Results 1  10
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22
Computing a glimpse of randomness
 Experimental Mathematics
"... A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In ..."
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A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly noncomputable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 63 bits of a Chaitin Omega: 000000100000010000100000100001110111001100100111100010010011100. Full description of programs and proofs will be given elsewhere. 1
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 7 (4 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.
Five Lectures on Algorithmic Randomness
 in Computational Prospects of Infinity, ed. C.T. Chong, Proc. 2005 Singapore meeting
, 2007
"... This paper follows on from the author’s Five Lectures on Algorithmic Randomness. It is concerned with material not found in that long paper, concentrating on MartinLöf lowness and triviality. We present a hopefully userfriendly account of the decanter method, and discuss recent results of the auth ..."
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This paper follows on from the author’s Five Lectures on Algorithmic Randomness. It is concerned with material not found in that long paper, concentrating on MartinLöf lowness and triviality. We present a hopefully userfriendly account of the decanter method, and discuss recent results of the author with Peter Cholak and Noam Greenberg concerning the class of strongly jump traceable reals introduced by
Monotonically Computable Real Numbers
, 2001
"... A real number x is called kmonotonically computable (kmc), for constant k> 0, if there is a computable sequence (xn)n∈N of rational numbers which converges to x such that the convergence is kmonotonic in the sense that k · x−xn  ≥ x−xm  for any m> n and x is monotonically computable (m ..."
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Cited by 6 (4 self)
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A real number x is called kmonotonically computable (kmc), for constant k> 0, if there is a computable sequence (xn)n∈N of rational numbers which converges to x such that the convergence is kmonotonic in the sense that k · x−xn  ≥ x−xm  for any m> n and x is monotonically computable (mc) if it is kmc for some k> 0. x is weakly computable if there is a computable sequence (xs)s∈N of rational numbers converging to x such that the sum s∈N xs − xs+1  is finite. In this paper we show that all mc real numbers are weakly computable but the converse fails. Furthermore, we show also an infinite hierarchy of mc real numbers.
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.
Algorithmic Randomness and Computability
"... Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability. ..."
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Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability.
Superbranching degrees
 Proceedings Oberwolfach 1989, Springer Verlag Lecture Notes in Mathematics
, 1990
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Limit computability and constructive measure
 National University of Singapore, 2008. Institute of Mathematical Sciences
"... Abstract. In this paper we study constructive measure and dimension in the class 02 of limit computable sets. We prove that the lower cone of any Turingincomplete set in 02 has 0 2dimension 0, and in contrast, that although the upper cone of a noncomputable set in 02 always has 0 2measure 0, up ..."
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Abstract. In this paper we study constructive measure and dimension in the class 02 of limit computable sets. We prove that the lower cone of any Turingincomplete set in 02 has 0 2dimension 0, and in contrast, that although the upper cone of a noncomputable set in 02 always has 0 2measure 0, upper cones in 02 have nonzero