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50
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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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 29 (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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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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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.
Lowness for Kurtz randomness
 J. Symbolic Logic
"... Abstract. We prove that degrees that are low for Kurtz randomness cannot be diagonally nonrecursive. Together with the work of Stephan and Yu [16], this proves that they coincide with the hyperimmunefree nonDNR degrees, which are also exactly the degrees that are low for weak 1genericity. We als ..."
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Abstract. We prove that degrees that are low for Kurtz randomness cannot be diagonally nonrecursive. Together with the work of Stephan and Yu [16], this proves that they coincide with the hyperimmunefree nonDNR degrees, which are also exactly the degrees that are low for weak 1genericity. We also consider Low(M,Kurtz), the class of degrees a such that every element of M is aKurtz random. These are characterised when M is the class of MartinLöf random, computably random, or Schnorr random reals. We show that Low(ML,Kurtz) coincides with the nonDNR degrees, while both Low(CR,Kurtz) and Low(Schnorr,Kurtz) are exactly the nonhigh, nonDNR degrees. 1.
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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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 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 11 (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.
DEMUTH RANDOMNESS AND COMPUTATIONAL COMPLEXITY
"... Demuth tests generalize MartinLöf tests (Gm)m∈N in that one can exchange the mth component for a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that Gm ⊇ Gm+1 for each m, we have weak Demu ..."
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Cited by 8 (3 self)
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Demuth tests generalize MartinLöf tests (Gm)m∈N in that one can exchange the mth component for a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that Gm ⊇ Gm+1 for each m, we have weak Demuth randomness. We show that a weakly Demuth random set can be high, yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly jumptraceable. We also prove a basis theorem for nonempty Π 0 1 classes P. It extends the JockuschSoare basis theorem that some member of P is computably dominated. We use the result to show that some weakly 2random set does not compute a 2fixed point free function.
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 7 (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.