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Lowness Properties and Randomness
 ADVANCES IN MATHEMATICS
"... The set A is low for MartinLof random if each random set is already random relative to A. A is Ktrivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of AmbosSpies and Kucera [2 ..."
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Cited by 79 (21 self)
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The set A is low for MartinLof random if each random set is already random relative to A. A is Ktrivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of AmbosSpies and Kucera [2], showing that each low for MartinLof random set is # 2 . Our class induces a natural intermediate # 3 ideal in the r.e. Turing degrees (which generates the whole class under downward closure). Answering
Degrees of random sets
, 1991
"... An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrando ..."
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Cited by 46 (4 self)
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An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrandom and weakly nrandom sequences with an emphasis on the structure of their Turing degrees. After an introduction and summary, in Chapter II we present several equivalent definitions of nrandomness and weak nrandomness including a new definition in terms of a forcing relation analogous to the characterization of ngeneric sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than nrandomness. Chapter III is concerned with intrinsic properties of nrandom sequences. The main results are that an (n + 1)random sequence A satisfies the condition A (n) ≡T A⊕0 (n) (strengthening a result due originally to Sacks) and that nrandom sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is nrandom then A is nrandom relative to B. It follows that any countable distributive lattice can be embedded
Computational Randomness and Lowness
, 2001
"... . We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0 . This contrasts with a result of Kucera and Terwijn [5] on sets tha ..."
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Cited by 26 (1 self)
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. We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0 . This contrasts with a result of Kucera and Terwijn [5] on sets that are low for the class of MartinLof random reals. The Cantor space 2 ! is the set of infinite binary sequences; these are called reals and are identified with subsets of !. If oe 2 2 !! , that is, oe is a finite binary sequence, we denote by [oe] the set of reals that extend oe. These form a basis of clopen sets for the usual discrete topology on 2 ! . Write joej for the length of oe 2 2 !! . The Lebesgue measure on 2 ! is defined by stipulating that [oe] = 2 \Gammajoej . With every set U ` 2 !! we associate the open set S oe2U [oe]. When it is convenient, we confuse U with the open set associated to it, in particular we write U for the measure of the open set correspondi...
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.
Algorithmic Randomness and Lowness
, 1997
"... . We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0 . This contrasts with a result of Kucera and Terwijn [2] on sets tha ..."
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Cited by 11 (2 self)
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. We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0 . This contrasts with a result of Kucera and Terwijn [2] on sets that are low for the class of MartinLof random reals. The Cantor space 2 ! is the set of infinite binary sequences; these are called reals and are identified with subsets of !. If oe 2 2 !! , that is, oe is a finite binary sequence, we denote by [oe] the set of reals that extend oe. These form a basis of clopens for the usual discrete topology on 2 ! . Write joej for the length of oe 2 2 !! . The Lebesgue measure ¯ on 2 ! is defined by stipulating that ¯[oe] = 2 \Gammajoej . With every set U ` 2 !! we associate the open set S oe2U [oe]. When it is convenient, we confuse U with the open set associated to it, in particular we write ¯U for the measure of the open set correspondin...
Lowness Properties of Reals and HyperImmunity
 In Wollic 2003
, 2003
"... AmbosSpies and Kucera [1, Problem 4.5] asked if there is a noncomputable set A which is low for the computably random reals. We show that no such A is of hyperimmune degree. Thus, each g T A is dominated by a computable function. AmbosSpies and Kucera [1, Problem 4.8] also asked if every Slo ..."
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Cited by 9 (2 self)
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AmbosSpies and Kucera [1, Problem 4.5] asked if there is a noncomputable set A which is low for the computably random reals. We show that no such A is of hyperimmune degree. Thus, each g T A is dominated by a computable function. AmbosSpies and Kucera [1, Problem 4.8] also asked if every Slow set is S 0 low. We give a partial solution to this problem, showing that no Slow set is of hyperimmune degree.
Local initial segments of the Turing degrees
 Bull. Symbolic Logic
, 2002
"... Abstract. Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of nontrivial automorphisms of the Turing degrees are indicated. §1. Introduction. This article concerns the algebraic study of the u ..."
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Cited by 6 (2 self)
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Abstract. Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of nontrivial automorphisms of the Turing degrees are indicated. §1. Introduction. This article concerns the algebraic study of the upper semilattice of Turing degrees. Upper semilattices of interest in this regard tend to have a least element, hence for convenience the following definition is made. Definition 1.1. A unital semilattice (usl) is a structure L = (L, ∗, e) satisfying the following equalities for all a, b, c ∈ L.
Countable thin Π0 1 classes
 Annals of Pure and Applied Logic
, 1993
"... Abstract. AΠ0 1 class P ⊂ {0,1}ωis thin if every Π0 1 subclass Q of P is the intersection of P with some clopen set. Countable thin Π0 1 classes are constructed having arbitrary recursive CantorBendixson rank. A thin Π0 1 class P is constructed with a unique nonisolated point A of degree 0 ′. It is ..."
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Cited by 5 (4 self)
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Abstract. AΠ0 1 class P ⊂ {0,1}ωis thin if every Π0 1 subclass Q of P is the intersection of P with some clopen set. Countable thin Π0 1 classes are constructed having arbitrary recursive CantorBendixson rank. A thin Π0 1 class P is constructed with a unique nonisolated point A of degree 0 ′. It is shown that, for all ordinals α>1, no set of degree ≥ 0 ′ ′ can be a member of any thin Π0 1 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π0 1 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree (not necessarily r.e.) that contains a set which is of rank one in some thin Π0 1 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π0 1 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π0 1 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree 0 ′ ′. Introduction.
Generalized high degrees have the complementation property
 Journal of Symbolic Logic
"... Abstract. We show that if d ∈ GH1 then D( ≤ d) has the complementation property, i.e. for all a < d there is some b < d such that a ∧ b = 0 and a ∨ b = d. §1. Introduction. A major theme in the investigation of the structure of the Turing degrees, (D, ≤T), has been the relationship between the order ..."
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Cited by 3 (0 self)
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Abstract. We show that if d ∈ GH1 then D( ≤ d) has the complementation property, i.e. for all a < d there is some b < d such that a ∧ b = 0 and a ∨ b = d. §1. Introduction. A major theme in the investigation of the structure of the Turing degrees, (D, ≤T), has been the relationship between the order theoretic properties of a degree and its complexity of definition in arithmetic as expressed by the Turing jump operator which embodies a single step in the hierarchy of quantification. For example, there is a long history of results showing that 0 ′