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29
On initial segment complexity and degrees of randomness
 Trans. Amer. Math. Soc
"... Abstract. One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define X ≤K Y to mean that (∀n) K(X ↾ n) ≤ K(Y ↾ n) +O(1). The equivalence classes under this relation are the Kdegrees. We prove that if X ⊕ Y is 1rand ..."
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Cited by 32 (6 self)
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Abstract. One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define X ≤K Y to mean that (∀n) K(X ↾ n) ≤ K(Y ↾ n) +O(1). The equivalence classes under this relation are the Kdegrees. We prove that if X ⊕ Y is 1random, then X and Y have no upper bound in the Kdegrees (hence, no join). We also prove that nrandomness is closed upward in the Kdegrees. Our main tool is another structure intended to measure the degree of randomness of real numbers: the vLdegrees. Unlike the Kdegrees, many basic properties of the vLdegrees are easy to prove. We show that X ≤K Y implies X ≤vL Y, so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for ≤C, the analogue of ≤K for plain Kolmogorov complexity. Two other interesting results are included. First, we prove that for any Z ∈ 2ω, a 1random real computable from a 1Zrandom real is automatically 1Zrandom. Second, we give a plain Kolmogorov complexity characterization of 1randomness. This characterization is related to our proof that X ≤C Y implies X ≤vL Y. 1.
Low for random reals and positivemeasure domination
 Proceedings of the American Mathematical Society, 2007. Preprint
, 2005
"... Abstract. The low for random reals are characterized topologically, as well as in terms of domination of Turing functionals on a set of positive measure. 1. ..."
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Cited by 22 (1 self)
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Abstract. The low for random reals are characterized topologically, as well as in terms of domination of Turing functionals on a set of positive measure. 1.
Relativizing Chaitin’s halting probability
 J. Math. Log
"... Abstract. As a natural example of a 1random real, Chaitin proposed the halting probability Ω of a universal prefixfree machine. We can relativize this example by considering a universal prefixfree oracle machine U. Let Ω A U be the halting probability of U A; this gives a natural uniform way of p ..."
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Cited by 21 (7 self)
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Abstract. As a natural example of a 1random real, Chaitin proposed the halting probability Ω of a universal prefixfree machine. We can relativize this example by considering a universal prefixfree oracle machine U. Let Ω A U be the halting probability of U A; this gives a natural uniform way of producing an Arandom real for every A ∈ 2 ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory and this Omega operator. But unlike the jump, which is invariant (up to computable permutation) under the choice of an effective enumeration of the partial computable functions, Ω A U can be vastly different for different choices of U. Even for a fixed U, there are oracles A = ∗ B such that Ω A U and Ω B U are 1random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness. 1.
Lowness for the class of Schnorr random reals
 SIAM Journal on Computing
, 2005
"... We answer a question of AmbosSpies and Kučera in the affirmative. They asked whether, when a real is low for Schnorr randomness, it is already low for Schnorr tests. ..."
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Cited by 12 (5 self)
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We answer a question of AmbosSpies and Kučera in the affirmative. They asked whether, when a real is low for Schnorr randomness, it is already low for Schnorr tests.
Constructive dimension and weak truthtable degrees
 In Computation and Logic in the Real World  Third Conference of Computability in Europe. SpringerVerlag Lecture Notes in Computer Science #4497
, 2007
"... Abstract. This paper examines the constructive Hausdorff and packing dimensions of weak truthtable degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truthtable equivalent to a sequence R with ..."
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Cited by 12 (3 self)
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Abstract. This paper examines the constructive Hausdorff and packing dimensions of weak truthtable degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truthtable equivalent to a sequence R with dimH(R) ≥ dimH(S)/dimP(S) − ɛ, for arbitrary ɛ> 0. Furthermore, if dimP(S)> 0, then dimP(R) ≥ 1−ɛ. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of wtt degrees (and, by extension, Turing degrees). A lower bound of dimH(S)/dimP(S) is shown to hold for the wtt degree of any sequence S. A new proof is given of a previouslyknown zeroone law for the constructive packing dimension of wtt degrees. It is also shown that, for any regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S)> 0, the wtt degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor.
