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SUPERHIGHNESS AND STRONG JUMP TRACEABILITY
"... Abstract. Let A be c.e. Then A is strongly jump traceable if and only if A is Turing below each superhigh MartinLöfrandom set. 1. ..."
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Abstract. Let A be c.e. Then A is strongly jump traceable if and only if A is Turing below each superhigh MartinLöfrandom set. 1.
Superhighness and strong jump traceability
"... Abstract. Let A be a c.e. set. Then A is strongly jump traceable if and only if A is Turing below each superhigh MartinLöf random set. The proof combines priority with measure theoretic arguments. 1 ..."
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Abstract. Let A be a c.e. set. Then A is strongly jump traceable if and only if A is Turing below each superhigh MartinLöf random set. The proof combines priority with measure theoretic arguments. 1
superhighness
"... We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2random. We also study the class superhigh ✸ , and show that it contains some, but not all, of the noncomputable Ktrivial sets. ..."
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We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2random. We also study the class superhigh ✸ , and show that it contains some, but not all, of the noncomputable Ktrivial sets.
STRONG JUMP TRACEABILITY AND DEMUTH RANDOMNESS
"... Abstract. We solve the covering problem for Demuth randomness, showing that a computably enumerable set is computable from a Demuth random set if and only if it is strongly jumptraceable. We show that on the other hand, the class of sets which form a base for Demuth randomness is a proper subclass ..."
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Abstract. We solve the covering problem for Demuth randomness, showing that a computably enumerable set is computable from a Demuth random set if and only if it is strongly jumptraceable. We show that on the other hand, the class of sets which form a base for Demuth randomness is a proper subclass
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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random 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
STRONG JUMPTRACEABILITY II: KTRIVIALITY
, 2010
"... We show that every strongly jumptraceable set is Ktrivial. Unlike other results, we do not assume that the sets in question are computably enumerable. ..."
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We show that every strongly jumptraceable set is Ktrivial. Unlike other results, we do not assume that the sets in question are computably enumerable.
SUPERHIGHNESS BJØRN KJOSHANSSEN AND ANDRE ́ NIES
"... Abstract. We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2random. We also study the class superhigh3, and show that it contains some, but not all, of the noncomputable Ktrivial sets. 1. ..."
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Abstract. We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2random. We also study the class superhigh3, and show that it contains some, but not all, of the noncomputable Ktrivial sets. 1.
COMPUTABILITY, TRACEABILITY AND BEYOND
"... This thesis is concerned with the interaction between computability and randomness. In the first part, we study the notion of traceability. This combinatorial notion has an increasing influence in the study of algorithmic randomness. We prove a separation result about the bounds on jump traceability ..."
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traceability, and show that the index set of the strongly jump traceable computably enumerable (c.e.) sets is Π0 4complete. This shows that the problem of deciding if a c.e. set is strongly jump traceable, is as hard as it can be. We define a strengthening of strong jump traceability, called hyper jump
Computability and Randomness
"... 2008 To my parents Christel and Otfrid with gratitudePREFACE The complexity and randomness aspects of sets of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathema ..."
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2008 To my parents Christel and Otfrid with gratitudePREFACE The complexity and randomness aspects of sets of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts and methods originating from randomness enrich computability theory. This book is about the two aspects of sets of natural numbers and about their interplay. Sets of natural numbers are identified with infinite sequences of zeros and ones, and simply called sets. Chapters 1 and 6 are mostly about the complexity aspect. We introduce lowness and highness properties of sets. Chapters 2, 3, and 7 are mostly about the randomness aspect. Firstly we study randomness of finite objects. Then we proceed to sets. We establish a hierarchy of mathematical randomness notions. Each notion matches our intuition of
Results 1  10
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