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15
Mass problems and hyperarithmeticity
, 2006
"... A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of we ..."
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A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of the Cantor space. The lattice Pw has been studied in previous publications. The purpose of this paper is to show that Pw partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in Pw which are indexed by the ordinal numbers less than ω CK 1 and which correspond to the hyperarithmetical hierarchy. Namely, for each α < ω CK 1 let hα be the weak degree of 0 (α) , the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p ∗ be the weak degree
Benign cost functions and lowness properties
"... Abstract. We show that the class of strongly jumptraceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of wellbehaved cost function, called benign. This characterisation implies the containment of the class of strongly jumptraceable c.e. ..."
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Cited by 9 (5 self)
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Abstract. We show that the class of strongly jumptraceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of wellbehaved cost function, called benign. This characterisation implies the containment of the class of strongly jumptraceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LRhard random degrees, and all ωc.e. random degrees. The last result implies recent results of Diamondstone’s and Ng’s regarding cupping with supwerlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees. 1.
Five Lectures on Algorithmic Randomness
 in Computational Prospects of Infinity, ed. C.T. Chong, Proc. 2005 Singapore meeting
, 2007
"... This paper follows on from the author’s Five Lectures on Algorithmic Randomness. It is concerned with material not found in that long paper, concentrating on MartinLöf lowness and triviality. We present a hopefully userfriendly account of the decanter method, and discuss recent results of the auth ..."
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Cited by 7 (2 self)
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This paper follows on from the author’s Five Lectures on Algorithmic Randomness. It is concerned with material not found in that long paper, concentrating on MartinLöf lowness and triviality. We present a hopefully userfriendly account of the decanter method, and discuss recent results of the author with Peter Cholak and Noam Greenberg concerning the class of strongly jump traceable reals introduced by
Lowness and Π0 2 nullsets
 J. Symbolic Logic
, 2006
"... Abstract. We prove that there exists a noncomputable c.e. real which is low for weak 2randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2randomness are low for MartinLöf randomness. 1. ..."
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Cited by 5 (4 self)
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Abstract. We prove that there exists a noncomputable c.e. real which is low for weak 2randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2randomness are low for MartinLöf randomness. 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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Cited by 4 (1 self)
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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.
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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Cited by 4 (0 self)
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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
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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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.
Interactions of Computability and Randomness
"... We survey results relating the computability and randomness aspects of sets of natural numbers. Each aspect corresponds to several mathematical properties. Properties originally defined in very different ways are shown to coincide. For instance, lowness for MLrandomness is equivalent to Ktrivialit ..."
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Cited by 2 (0 self)
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We survey results relating the computability and randomness aspects of sets of natural numbers. Each aspect corresponds to several mathematical properties. Properties originally defined in very different ways are shown to coincide. For instance, lowness for MLrandomness is equivalent to Ktriviality. We include some interactions of randomness with computable analysis. Mathematics Subject Classification (2010). 03D15, 03D32. Keywords. Algorithmic randomness, lowness property, Ktriviality, cost function.
Computability and randomness: Five questions
"... 1 How were you initially drawn to the study of computation and randomness? My first contact with the area was in 1996 when I still worked at the University of Chicago. Back then, my main interest was in structures from computability theory, such as the Turing degrees of computably enumerable sets. I ..."
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1 How were you initially drawn to the study of computation and randomness? My first contact with the area was in 1996 when I still worked at the University of Chicago. Back then, my main interest was in structures from computability theory, such as the Turing degrees of computably enumerable sets. I analyzed them via coding with firstorder formulas. During a visit to New Zealand, Cris Calude in Auckland introduced me to algorithmic information theory, a subject on which he had just finished a book [3]. We wrote a paper [4] showing that a set truthtable above the halting problem is not MartinLöf random (in fact the proof showed that it is not even weakly random [33, 4.3.9]). I also learned about Solovay reducibility, which is a way to gauge the relative randomness of real numbers with a computably enumerable left cut. These topics, and many more, were studied either in Chaitin’s work [6] or in Solovay’s visionary, but never published, manuscript [35], of which Cris possessed a copy. l In April 2000 I returned to New Zealand. I worked with Rod Downey and Denis Hirschfeldt on the Solovay degrees of real numbers with computably enumerable left cut. We proved that this degree structure is dense, and that the top degree, the degree of Chaitin’s Ω, cannot be split into two lesser degrees [9]. During this visit I learned about Ktriviality, a notion formalizing the intuitive idea of a set of natural numbers that is far from random. To understand Ktriviality, we first need a bit of background. Sets of natural numbers (simply called sets below) are a main topic of study in computability theory. Sets can be “identified ” with infinite sequences of bits. Given a set A, the bit in position n has value 1 if n is in A, otherwise its value is 0. A string is a finite sequence of bits, such as 11001110110. Let K(x) denote the length of a shortest prefixfree description of a string x (sometimes called the prefixfree Kolmogorov complexity of x even though Kolmogorov didn’t introduce it). We say that K(x) is the prefixfree complexity of x. Chaitin [6] defined a set A ⊆ N to be Ktrivial if each initial segment of A has prefixfree complexity no greater than the prefixfree complexity of its length. That is, there is b ∈ N such that, for each n,
STRONG JUMPTRACEABILITY II: KTRIVIALITY
, 2010
"... Abstract. We show that every strongly jumptraceable set is Ktrivial. Unlike other results, we do not assume that the sets in question are computably enumerable. 1. ..."
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Abstract. We show that every strongly jumptraceable set is Ktrivial. Unlike other results, we do not assume that the sets in question are computably enumerable. 1.