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22
Using random sets as oracles
"... Let R be a notion of algorithmic randomness for individual subsets of N. We say B is a base for R randomness if there is a Z �T B such that Z is R random relative to B. We show that the bases for 1randomness are exactly the Ktrivial sets and discuss several consequences of this result. We also sho ..."
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Cited by 34 (15 self)
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Let R be a notion of algorithmic randomness for individual subsets of N. We say B is a base for R randomness if there is a Z �T B such that Z is R random relative to B. We show that the bases for 1randomness are exactly the Ktrivial sets and discuss several consequences of this result. We also show that the bases for computable randomness include every ∆ 0 2 set that is not diagonally noncomputable, but no set of PAdegree. As a consequence, we conclude that an nc.e. set is a base for computable randomness iff it is Turing incomplete. 1
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 33 (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.
Measures and their random reals
 IN PREPARATION
"... We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every nonrecursive real is nontrivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of ..."
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Cited by 11 (2 self)
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We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every nonrecursive real is nontrivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every nonhyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for a continuous measure can be found throughout the hyperarithmetical Turing degrees.
Effectively closed sets of measures and randomness
 Ann. Pure Appl. Logic
"... We show that if a real x ∈ 2ω is strongly Hausdorff Hhrandom, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure µ such that the µmeasure of the basic open cylinders shrinks according to h. The proof uses a new method to con ..."
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Cited by 7 (1 self)
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We show that if a real x ∈ 2ω is strongly Hausdorff Hhrandom, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure µ such that the µmeasure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for Π0 1classes applied to closed sets of probability measures. We use the main result to give a new proof of Frostman’s Lemma, to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman’s Theorem. 1
On the gap between trivial and nontrivial initial segment prefixfree complexity
, 2010
"... Abstract. An infinite sequence X is said to have trivial (prefixfree) initial segment complexity if K(X ↾n) ≤ + K(0n) for all n, where K is the prefixfree complexity and ≤ + denotes inequality modulo a constant. In other words, if the information in any initial segment of it is merely the informa ..."
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Cited by 5 (4 self)
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Abstract. An infinite sequence X is said to have trivial (prefixfree) initial segment complexity if K(X ↾n) ≤ + K(0n) for all n, where K is the prefixfree complexity and ≤ + denotes inequality modulo a constant. In other words, if the information in any initial segment of it is merely the information in a sequence of 0s of the same length. We study the gap between the trivial complexity K(0n) and the complexity of a nontrivial sequence, i.e. the functions f such that (⋆) K(X ↾n) ≤ + K(0 n) + f(n) for all n for a nontrivial (in terms of initial segment complexity) sequence X. We show that given any ∆0 2 unbounded nondecreasing function f there exist uncountably many sequences X which satisfy (⋆). On the other hand there exists a ∆0 3 unbounded nondecreasing function f which does not satisfy (⋆) for any X with nontrivial initial segment complexity. This improves the bound ∆0 4 that was known from [CM06]. Finally we give some applications of these results. 1.
Eliminating concepts
 Proceedings of the IMS workshop on computational prospects of infinity
, 2008
"... Four classes of sets have been introduced independently by various researchers: low for K, low for MLrandomness, basis for MLrandomness and Ktrivial. They are all equal. This survey serves as an introduction to these coincidence results, obtained in [24] and [10]. The focus is on providing backdo ..."
