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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.
Lowness properties and approximations of the jump
 Proceedings of the Twelfth Workshop of Logic, Language, Information and Computation (WoLLIC 2005). Electronic Lecture Notes in Theoretical Computer Science 143
, 2006
"... ..."
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.
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 ..."
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Cited by 2 (2 self)
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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
Chaitin\Omega numbers, Solovay machines and Godel incompleteness. Theoretical Computer Science
 Theoretical Computer Science
, 2000
"... Computably enumerable (c.e.) reals can be coded by Chaitin machines through their halting probabilities. Tuning Solovay’s construction of a Chaitin universal machine for which ZFC (if arithmetically sound) cannot determine any single bit of the binary expansion of its halting probability, we show th ..."
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Cited by 2 (0 self)
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Computably enumerable (c.e.) reals can be coded by Chaitin machines through their halting probabilities. Tuning Solovay’s construction of a Chaitin universal machine for which ZFC (if arithmetically sound) cannot determine any single bit of the binary expansion of its halting probability, we show that every c.e. random real is the halting probability of a universal Chaitin machine for which ZFC cannot determine more than its initial block of 1 bits—as soon as you get a 0, it is all over. Finally, a constructive version of Chaitin informationtheoretic incompleteness
Chaitin Ω numbers, Solovay machines and Gödel incompleteness
 THEORETICAL COMPUTER SCIENCE
, 2002
"... Computably enumerable (c.e.) reals can be coded by Chaitin machines through their halting probabilities. Tuning Solovay’s construction of a Chaitin universal machine for which ZFC (if arithmetically sound) cannot determine any single bit of the binary expansion of its halting probability, we show th ..."
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Cited by 2 (0 self)
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Computably enumerable (c.e.) reals can be coded by Chaitin machines through their halting probabilities. Tuning Solovay’s construction of a Chaitin universal machine for which ZFC (if arithmetically sound) cannot determine any single bit of the binary expansion of its halting probability, we show that every c.e. random real is the halting probability of a universal Chaitin machine for which ZFC cannot determine more than its initial block of 1 bits—as soon as you get a 0, it is all over. Finally, a constructive version of Chaitin informationtheoretic incompleteness
Computably enumerable reals and uniformly presentable ideals
 Mathematical Logic Quarterly
"... We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefixfree set of strings such that α = ∑ σ∈A 2−σ . Note that ∑ σ∈A 2−σ  is precisely the measure of the set of reals that have a ..."
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Cited by 2 (2 self)
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We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefixfree set of strings such that α = ∑ σ∈A 2−σ . Note that ∑ σ∈A 2−σ  is precisely the measure of the set of reals that have a string in A as an initial segment. So we will simply abbreviate ∑ σ∈A 2−σ  by µ(A). It is known that whenever A so presents α then A ≤wtt α, where ≤wtt denotes weak truth table reducibility, and that the wtt degrees of presentations form an ideal I(α) in the computably enumerable wtt degrees. We prove that any such ideal is Σ 0 3, and conversely that if I is any Σ 0 3 ideal in the computably enumerable wtt degrees then there is a computable enumerable real α such that I = I(α). 1