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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.
ON THE NUMBER OF INFINITE SEQUENCES WITH TRIVIAL INITIAL SEGMENT COMPLEXITY
"... Abstract. The sequences which have trivial prefixfree initial segment complexity are known as Ktrivial sets, and form a cumulative hierarchy of length ω. We show that the problem of finding the number of Ktrivial sets in the various levels of the hierarchy is ∆0 3. This answers a question of Down ..."
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Abstract. The sequences which have trivial prefixfree initial segment complexity are known as Ktrivial sets, and form a cumulative hierarchy of length ω. We show that the problem of finding the number of Ktrivial sets in the various levels of the hierarchy is ∆0 3. This answers a question of Downey/Miller/Yu (see [DH10, Section 10.1.4]) which also appears in [Nie09, Problem 5.2.16]. We also show the same for the hierarchy of the low for K sequences, which are the ones that (when used as oracles) do not give shorter initial segment complexity compared to the computable oracles. In both cases the classification ∆0 3 is sharp. 1.
Algorithmic Randomness and Computability
"... Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability. ..."
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Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability.
RANDOMNESS – BEYOND LEBESGUE MEASURE
"... Much of the recent research on algorithmic randomness has focused on randomness for Lebesgue measure. While, from a computability theoretic point of view, the picture remains unchanged if one passes to arbitrary computable measures, interesting phenomena occur if one studies the the set of reals wh ..."
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Much of the recent research on algorithmic randomness has focused on randomness for Lebesgue measure. While, from a computability theoretic point of view, the picture remains unchanged if one passes to arbitrary computable measures, interesting phenomena occur if one studies the the set of reals which are random for an arbitrary (continuous) probability measure or a generalized Hausdorff measure on Cantor space. This paper tries to give a survey of some of the research that has been done on randomness for nonLebesgue measures.
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.
Randomness, Computation and Mathematics
"... Abstract. This article examines some of the recent advances in our understanding of algorithmic randomness. It also discusses connections with various areas of mathematics, computer science and other areas of science. Some questions and speculations will be discussed. 1 ..."
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Abstract. This article examines some of the recent advances in our understanding of algorithmic randomness. It also discusses connections with various areas of mathematics, computer science and other areas of science. Some questions and speculations will be discussed. 1
LECTURES ON EFFECTIVE RANDOMNESS
, 2006
"... This class will be dealing with the intersection of randomness and computability theory, or how the tools of randomness can be applied to computabilitytheoretic problems. This class will not cover the history of the subject, which can be ..."
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This class will be dealing with the intersection of randomness and computability theory, or how the tools of randomness can be applied to computabilitytheoretic problems. This class will not cover the history of the subject, which can be
CHARACTERIZING THE STRONGLY JUMPTRACEABLE SETS VIA RANDOMNESS
"... Abstract. We show that if a set A is computable from every superlow 1random set, then A is strongly jumptraceable. Together with a result from [9], this theorem shows that the computably enumerable jumptraceable sets are exactly the computably enumerable sets computable from every superlow 1rand ..."
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Abstract. We show that if a set A is computable from every superlow 1random set, then A is strongly jumptraceable. Together with a result from [9], this theorem shows that the computably enumerable jumptraceable sets are exactly the computably enumerable sets computable from every superlow 1random set. We also prove the analogous result for superhighness: a c.e. set is strongly jumptraceable if and only if it is computable from any superhigh random set. Finally, we show that for each cost function c with the limit condition there is a random ∆ 0 2 set Y such that each c.e. set A �T Y obeys c. 1.