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13
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
Monotonically computable real numbers
 Math. Log. Quart
, 2002
"... Key words hmonotone computable real, ωmonotone computable real. ..."
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Cited by 7 (5 self)
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Key words hmonotone computable real, ωmonotone computable real.
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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Cited by 7 (2 self)
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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
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
The Kolmogorov Complexity of Random Reals
 Ann. Pure Appl. Logic
, 2003
"... We investigate the initial segment complexity of random reals. Let K(... ..."
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Cited by 5 (1 self)
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We investigate the initial segment complexity of random reals. Let K(...
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.
Ktrivial closed sets and continuous functions
 in Proceedings of CIE 2007
"... Abstract. We investigate the notion of Ktriviality for closed sets and continuous functions. Every Ktrivial closed set contains a Ktrivial real. There exists a Ktrivial Π 0 1 class with no computable elements. For any Ktrivial degree d, there is a Ktrivial continuous function of degree d. 1 ..."
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Abstract. We investigate the notion of Ktriviality for closed sets and continuous functions. Every Ktrivial closed set contains a Ktrivial real. There exists a Ktrivial Π 0 1 class with no computable elements. For any Ktrivial degree d, there is a Ktrivial continuous function of degree d. 1
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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Cited by 2 (2 self)
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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.
Computability and applications to analysis
, 2004
"... The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be publiced without prope ..."
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The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be publiced without proper acknowledgement. We study real numbers from the point of view of effectiveness and computability, especially regarding their approximations by ‘effective ’ sequences of rational numbers. For this study we employ the two main classification methods from computability theory: hierarchies and degrees. We are especially interested in establishing connections with the classical theory. In chapter 1 we extend the hierarchy defined in Weihrauch and Zheng [33] (classifying the arithmetical reals) to cover all hyperarithmetical real numbers. In chapter 2 we start the study of approximations of reals by means of degree structures. This is a new approach for the classification of the computably approximable (i.e. ∆2) reals which yields a rich and interesting theory with many connections to the