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20
Randomness and reducibility
 J. Comput. System Sci
, 2001
"... How random is a real? Given two reals, which is more random? If we partition reals into equivalence classes of reals of the “same degrees of randomness”, what does the resulting structure look like? The goal of this paper is to look at questions like these, specifically by studying the properties of ..."
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Cited by 26 (4 self)
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How random is a real? Given two reals, which is more random? If we partition reals into equivalence classes of reals of the “same degrees of randomness”, what does the resulting structure look like? The goal of this paper is to look at questions like these, specifically by studying the properties of reducibilities that act as
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.
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.
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.
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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Cited by 2 (2 self)
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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. Downey is partially supported by the New Zealand Marsden Fund. Wu is supported by the New Zealand FRST PostDoctoral Fellowship. Downey and Wu are partially supported by the International Joint Project No. 00310308 of NSFC of China. 1 1
Asymptotic density and the Ershov hierarchy, in preparation
"... Abstract. We classify the asymptotic densities of the ∆02 sets according to their level in the Ershov hierarchy. In particular, it is shown that for n ≥ 2, a real r ∈ [0, 1] is the density of an nc.e. set if and only if it is a difference of leftΠ02 reals. Further, we show that the densities of th ..."
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Cited by 1 (1 self)
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Abstract. We classify the asymptotic densities of the ∆02 sets according to their level in the Ershov hierarchy. In particular, it is shown that for n ≥ 2, a real r ∈ [0, 1] is the density of an nc.e. set if and only if it is a difference of leftΠ02 reals. Further, we show that the densities of the ωc.e. sets coincide with the densities of the ∆02 sets, and there are ωc.e. sets whose density is not the density of an nc.e. set for any n ∈ ω. 1.
Abstract On the Hierarchy of ∆ 0 2Real Numbers
"... 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. Key words: Computably approximable reals, ∆0 2reals, hierarchy 1
1 RECOGNIZING STRONG RANDOM REALS
"... Abstract. The class of strong random reals can be defined via a natural conception of effective null set. We show that the same class is also characterized by a learningtheoretic criterion of ‘recognizability’. 1. Characterizing randomness. Consider a physical process that, if suitably idealized, g ..."
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Abstract. The class of strong random reals can be defined via a natural conception of effective null set. We show that the same class is also characterized by a learningtheoretic criterion of ‘recognizability’. 1. Characterizing randomness. Consider a physical process that, if suitably idealized, generates an indefinite sequence of independent random bits. One such process might be radioactive decay of a lump of uranium whose mass is kept at just the level needed to ensure that the probability is onehalf that no alpha particle is emitted in the nth microsecond of the experiment. Let us think of the bits as drawn from {0, 1} and denote the resulting sequence by x with coordinates x0, x1,.... Now wouldn’t it be odd if there were a computer program P with the following property? 1. For any input i, P enters a nonterminating routine that writes a nonempty, finite sequence b1,...,bm with bm = xi (m depends on i). 1 The program will not, in general, allow prediction of xi inasmuch as there is no requirement that the ultimate bit bm written by P(i) be marked as final. Nonetheless, shouldn’t randomness exclude any computational process from having the kind of intimate knowledge of xi described in 1? The tension engendered by 1 afflicts a celebrated theory of randomness developed over the last half century. 2 The theory offers diverse criteria, each well motivated, for the concept ‘infinite sequence of random bits’. Remarkably, the criteria yield the same collection of sequences – a collection, moreover, of measure 1 with respect to the ‘coin flip ’ measure on the collection of infinite binary sequences. Despite this evidence for theoretical adequacy, some of the sequences labeled ‘random ’ can be associated with a program P satisfying 1. One response to this state of affairs has been to modify (in a simple and satisfying way) the randomness criteria originally proposed by MartinLöf (1966). The resulting collection Received xxxxx, 200x.