstract. The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe assert the existence of nonempty cor.e. closed sets devoid of computable points: sets which are even ‘large ’ in the sense of positive Lebesgue mea ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe assert the existence of nonempty cor.e. closed sets devoid of computable points: sets which are even ‘large ’ in the sense of positive Lebesgue measure. We observe that a certain size is in fact necessary: every nonempty cor.e. closed real set without computable points has continuum cardinality. This initiates a comparison of different notions of computability for closed real subsets nonuniformly like, e.g., for sets of fixed cardinality or sets containing a (not necessarily effectively findable) computable point. By relativization we obtain a bounded recursive rational sequence of which every accumulation point is not even computable with support of a Halting oracle. Finally the question is treated whether compact sets have cor.e. closed connected components; and every starshaped cor.e. closed set is asserted to contain a computable point. 1
Singular Coverings and NonUniform Notions of Closed Set Computability
, 2006
"... Abstract. The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe assert the existence of nonempty cor.e. closed sets devoid of computable points: sets which are even ‘large ’ in the sense of positive Lebesgue mea ..."
Abstract
 Add to MetaCart
Abstract. The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe assert the existence of nonempty cor.e. closed sets devoid of computable points: sets which are even ‘large ’ in the sense of positive Lebesgue measure. We observe that a certain size is in fact necessary: every nonempty cor.e. closed real set without computable points has continuum cardinality. More generally, the present paper exhibits a comparison of different notions of computability for closed real subsets nonuniformly like, e.g., for sets of fixed cardinality or sets containing a (not necessarily effectively findable) computable point. By relativization we finally obtain a bounded recursive rational sequence of which every accumulation point is not even computable with support of a Halting oracle. 1