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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.
Degrees of d.c.e. reals
 MATHEMATICAL LOGIC QUARTELY
, 2004
"... A real is called c.e. if it is the halting probability of a prefix free Turing machine. Equivalently, a real 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 co ..."
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A real is called c.e. if it is the halting probability of a prefix free Turing machine. Equivalently, a real 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 reals of the form α − β where α 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. 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.
Weak Computability and Representation of Real Numbers
, 2003
"... Analogous to Ershov’s hierarchy for ∆02subsets of natural numbers we discuss the similar hierarchy for recursively approximable real numbers. Namely, we define the kcomputability for natural number k and fcomputability for function f. We will show that these notions are not equivalent for differe ..."
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Analogous to Ershov’s hierarchy for ∆02subsets of natural numbers we discuss the similar hierarchy for recursively approximable real numbers. Namely, we define the kcomputability for natural number k and fcomputability for function f. We will show that these notions are not equivalent for different representations of real numbers based on Cauchy sequence, Dedekind cut and binary expansion.
On the Hierarchy of ∆ 0 2Real Numbers
, 2004
"... A real number x is called ∆0 2 if its binary expansion corresponds to a ∆02set of natural numbers. Such reals are just the limits of computable sequences of rational numbers and hence also called computably approximable. Depending on how fast the sequences converge, ∆0 2reals have different levels ..."
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A real number x is called ∆0 2 if its binary expansion corresponds to a ∆02set of natural numbers. Such reals are just the limits of computable sequences of rational numbers and hence also called computably approximable. Depending on how fast the sequences converge, ∆0 2reals have different levels of effectiveness. This leads to various hierarchies of ∆0 2 reals. In this paper we summarize several recent developments related to such kind of hierarchies.
6th Conference on Real Numbers and Computers
, 2004
"... Setting up the 6th conference on Real Numbers and Computers has been a great pleasure. Once again, we will bring together specialists from various research areas, all concerned with problems related to theoretical calculability or actual computations based on real numbers. These computations use man ..."
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Setting up the 6th conference on Real Numbers and Computers has been a great pleasure. Once again, we will bring together specialists from various research areas, all concerned with problems related to theoretical calculability or actual computations based on real numbers. These computations use many
SOME PROPERTIES OF D.C.E. REALS AND THEIR DEGREES
, 2005
"... Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ..."
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Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not.