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Automorphisms of the lattice of Π 0 1 classes: perfect thin classes and anc degrees
 Trans. Amer. Math. Soc
"... Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, o ..."
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Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin’s work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare’s work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of Π0 1 classes) forms an orbit in the lattice of Π01 classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of Π0 1 classes. We remark that the automorphism result is proven via a ∆0 3 automorphism, and demonstrate that this complexity is necessary. 1.
Oscillation in the initial segment complexity of random reals
 Adv. Math
, 2010
"... Abstract. We study oscillation in the prefixfree complexity of initial segments of 1random reals. For upward oscillations, we prove that P n∈ω 2 −g(n) diverges iff (∃∞n) K(X n)> n+g(n) for every 1random X ∈ 2ω. For downward oscillations, we characterize the functions g such that (∃∞n) K(X n) ..."
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Abstract. We study oscillation in the prefixfree complexity of initial segments of 1random reals. For upward oscillations, we prove that P n∈ω 2 −g(n) diverges iff (∃∞n) K(X n)> n+g(n) for every 1random X ∈ 2ω. For downward oscillations, we characterize the functions g such that (∃∞n) K(X n) < n+g(n) for almost every X ∈ 2ω. The proof of this result uses an improvement of Chaitin’s counting theorem—we give a tight upper bound on the number of strings σ ∈ 2n such that K(σ) < n+K(n)−m. The work on upward oscillations has applications to the Kdegrees. Write X ≤K Y to mean that K(X n) ≤ K(Y n) + O(1). The induced structure is called the Kdegrees. We prove that there are comparable (∆02) 1random Kdegrees. We also prove that every lower cone and some upper cones in the 1random Kdegrees have size continuum. Finally, we show that it is independent of ZFC, even assuming that the Continuum Hypothesis fails, whether all chains of 1random Kdegrees of size less than 2ℵ0 have a lower bound in the 1random Kdegrees. “Although this oscillatory behaviour is usually considered to be a nasty feature, we believe that it illustrates one of the great advantages of complexity: the possibility to study degrees of randomness.” Michiel van Lambalgen, Ph.D. Dissertation [31, p. 145]. 1.
Randomness and universal machines
 CCA 2005, Second International Conference on Computability and Complexity in Analysis, Fernuniversität Hagen, Informatik Berichte 326:103–116
, 2005
"... The present work investigates several questions from a recent survey of Miller and Nies related to Chaitin’s Ω numbers and their dependence on the underlying universal machine. It is shown that there are universal machines for which ΩU is just x 21−H(x). For such a universal machine there exists a c ..."
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The present work investigates several questions from a recent survey of Miller and Nies related to Chaitin’s Ω numbers and their dependence on the underlying universal machine. It is shown that there are universal machines for which ΩU is just x 21−H(x). For such a universal machine there exists a cor.e. set X such that ΩU[X] = � p:U(p)↓∈X 2−p  is neither leftr.e. nor MartinLöf random. Furthermore, one of the open problems of Miller and Nies is answered completely by showing that there is a sequence Un of universal machines such that the truthtable degrees of the ΩUn form an antichain. Finally it is shown that the members of hyperimmunefree Turing degree of a given Π0 1class are not low for Ω unless this class contains a recursive set. 1
The Baire category theorem in weak subsystems of secondorder arithmetic
 THE JOURNAL OF SYMBOLIC LOGIC
, 1993
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Mass problems and almost everywhere domination
 Mathematical Logic Quarterly
, 2007
"... We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and MartinLöf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED ..."
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Cited by 10 (7 self)
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We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and MartinLöf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED, MLR×AED, MLR∩AED respectively. Let Pw be the lattice of degrees of unsolvability of mass problems associated with nonempty Π 0 1 subsets of 2 ω. Let 1 and 0 be the top and bottom elements of Pw. We show that inf(b1,1) and inf(b2,1) and inf(b3,1) belong to Pw and that 0 < inf(b1,1) < inf(b2,1) < inf(b3,1) < 1. Under the natural embedding of the recursively enumerable Turing degrees into Pw, we show that inf(b1,1) and inf(b3,1) but not inf(b2,1) are comparable with some recursively enumerable Turing degrees other than 0 and 0 ′. In order to make this paper more selfcontained, we exposit the proofs of some recent theorems due to Hirschfeldt, Miller, Nies, and Stephan.
