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12
Some ComputabilityTheoretical Aspects of Reals and Randomness
, 2001
"... We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. Then we study such objects in terms of algorithmic randomness, culminating in some recent work of the author with Hirschfeldt, Laforte, and Nies conce ..."
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We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. Then we study such objects in terms of algorithmic randomness, culminating in some recent work of the author with Hirschfeldt, Laforte, and Nies concerning methods of calibrating randomness.
Totally ωcomputably enumerable degrees and bounding critical triples, preprint
"... Abstract. We characterize the class of c.e. degrees that bound a critical triple (equivalently, a weak critical triple) as those degrees that compute a function that has no ωc.e. approximation. 1. ..."
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Abstract. We characterize the class of c.e. degrees that bound a critical triple (equivalently, a weak critical triple) as those degrees that compute a function that has no ωc.e. approximation. 1.
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 ..."
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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 ′
ON STRONGLY JUMP TRACEABLE REALS
"... Abstract. In this paper we show that there is no minimal bound for jump traceability. In particular, there is no single order function such that strong jump traceability is equivalent to jump traceability for that order. The uniformity of the proof method allows us to adapt the technique to showing ..."
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Abstract. In this paper we show that there is no minimal bound for jump traceability. In particular, there is no single order function such that strong jump traceability is equivalent to jump traceability for that order. The uniformity of the proof method allows us to adapt the technique to showing that the index set of the c.e. strongly jump traceables is Π 0 4complete. §1. Introduction. One of the fundamental concerns of computability theory is in understanding the relative difficulty of computational problems as measured by Turing reducubility (≤T). The equivalence classes of the preordering ≤T are called Turing degrees, and it is long recognized that the fundamental operator on the structure of the Turing degrees is the jump operator. For a set A, the
TURING DEGREES OF REALS OF POSITIVE EFFECTIVE PACKING DIMENSION
"... Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1random real, yet contains elements of nonzero effective Hausdorff Dimension? We show tha ..."
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Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1random real, yet contains elements of nonzero effective Hausdorff Dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We construct a minimal degree of effective packing dimension 1. This leads us to examine the Turing degrees of reals with positive effective packing dimension. Unlike effective Hausdorff dimension, this is a notion of complexity which is shared by both random and sufficiently generic reals. We provide a characterization of the c.e. array noncomputable degrees in terms of effective packing dimension. 1.
Beyond strong jump traceability
"... Abstract. Strong jump traceability has been studied by various authors. In this paper we study a variant of strong jump traceability by looking at a partial relativization of traceability. We discover a new subclass H of the c.e. Ktrivials with some interesting properties. These sets are computatio ..."
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Abstract. Strong jump traceability has been studied by various authors. In this paper we study a variant of strong jump traceability by looking at a partial relativization of traceability. We discover a new subclass H of the c.e. Ktrivials with some interesting properties. These sets are computationally very weak, but yet contains a cuppable member. Surprisingly they cannot be constructed using cost functions, and is the first known example of a subclass of the Ktrivials which does not contain any promptly simple member. Furthermore there is a single c.e. set which caps every member of H, demonstrating that they are in fact very far away from being promptly simple. 1.
WORKING WITH STRONG REDUCIBILITIES ABOVE TOTALLY ωC.E. DEGREES
"... Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ω ..."
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Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ωc.e. iff every set in it is weak truthtable reducible to a hypersimple, or ranked, set. We also show that a c.e. degree is array computable iff every leftc.e. real of that degree is reducible in a computable Lipschitz way to a random leftc.e. real (an Ωnumber). 1.
PROMPT SIMPLICITY, ARRAY COMPUTABILITY AND CUPPING
"... Abstract. We show that the class of c.e. degrees that can be joined to 0 ′ by an array computable c.e. degree properly contains the class of promptly simple degrees. 1. ..."
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Abstract. We show that the class of c.e. degrees that can be joined to 0 ′ by an array computable c.e. degree properly contains the class of promptly simple degrees. 1.
EFFECTIVE PACKING DIMENSION AND TRACEABILITY
"... The concern of this paper is with effective packing dimension. This concept can be traced back to the work of Borel and Lebesgue who studied measure as a way of specifying the size of sets. Carathéodory later generalized Lebesgue measure to the ndimensional Euclidean space, and this was taken furth ..."
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The concern of this paper is with effective packing dimension. This concept can be traced back to the work of Borel and Lebesgue who studied measure as a way of specifying the size of sets. Carathéodory later generalized Lebesgue measure to the ndimensional Euclidean space, and this was taken further by Hausdorff [Hau19]