Randomness and universal machines
 CCA 2005, Second International Conference on Computability and Complexity in Analysis, Fernuniversität Hagen, Informatik Berichte 326:103–116
, 2005
"... The present work investigates several questions from a recent survey of Miller and Nies related to Chaitin’s Ω numbers and their dependence on the underlying universal machine. It is shown that there are universal machines for which ΩU is just x 21−H(x). For such a universal machine there exists a c ..."
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Cited by 10 (6 self)
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The present work investigates several questions from a recent survey of Miller and Nies related to Chaitin’s Ω numbers and their dependence on the underlying universal machine. It is shown that there are universal machines for which ΩU is just x 21−H(x). For such a universal machine there exists a cor.e. set X such that ΩU[X] = � p:U(p)↓∈X 2−p  is neither leftr.e. nor MartinLöf random. Furthermore, one of the open problems of Miller and Nies is answered completely by showing that there is a sequence Un of universal machines such that the truthtable degrees of the ΩUn form an antichain. Finally it is shown that the members of hyperimmunefree Turing degree of a given Π0 1class are not low for Ω unless this class contains a recursive set. 1
Constructive dimension and Turing degrees
"... This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is Turing equivalent to a sequence R with dimH(R) ≥ (dimH(S)/dimP(S)) ..."
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Cited by 6 (0 self)
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This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is Turing equivalent to a sequence R with dimH(R) ≥ (dimH(S)/dimP(S)) − ɛ, for arbitrary ɛ> 0. Furthermore, if dimP(S)> 0, then dimP(R) ≥ 1 − ɛ. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dimH(S)/dimP(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previouslyknown zeroone law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S)> 0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truthtable and bounded Turing reductions differ in their ability to extract dimension.
Effective randomness for computable probability measures
, 2006
"... Any notion of effective randomness that is defined with respect to arbitrary computable probability measures canonically induces an equivalence relation on such measures for which two measures are considered equivalent if their respective classes of random elements coincide. Elaborating on work of B ..."
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Cited by 4 (0 self)
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Any notion of effective randomness that is defined with respect to arbitrary computable probability measures canonically induces an equivalence relation on such measures for which two measures are considered equivalent if their respective classes of random elements coincide. Elaborating on work of Bienvenu [1], we determine all the implications that hold between the equivalence relations induced by MartinLöf randomness, computable randomness, Schnorr randomness, and weak randomness, and the equivalence and consistency relations of probability measures, except that we do not know whether two computable probability measures need to be equivalent in case their respective concepts of weakly randomness coincide. Keywords: computable probability measures, MartinLöf randomness, computable randomness, Schnorr randomness, weak randomness, equivalence of probability measures, consistency of probability measures.
Constructive equivalence relations on computable probability measures
 International Computer Science Symposium in Russia, Lecture Notes in Computer Science
, 2006
"... Abstract. We study the equivalence relations on probability measures corresponding respectively to having the same MartinLöf random reals, having the same KolmogorovLoveland random reals, and having the same computably random reals. In particular, we show that, when restricted to the class of stro ..."
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Cited by 3 (2 self)
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Abstract. We study the equivalence relations on probability measures corresponding respectively to having the same MartinLöf random reals, having the same KolmogorovLoveland random reals, and having the same computably random reals. In particular, we show that, when restricted to the class of strongly positive generalized Bernoulli measures, they all coincide with the classical equivalence, which requires that two measures have the same nullsets. 1
HIGHER RANDOMNESS NOTIONS AND THEIR LOWNESS PROPERTIES
, 2007
"... Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1randomness ⊂ Π 1 1MartinLöf randomness ⊂ ∆ 1 1randomness = ∆ 1 1MartinLöf randomness. We characterize the set of reals that are low for ∆ 1 1 randomness as precisely those that are ∆ 1 ..."
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Cited by 2 (2 self)
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Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1randomness ⊂ Π 1 1MartinLöf randomness ⊂ ∆ 1 1randomness = ∆ 1 1MartinLöf randomness. We characterize the set of reals that are low for ∆ 1 1 randomness as precisely those that are ∆ 1 1traceable. We prove that there is a perfect set of such reals. 1.