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94
POST’S PROGRAMME FOR THE ERSHOV HIERARCHY BAHAREH AFSHARI, GEORGE BARMPALIAS,
"... Abstract. This paper extends Post’s programme to finite levels of the Ershov hierarchy of ∆2 sets. Our initial characterisation, in the spirit of Post [27], of the degrees of the immune and hyperimmune nenumerable sets leads to a number of results setting other immunity properties in the context of ..."
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Abstract. This paper extends Post’s programme to finite levels of the Ershov hierarchy of ∆2 sets. Our initial characterisation, in the spirit of Post [27], of the degrees of the immune and hyperimmune nenumerable sets leads to a number of results setting other immunity properties in the context of the Turing and wttdegrees derived from the Ershov hierarchy. For instance, we show that any nenumerable hyperhyperimmune set must be coenumerable, for each n ≥ 2. The situation with regard to the wttdegrees is particularly interesting, as demonstrated by a range of results concerning the wttpredecessors of hypersimple sets. Finally, we give a number of results directed at characterising basic classes of nenumerable degrees in terms of natural information content. For example, a 2enumerable degree contains a 2enumerable dense immune set iff it contains a 2enumerable rcohesive set iff it bounds a high enumerable set. This result is extended to a characterisation of nenumerable degrees which bound high enumerable degrees. Furthermore, a characterisation for nenumerable degrees bounding only low2 enumerable degrees is given. 1.
The computable Lipschitz degrees of computably enumerable sets are not dense
 Ann. Pure Appl. Logic
"... Abstract. The computable Lipschitz reducibility was introduced by Downey, Hirschfeldt and LaForte under the name of strong weak truthtable reducibility [6]. This reducibility measures both the relative randomness and the relative computational power of real numbers. This paper proves that the comp ..."
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Cited by 2 (0 self)
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that the computable Lipschitz degrees of computably enumerable sets are not dense. An immediate corollary is that the Solovay degrees of strongly c.e. reals are not dense. There are similarities to Barmpalias and Lewis’s proof that the identity bounded Turing degrees of c.e. sets are not dense [2], however
Π 0 1 CLASSES, LR DEGREES AND TURING DEGREES
"... Abstract. We say that A ≤LR B if every Brandom set is Arandom with respect to MartinLöf randomness. We study this reducibility and its interactions with the Turing reducibility, Π0 1 classes, hyperimmunity and other recursion theoretic notions. A natural variant of the Turing reducibility from th ..."
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Cited by 6 (5 self)
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studied by Barmpalias, Lewis, Soskova [2] and Simpson [15]. In this paper we study ≤LR and its interactions with ≤T. In Section 1 we lay out the basic framework and facts which are used throughout the rest of
Random noncupping revisited
 J. Complexity
, 2006
"... Abstract. Say that Y has the strong random anticupping property if there is a set A such that for every MartinLöf random set R Y ≤T A ⊕ R ⇒ Y ≤T R (in this case A is an anticupping witness for Y). Nies has shown that every random ∆ 0 2 set has the strong random anticupping property via a promptly s ..."
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Cited by 6 (2 self)
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Abstract. Say that Y has the strong random anticupping property if there is a set A such that for every MartinLöf random set R Y ≤T A ⊕ R ⇒ Y ≤T R (in this case A is an anticupping witness for Y). Nies has shown that every random ∆ 0 2 set has the strong random anticupping property via a promptly simple anticupping witness. We show that every ∆ 0 2 set has the random anticupping property via a promptly simple anticupping witness. Moreover, we prove the following stronger statement: for every noncomputable Y ≤T ∅ ′ there exists a promptly simple A such that Y ≤T A ⊕ R ⇒ A ≤T R for all MartinLöf random sets R. 1.
Computably enumerable sets in the Solovay and the strong weak truth table degrees
 in New Computational Paradigms: First Conference on Computability in Europe, CiE 2005
, 2005
"... Abstract. The strong weak truth table reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky a ..."
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Cited by 9 (7 self)
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Abstract. The strong weak truth table reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky and Weinberger on applications of computability to differential geometry. Yu and Ding showed that the relevant degree structure restricted to the c.e. reals has no greatest element, and asked for maximal elements. We answer this question for the case of c.e. sets. Using a doubly nonuniform argument we show that there are no maximal elements in the sw degrees of the c.e. sets. We note that the same holds for the Solovay degrees of c.e. sets. 1
Tracing and domination in the Turing degrees
 Ann. Pure Appl. Logic
"... Abstract. We show that if 0 ′ is c.e. traceable by a, then a is array noncomputable. It follows that there is no minimal almost everywhere dominating degree, in the sense of Dobrinen and Simpson [DS04]. This answers a question of Simpson and a question of Nies [Nie09, Problem 8.6.4]. Moreover, it a ..."
