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Mass problems and hyperarithmeticity
, 2006
"... A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of we ..."
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Cited by 24 (16 self)
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A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of the Cantor space. The lattice Pw has been studied in previous publications. The purpose of this paper is to show that Pw partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in Pw which are indexed by the ordinal numbers less than ω CK 1 and which correspond to the hyperarithmetical hierarchy. Namely, for each α < ω CK 1 let hα be the weak degree of 0 (α) , the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p ∗ be the weak degree
Almost everywhere domination and superhighness
 Mathematical Logic Quarterly
"... Let ω denote the set of natural numbers. For functions f, g: ω → ω, we say that f is dominated by g if f(n) < g(n) for all but finitely many n ∈ ω. We consider the standard “fair coin ” probability measure on the space 2 ω of infinite sequences of 0’s and 1’s. A Turing oracle B is said to be almo ..."
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Cited by 18 (9 self)
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Let ω denote the set of natural numbers. For functions f, g: ω → ω, we say that f is dominated by g if f(n) < g(n) for all but finitely many n ∈ ω. We consider the standard “fair coin ” probability measure on the space 2 ω of infinite sequences of 0’s and 1’s. A Turing oracle B is said to be almost everywhere dominating if, for measure one many X ∈ 2 ω, each function which is Turing computable from X is dominated by some function which is Turing computable from B. Dobrinen and Simpson have shown that the almost everywhere domination property and some of its variant properties are closely related to the reverse mathematics of measure theory. In this paper we exposit some recent results of KjosHanssen, KjosHanssen/Miller/Solomon, and others concerning LRreducibility and almost everywhere domination. We also prove the following new result: If B is almost everywhere dominating, then B is superhigh, i.e., 0 ′′ is
Lowness notions, measure and domination
, 2008
"... Abstract. We show that positive measure domination implies uniform almost everywhere domination and that this proof translates into a proof in the subsystem WWKL0 (but not in RCA0) of the equivalence of various Lebesgue measure regularity statements introduced by Dobrinen and Simpson. This work also ..."
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Cited by 15 (1 self)
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Abstract. We show that positive measure domination implies uniform almost everywhere domination and that this proof translates into a proof in the subsystem WWKL0 (but not in RCA0) of the equivalence of various Lebesgue measure regularity statements introduced by Dobrinen and Simpson. This work also allows us to prove that low for weak 2randomness is the same as low for MartinLöf randomness (a result independently obtained by Nies). Using the same technique, we show that ≤LR implies ≤LK, generalizing the fact that low for MartinLöf randomness implies low for K. 1.
KTRIVIAL DEGREES AND THE JUMPTRACEABILITY HIERARCHY
"... Abstract. For every order h such that P n 1/h(n) is finite, every Ktrivial degree is hjumptraceable. This motivated Cholak, Downey and Greenberg [2] to ask whether this traceability property is actually equivalent to Ktriviality, thereby giving the hoped for combinatorial characterisation of low ..."
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Cited by 5 (2 self)
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Abstract. For every order h such that P n 1/h(n) is finite, every Ktrivial degree is hjumptraceable. This motivated Cholak, Downey and Greenberg [2] to ask whether this traceability property is actually equivalent to Ktriviality, thereby giving the hoped for combinatorial characterisation of lowness for MartinLöf randomness. We show however that the Ktrivial degrees are properly contained in those that are hjumptraceable for every convergent order h. 1.
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.
Kolmogorov complexity and computably enumerable sets
, 2011
"... Abstract. We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments ..."
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Cited by 2 (2 self)
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Abstract. We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefixfree complexity. A computably enumerable set A is Ktrivial if and only if the family of sets with complexity bounded by the complexity of A is uniformly computable from the halting problem. 1.
RANDOMNESS NOTIONS AND PARTIAL RELATIVIZATION
"... Abstract. We study weak 2 randomness, weak randomness relative to ∅ ′ and Schnorr randomness relative to ∅ ′. One major theme is characterizing the oracles A such that ML[A] ⊆ C, where C is a randomness notion and ML[A] denotes the MartinLöf random reals relative to A. We discuss the connections ..."
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Cited by 2 (2 self)
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Abstract. We study weak 2 randomness, weak randomness relative to ∅ ′ and Schnorr randomness relative to ∅ ′. One major theme is characterizing the oracles A such that ML[A] ⊆ C, where C is a randomness notion and ML[A] denotes the MartinLöf random reals relative to A. We discuss the connections with LRreducibility and also study the reducibility associated with weak 2randomness. 1.
COMPUTABILITY, TRACEABILITY AND BEYOND
"... This thesis is concerned with the interaction between computability and randomness. In the first part, we study the notion of traceability. This combinatorial notion has an increasing influence in the study of algorithmic randomness. We prove a separation result about the bounds on jump traceability ..."
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Cited by 1 (0 self)
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This thesis is concerned with the interaction between computability and randomness. In the first part, we study the notion of traceability. This combinatorial notion has an increasing influence in the study of algorithmic randomness. We prove a separation result about the bounds on jump traceability, and show that the index set of the strongly jump traceable computably enumerable (c.e.) sets is Π0 4complete. This shows that the problem of deciding if a c.e. set is strongly jump traceable, is as hard as it can be. We define a strengthening of strong jump traceability, called hyper jump traceability, and prove some interesting results about this new class. Despite the fact that the hyper jump traceable sets have their origins in algorithmic randomness, we are able to show that they are natural examples of several Turing degree theoretic properties. For instance, we show that the hyper jump traceable sets are the first example of a lowness class with no promptly simple members. We also study the dual highness notions obtained from strong jump traceability, and explore their degree theoretic properties.
ALGORITHMIC RANDOMNESS AND MEASURES OF COMPLEXITY
"... Abstract. We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress s ..."
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Abstract. We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability. 1.