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13
Some ComputabilityTheoretical Aspects of Reals and Randomness
, 2001
"... We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. Then we study such objects in terms of algorithmic randomness, culminating in some recent work of the author with Hirschfeldt, Laforte, and Nies conce ..."
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Cited by 21 (7 self)
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We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. Then we study such objects in terms of algorithmic randomness, culminating in some recent work of the author with Hirschfeldt, Laforte, and Nies concerning methods of calibrating randomness.
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 20 (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.
Effectively Closed Sets
 ASL Lecture Notes in Logic
"... Abstract. We investigate notions of randomness in the space C[2 IN] of nonempty closed subsets of {0, 1} IN. A probability measure is given and a version of the MartinLöf Test for randomness is defined. Π 0 2 random closed sets exist but there are no random Π 0 1 closed sets. It is shown that a ran ..."
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Cited by 12 (5 self)
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Abstract. We investigate notions of randomness in the space C[2 IN] of nonempty closed subsets of {0, 1} IN. A probability measure is given and a version of the MartinLöf Test for randomness is defined. Π 0 2 random closed sets exist but there are no random Π 0 1 closed sets. It is shown that a random closed set is perfect, has measure 0, and has no computable elements. A closed subset of 2 IN may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. This leads to some results on a Chaitinstyle notion of randomness for closed sets. 1
Presentations of computably enumerable reals
 Theoretical Computer Science
, 2002
"... Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefixfree set of binary strings. ..."
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Cited by 9 (5 self)
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Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefixfree set of binary strings.
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.
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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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.
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 2 (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.
Closed leftr.e. sets
, 2011
"... Abstract. A set is called rclosed leftr.e. iff every set rreducible to it is also a leftr.e. set. It is shown that some but not all leftr.e. cohesive sets are manyone closed leftr.e. sets. Ascending reductions are manyone reductions via an ascending function; leftr.e. cohesive sets are als ..."
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Cited by 1 (1 self)
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Abstract. A set is called rclosed leftr.e. iff every set rreducible to it is also a leftr.e. set. It is shown that some but not all leftr.e. cohesive sets are manyone closed leftr.e. sets. Ascending reductions are manyone reductions via an ascending function; leftr.e. cohesive sets are also ascending closed leftr.e. sets. Furthermore, it is shown that there is a weakly 1generic manyone closed leftr.e. set. We also consider initial segment complexity of closed leftr.e. sets. We show that initial segment complexity of ascending closed leftr.e. sets is of sublinear order. Furthermore, this is near optimal as for any nondecreasing unbounded recursive function g, there are ascending closed leftr.e. sets A with initial segment complexity C(A(0)A(1)... A(n)) ≥ cn/g(n) for some constant c and all n. The initial segment complexity of a conjunctively (or disjunctively) closed leftr.e. set satisfies, for all ε, for almost all n, C(A(0)A(1)... A(n)) ≤ (2 + ε) log(n). 1
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.