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Cited by 8 (2 self)
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Abstract. We show that if 0 ′ is c.e. traceable by a, then a is array noncomputable. It follows that there is no minimal almost everywhere dominating degree, in the sense of Dobrinen and Simpson [DS04]. This answers a question of Simpson and a question of Nies [Nie09, Problem 8.6.4]. Moreover, it adds a new arrow in [Nie09, Figure 8.1], which is a diagram depicting the relations of various ‘computational lowness’ properties. Finally, it gives a natural definable property, namely nonminimality, which separates almost everywhere domination from highness. 1.
Relative randomness and cardinality
 Notre Dame J. Formal Logic
"... Abstract. A set B ⊆ N is called low for MartinLöf random if every MartinLöf random set is also MartinLöf random relative to B. We show that a ∆02 set B is low for MartinLöf random iff the class of oracles which compress less efficiently than B, namely the class CB = {A  ∀n KB(n) ≤+ KA(n)} i ..."
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Cited by 6 (5 self)
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Abstract. A set B ⊆ N is called low for MartinLöf random if every MartinLöf random set is also MartinLöf random relative to B. We show that a ∆02 set B is low for MartinLöf random iff the class of oracles which compress less efficiently than B, namely the class CB = {A  ∀n KB(n) ≤+ KA(n)} is countable (where K denotes the prefix free complexity and ≤+ denotes inequality modulo a constant). It follows that ∆02 is the largest arithmetical class with this property and if CB is uncountable, it contains a perfect Π01 set of reals. The proof introduces a new method for constructing nontrivial reals below a ∆02 set which is not low for MartinLöf random. 1.
RANDOMNESS AND THE LINEAR DEGREES OF COMPUTABILITY
"... Abstract. We show that there exists a real α such that, for all reals β, if α is linear reducible to β (α ≤ℓ β, previously denoted α ≤sw β) then β ≤T α. In fact, every random real satisfies this quasimaximality property. As a corollary we may conclude that there exists no ℓcomplete ∆2 real. Upon r ..."
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Abstract. We show that there exists a real α such that, for all reals β, if α is linear reducible to β (α ≤ℓ β, previously denoted α ≤sw β) then β ≤T α. In fact, every random real satisfies this quasimaximality property. As a corollary we may conclude that there exists no ℓcomplete ∆2 real. Upon realizing that quasimaximality does not characterize the random reals—there exist reals which are not random but which are of quasimaximal ℓdegree—it is then natural to ask whether maximality could provide such a characterization. Such hopes, however, are in vain since no real is of maximal ℓdegree. 1. introduction In the process of computing a real α given an oracle for β it is natural to consider the condition that for the computation of the first n bits of α we are only allowed to use the information in the first n bits of β. It is not difficult to see that this notion of oracle computation is complexity sensitive in many ways. We can then generalize this definition in a straightforward
KOLMOGOROV COMPLEXITY OF INITIAL SEGMENTS OF SEQUENCES AND ARITHMETICAL DEFINABILITY
"... Abstract. The structure of the Kdegrees provides a way to classify sets of natural numbers or infinite binary sequences with respect to the level of randomness of their initial segments. In the Kdegrees of infinite binary sequences, X is below Y if the prefixfree Kolmogorov complexity of the firs ..."
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Cited by 8 (6 self)
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Abstract. The structure of the Kdegrees provides a way to classify sets of natural numbers or infinite binary sequences with respect to the level of randomness of their initial segments. In the Kdegrees of infinite binary sequences, X is below Y if the prefixfree Kolmogorov complexity of the first n bits of X is less than the complexity of the first n bits of Y, for each n. Identifying infinite binary sequences with subsets of N, we study the Kdegrees of arithmetical sets and explore the interactions between arithmetical definability and prefix free Kolmogorov complexity. We show that in the Kdegrees, for each n> 1 there exists a Σ0 n nonzero degree which does not bound any ∆0 n nonzero degree. An application of this result is that in the Kdegrees there exists a Σ0 2 degree which forms a minimal pair with all Σ0 1 degrees. This extends work of Csima/Montalbán [CM06] and Merkle/Stephan [MS07]. Our main result is that, given any ∆0 2 family C of sequences, there is a ∆0 2 sequence of nontrivial initial segment complexity which is not larger than the initial segment complexity of any nontrivial member of C. This general theorem has the following surprising consequence. There is a 0 ′computable sequence of nontrivial initial segment complexity which is not larger than the initial segment complexity of any nontrivial computably enumerable set. Our analysis and results demonstrate that, examining the extend to which arithmetical definability interacts with the K reducibility (and in general any ‘weak reducibility’) is a fruitful way of studying the induced structure. 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 7 (4 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.
Results 1  10
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