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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.
Π 0 1 CLASSES AND STRONG DEGREE SPECTRA OF RELATIONS
"... Abstract. We study the weak truthtable and truthtable degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truthtable reducible to any initial segment of any scattered compu ..."
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Abstract. We study the weak truthtable and truthtable degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truthtable reducible to any initial segment of any scattered computable linear order. Countable Π 0 1 subsets of 2 ω and Kolmogorov complexity play a major role in the proof.
Bounded Randomness ⋆
"... Abstract. We introduce some new variations of the notions of being MartinLöf random where the tests are all clopen sets. We explore how these randomness notions relate to classical randomness notions and to degrees of unsolvability. 1 ..."
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Abstract. We introduce some new variations of the notions of being MartinLöf random where the tests are all clopen sets. We explore how these randomness notions relate to classical randomness notions and to degrees of unsolvability. 1
1 Bounded Limit Recursiveness
, 2007
"... Let X be a Turing oracle. A function f(n) issaidtobe boundedly limit recursive in X if it is the limit of an Xrecursive sequence of Xrecursive functions ˜f(n, s) such that the number of times ˜f(n, s) changes is bounded by a recursive function of n. Let us say that X is BLRlow if every function w ..."
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Let X be a Turing oracle. A function f(n) issaidtobe boundedly limit recursive in X if it is the limit of an Xrecursive sequence of Xrecursive functions ˜f(n, s) such that the number of times ˜f(n, s) changes is bounded by a recursive function of n. Let us say that X is BLRlow if every function which is boundedly limit recursive in X is boundedly limit recursive in 0. This is a lowness property in the sense of Nies. These notions were introduced by Joshua A. Cole and the speaker in a recently submitted paper on mass problems and hyperarithmeticity. The purpose of this talk is to compare BLRlowness to similar properties which have been considered in the recursiontheoretic literature. Among the properties discussed are: Ktriviality, superlowness, jumptraceability, weak jumptraceability, total ωrecursive enumerability, array recursiveness, array jumprecursiveness, and strong jumptraceability. 2 Definition. If X is a Turing oracle, let BLR(X) betheset of numbertheoretic functions f: ω → ω which are boundedly limit recursive in X. This means that there exist an Xrecursive approximating function ˜ f(n, s) and a recursive bounding function ̂ f(n) such that and for all n. f(n) = lims ˜ f(n, s) {s  ˜ f(n, s) ̸ = ˜ f(n, s +1)}  < ̂ f(n) In particular, BLR(0) = {f  f ≤ wtt 0 ′}. The BLR operator was introduced in Mass problems and hyperarithmeticity, by Joshua A. Cole and Stephen G. Simpson, 20 pages, submitted 2006 to JML. 3 Cole and Simpson used the BLR operator to construct a natural embedding of the hyperarithmetical hierarchy into P w. Namely, we proved that the Muchnik degrees inf(h ∗ α, 1) forα<ωCK 1 are distinct ∈Pw.
STRONG DEGREE SPECTRA OF RELATIONS
, 2008
"... For my husband Andrew, and children Prudence, Jancis, and Rutherford. One of the main areas of study in computable model theory is examining how certain aspects of a computable structure may change under an isomorphism to another computable structure. Let A be a computable structure, and let R be an ..."
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For my husband Andrew, and children Prudence, Jancis, and Rutherford. One of the main areas of study in computable model theory is examining how certain aspects of a computable structure may change under an isomorphism to another computable structure. Let A be a computable structure, and let R be an additional relation on the domain of A, so it is not named in the language of A. Harizanov defined the Turing degree spectrum of R on A to be the set of all Turing degrees of the images of R under all isomorphisms from A onto computable structures. Similarly, we define this notion for strong degrees such as weak truthtable degrees and truthtable degrees. We show that the conditions necessary for the Turing degree spectrum to contain all Turing degrees, found by Harizanov, are also enough to have the truthtable degree spectrum to contain all truthtable degrees. We further study the degreetheoretic complexity of initial segments of computable linear orderings. In particular, let L be a computable linear ordering of order type ω+ω ∗. Harizanov showed that the Turing degree spectrum of the ωpart of L is all of the limit computable
LOWNESS FOR BOUNDED RANDOMNESS
"... In [3], Brodhead, Downey and Ng introduced some new variations of the notions of being MartinLöf random where the tests are all clopen sets. We explore the lowness notions associated with these randomness notions. While these bounded notions seem far from classical notions with infinite tests like ..."
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In [3], Brodhead, Downey and Ng introduced some new variations of the notions of being MartinLöf random where the tests are all clopen sets. We explore the lowness notions associated with these randomness notions. While these bounded notions seem far from classical notions with infinite tests like MartinLöf and Demuth randomness, the lowness notions associated with bounded randomness turn out to be intertwined with the lowness notions for these two concepts. In fact, in one case, we get a new and likely very useful characterization of Ktriviality