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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.
mlq header will be provided by the publisher hMonotonically Computable Real Numbers
"... Received xx yyyyyyy xxxx, revised xx yyyyyyy xxxx, accepted xx yyyyyy xxxx Published online xx yyyyyy xxxx Key words hmonotone computable reals; ωmonotone computable reals. MSC (2000) 03F60,03D55 Let h: N → Q be a computable function. A real number x is called hmonotonically computable (hmc, for ..."
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Received xx yyyyyyy xxxx, revised xx yyyyyyy xxxx, accepted xx yyyyyy xxxx Published online xx yyyyyy xxxx Key words hmonotone computable reals; ωmonotone computable reals. MSC (2000) 03F60,03D55 Let h: N → Q be a computable function. A real number x is called hmonotonically computable (hmc, for short) if there is a computable sequence (xs) of rational numbers which converges to x hmonotonically in the sense that h(n)x − xn  ≥ x − xm  for all n and m> n. In this paper we investigate classes hMC of hmc real numbers for different computable functions h. Especially, for computable functions h: N → (0, 1)Q, we show that the class hMC coincides with the classes of computable and semicomputable real numbers if and only if � i∈N (1 − h(i)) = ∞ and the sum i∈N (1 − h(i)) is a computable real number, respectively. On the other hand, if h(n) ≥ 1 and h converges to 1, then hMC = SC no matter how fast h converges to 1. Furthermore, for any constant c> 1, if h is increasing and converges to c, then hMC = cMC. Finally, if h is monotone and unbounded, then hMC contains all ωmc real numbers which are gmc for some computable function g. 1
On the Monotonic Computability of SemiComputable Real Numbers
"... Abstract. Let h: N → Q be a computable function. A real number x is hmonotonically computable (hmc, for short) if there is a computable sequence (xs) of rational numbers which converges to x in such a way that the ratios of the approximation errors are bounded by h. In this paper we discuss the h ..."
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Abstract. Let h: N → Q be a computable function. A real number x is hmonotonically computable (hmc, for short) if there is a computable sequence (xs) of rational numbers which converges to x in such a way that the ratios of the approximation errors are bounded by h. In this paper we discuss the hmonotonic computability of semicomputable real numbers which are limits of monotone computable sequences of rational numbers. Especially, we show a sufficient and necessary condition for the function h such that the hmonotonic computability is simply equivalent to the normal computability. 1
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.
Weak Computability and Representation of Real Numbers
"... 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.