Effective Search Problems
 Mathematical Logic Quarterly
, 1994
"... The task of computing a function F with the help of an oracle X can be viewed as a search problem where the cost measure is the number of queries to X . We ask for the minimal number that can be achieved by a suitable choice of X and call this quantity the query complexity of F . This concept is s ..."
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Cited by 10 (5 self)
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The task of computing a function F with the help of an oracle X can be viewed as a search problem where the cost measure is the number of queries to X . We ask for the minimal number that can be achieved by a suitable choice of X and call this quantity the query complexity of F . This concept is suggested by earlier work of Beigel, Gasarch, Gill, and Owings on "Bounded query classes". We introduce a fault tolerant version and relate it with Ulam's game. For many natural classes of functions F we obtain tight upper and lower bounds on the query complexity of F . Previous results like the Nonspeedup Theorem and the Cardinality Theorem appear in a wider perspective. 1991 Mathematics Subject Classification: Primary 03D20; Secondary 68Q15, 68R05 Keywords: Search problems, bounded queries, query complexity, recursive functions 1 Introduction The task of computing a function F with the help of an oracle X ` ! (! is the set of all natural numbers) can be viewed as a search problem where t...
Π 0 1 sets and models of WKL0
"... We show that any two Medvedev complete Π 0 1 subsets of 2 ω are recursively homeomorphic. We obtain a Π 0 1 set Q ′ of countable coded ωmodels of WKL0 with a strong homogeneity property. We show that if G is a generic element of Q ′ , then the ωmodel of WKL0 coded by G satisfies ∀X∀Y (if X is de ..."
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We show that any two Medvedev complete Π 0 1 subsets of 2 ω are recursively homeomorphic. We obtain a Π 0 1 set Q ′ of countable coded ωmodels of WKL0 with a strong homogeneity property. We show that if G is a generic element of Q ′ , then the ωmodel of WKL0 coded by G satisfies ∀X∀Y (if X is definable from Y, then X is Turing reducible to Y). We use a result of Kučera to refute some plausible conjectures concerning ωmodels of WKL0. We generalize our results to nonωmodels of WKL0. We discuss the significance of our results for foundations of mathematics.
Some fundamental issues concerning degrees of unsolvability
 In [6], 2005. Preprint
, 2007
"... Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to f ..."
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Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a “smallness property ” of an infinite, corecursively enumerable set A ⊆ ω which ensures that the Turing degree deg T (A) = a ∈ RT is> 0 and < 0 ′. In order to address these issues, we embed RT into a slightly larger degree structure, Pw, which is much better behaved. Namely, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. We define a specific, natural embedding of RT into Pw, and we present some recent and new research results.
ComputabilityTheoretic and ProofTheoretic Aspects of Partial and Linear Orderings
 Israel Journal of mathematics
"... Szpilrajn's Theorem states that any partial order P = hS; <P i has a linear extension L = hS; <L i. This is a central result in the theory of partial orderings, allowing one to de ne, for instance, the dimension of a partial ordering. It is now natural to ask questions like \Does a we ..."
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Szpilrajn's Theorem states that any partial order P = hS; <P i has a linear extension L = hS; <L i. This is a central result in the theory of partial orderings, allowing one to de ne, for instance, the dimension of a partial ordering. It is now natural to ask questions like \Does a wellpartial ordering always have a wellordered linear extension?" Variations of Szpilrajn's Theorem state, for various (but not for all) linear order types , that if P does not contain a subchain of order type , then we can choose L so that L also does not contain a subchain of order type . In particular, a wellpartial ordering always has a wellordered extension.