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56
Randomness in effective descriptive set theory
 London. Math. Soc
"... Abstract. An analog of MLrandomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 1 counterparts. We prove the analogs of the KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyper ..."
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Cited by 9 (3 self)
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Abstract. An analog of MLrandomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 1 counterparts. We prove the analogs of the KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyperarithmetical, each low for random set is. Finally we study a very strong yet effective randomness notion: Z is strongly random if Z is in no null Π1 1 set of reals. We show that there is a greatest Π1 1 null set, that is, a universal test for this notion. 1.
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.
Randomness, lowness and degrees
 J. of Symbolic Logic
, 2006
"... Abstract. We say that A ≤LR B if every Brandom number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally a ..."
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Cited by 9 (4 self)
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Abstract. We say that A ≤LR B if every Brandom number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e. restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α is not GL2 the LR degree of α bounds 2 ℵ0 degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees. 1.
KOLMOGOROV COMPLEXITY AND SOLOVAY FUNCTIONS
, 2009
"... Solovay [19] proved that there exists a computable upper bound f of the prefixfree Kolmogorov complexity function K such that f(x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) ≤ f(x)+O(1) for all x and f(x) ≤ K(x) + O(1) for infinit ..."
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Cited by 8 (3 self)
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Solovay [19] proved that there exists a computable upper bound f of the prefixfree Kolmogorov complexity function K such that f(x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) ≤ f(x)+O(1) for all x and f(x) ≤ K(x) + O(1) for infinitely many x, which we call Solovay functions. We show that Solovay functions present interesting connections with randomness notions such as MartinLöf randomness and Ktriviality.
Lowness for weakly 1generic and Kurtzrandom
 in Theory and Applications of Models of Computation: Third Internationa l Conference, TAMC 2006
, 2006
"... Abstract. We prove that a set is low for weakly 1generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, we refute a recent conjecture on the characterization of these sets. Furthermore, we show that every set which is low for w ..."
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Cited by 7 (2 self)
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Abstract. We prove that a set is low for weakly 1generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, we refute a recent conjecture on the characterization of these sets. Furthermore, we show that every set which is low for weakly 1generic is also low for Kurtzrandom. 1
CALCULUS OF COST FUNCTIONS
"... Abstract. We study algebraic properties of cost functions. We give an application: building sets close to being Turing complete. 1. ..."
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Cited by 6 (4 self)
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Abstract. We study algebraic properties of cost functions. We give an application: building sets close to being Turing complete. 1.
Eliminating concepts
 Proceedings of the IMS workshop on computational prospects of infinity
, 2008
"... Four classes of sets have been introduced independently by various researchers: low for K, low for MLrandomness, basis for MLrandomness and Ktrivial. They are all equal. This survey serves as an introduction to these coincidence results, obtained in [24] and [10]. The focus is on providing backdo ..."
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Cited by 5 (2 self)
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Four classes of sets have been introduced independently by various researchers: low for K, low for MLrandomness, basis for MLrandomness and Ktrivial. They are all equal. This survey serves as an introduction to these coincidence results, obtained in [24] and [10]. The focus is on providing backdoor access to the proofs. 1. Outline of the results All sets will be subsets of N unless otherwise stated. K(x) denotes the prefix free complexity of a string x. A set A is Ktrivial if, within a constant, each initial segment of A has minimal prefix free complexity. That is, there is c ∈ N such that ∀n K(A ↾ n) ≤ K(0 n) + c. This class was introduced by Chaitin [5] and further studied by Solovay (unpublished). Note that the particular effective epresentation of a number n by a string (unary here) is irrelevant, since up to a constant K(n) is independent from the representation. A is low for MartinLöf randomness if each MartinLöf random set is already MartinLöf random relative to A. This class was defined in Zambella [28], and studied by Kučera and Terwijn [17]. In this survey we will see that the two classes are equivalent [24]. Further concepts have been introduced: to be a basis for MLrandomness (Kučera [16]), and to be low for K (Muchnik jr, in a seminar at Moscow State, 1999). They will also be eliminated, by showing equivalence with Ktriviality. All
KOLMOGOROV COMPLEXITY OF INITIAL SEGMENTS OF SEQUENCES AND ARITHMETICAL DEFINABILITY
"... Abstract. The structure of the Kdegrees provides a way to classify sets of natural numbers or infinite binary sequences with respect to the level of randomness of their initial segments. In the Kdegrees of infinite binary sequences, X is below Y if the prefixfree Kolmogorov complexity of the firs ..."
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Abstract. The structure of the Kdegrees provides a way to classify sets of natural numbers or infinite binary sequences with respect to the level of randomness of their initial segments. In the Kdegrees of infinite binary sequences, X is below Y if the prefixfree Kolmogorov complexity of the first n bits of X is less than the complexity of the first n bits of Y, for each n. Identifying infinite binary sequences with subsets of N, we study the Kdegrees of arithmetical sets and explore the interactions between arithmetical definability and prefix free Kolmogorov complexity. We show that in the Kdegrees, for each n> 1 there exists a Σ0 n nonzero degree which does not bound any ∆0 n nonzero degree. An application of this result is that in the Kdegrees there exists a Σ0 2 degree which forms a minimal pair with all Σ0 1 degrees. This extends work of Csima/Montalbán [CM06] and Merkle/Stephan [MS07]. Our main result is that, given any ∆0 2 family C of sequences, there is a ∆0 2 sequence of nontrivial initial segment complexity which is not larger than the initial segment complexity of any nontrivial member of C. This general theorem has the following surprising consequence. There is a 0 ′computable sequence of nontrivial initial segment complexity which is not larger than the initial segment complexity of any nontrivial computably enumerable set. Our analysis and results demonstrate that, examining the extend to which arithmetical definability interacts with the K reducibility (and in general any ‘weak reducibility’) is a fruitful way of studying the induced structure. 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.
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.