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Cited by 5 (2 self)
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Four classes of sets have been introduced independently by various researchers: low for K, low for MLrandomness, basis for MLrandomness and Ktrivial. They are all equal. This survey serves as an introduction to these coincidence results, obtained in [24] and [10]. The focus is on providing backdoor access to the proofs. 1. Outline of the results All sets will be subsets of N unless otherwise stated. K(x) denotes the prefix free complexity of a string x. A set A is Ktrivial if, within a constant, each initial segment of A has minimal prefix free complexity. That is, there is c ∈ N such that ∀n K(A ↾ n) ≤ K(0 n) + c. This class was introduced by Chaitin [5] and further studied by Solovay (unpublished). Note that the particular effective epresentation of a number n by a string (unary here) is irrelevant, since up to a constant K(n) is independent from the representation. A is low for MartinLöf randomness if each MartinLöf random set is already MartinLöf random relative to A. This class was defined in Zambella [28], and studied by Kučera and Terwijn [17]. In this survey we will see that the two classes are equivalent [24]. Further concepts have been introduced: to be a basis for MLrandomness (Kučera [16]), and to be low for K (Muchnik jr, in a seminar at Moscow State, 1999). They will also be eliminated, by showing equivalence with Ktriviality. All
CHAITIN’S HALTING PROBABILITY AND THE COMPRESSION OF STRINGS USING ORACLES
"... If a computer is given access to an oracle—the characteristic function of a set whose membership relation may or may not be algorithmically calculable—this may dramatically affect its ability to compress information and to determine structure in strings which might otherwise appear random. This lea ..."
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Cited by 4 (3 self)
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If a computer is given access to an oracle—the characteristic function of a set whose membership relation may or may not be algorithmically calculable—this may dramatically affect its ability to compress information and to determine structure in strings which might otherwise appear random. This leads to the basic question, “given an oracle A, how many oracles can compress information at most as well as A?” This question can be formalized using Kolmogorov complexity. We say that B ≤LK A if there exists a constant c such that K A (σ) < K B (σ) + c for all strings σ, where K X denotes the prefixfree Kolmogorov complexity relative to oracle X. The formal counterpart to the previous question now is, “what is the cardinality of the set of ≤LKpredecessors of A?” We completely determine the number of oracles that compress at most as well as any given oracle A, by answering a question of Miller [Mil10, Section 5] which also appears in Nies [Nie09, Problem 8.1.13]; the class of ≤LKpredecessors of a set A is countable if and only if Chaitin’s halting probability Ω is MartinLöf random relative to A.
Probability measures and effective randomness
, 2007
"... Abstract. We study the question, “For which reals x does there exist a measure µ such that x is random relative to µ? ” We show that for every nonrecursive x, there is a measure which makes x random without concentrating on x. We give several conditions on x equivalent to there being continuous meas ..."
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Cited by 2 (0 self)
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Abstract. We study the question, “For which reals x does there exist a measure µ such that x is random relative to µ? ” We show that for every nonrecursive x, there is a measure which makes x random without concentrating on x. We give several conditions on x equivalent to there being continuous measure which makes x random. We show that for all but countably many reals x these conditions apply, so there is a continuous measure which makes x random. There is a metamathematical aspect of this investigation. As one requires higher arithmetic levels in the degree of randomness, one must make use of more iterates of the power set of the continuum to show that for all but countably many x’s there is a continuous µ which makes x random to that degree. 1.
LIMIT COMPUTABILITY AND CONSTRUCTIVE MEASURE
"... In this paper we study constructive measure and dimension in the class ∆0 2 of limit computable sets. We prove that the lower cone of any Turingincomplete set in ∆0 2 has ∆0 2dimension 0, and in contrast, that although the upper cone of a noncomputable set in ∆0 2 always has ∆0 2measure 0, upper ..."
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Cited by 1 (0 self)
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In this paper we study constructive measure and dimension in the class ∆0 2 of limit computable sets. We prove that the lower cone of any Turingincomplete set in ∆0 2 has ∆0 2dimension 0, and in contrast, that although the upper cone of a noncomputable set in ∆0 2 always has ∆0 2measure 0, upper cones in ∆0 2 have nonzero ∆0 2dimension. In particular the ∆0 2dimension of the Turing degree of ∅ ′ (the Halting Problem) is 1. Finally, it is proved that the low sets do not have ∆0 2measure 0, which means that the low sets do not form a small subset of ∆0 2. This result has consequences for the existence of biimmune sets.