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Recursively Enumerable Reals and Chaitin Ω Numbers
"... A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from b ..."
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Cited by 34 (3 self)
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A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal selfdelimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number islike. In this paper we show that the converse implication is true as well: any Ωlike real in the unit interval is the halting probability of a universal selfdelimiting Turing machine.
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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Cited by 16 (5 self)
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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.
Turing degrees of certain isomorphic images of computable relations
 Ann. Pure Appl. Logic
, 1998
"... This paper is dedicated to Chris Ash, who invented αsystems. Abstract. A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let A be a computable model and let R be an extra relation on the domain of A. That is, R is not namedinthelanguag ..."
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Cited by 9 (2 self)
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This paper is dedicated to Chris Ash, who invented αsystems. Abstract. A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let A be a computable model and let R be an extra relation on the domain of A. That is, R is not namedinthelanguageofA. Wedefine DgA(R) to be the set of Turing degrees of the images f(R) under all isomorphisms f from A to computable models. We investigate conditions on A and R which are sufficient and necessary for DgA(R) to contain every Turing degree. These conditions imply that if every Turing degree ≤ 0 00 can be realized in DgA(R) via an isomorphism of the same Turing degree as its image of R, thenDgA(R) contains every Turing degree. We also discuss an example of A and R whose DgA(R) coincides with the Turing degrees which are ≤ 0 0. 1. Introduction and
Intervals of the Lattice of Computably Enumerable Sets and Effective Boolean Algebras
, 1997
"... We prove that each interval of the lattice E of c.e. sets under inclusion is either a boolean algebra or has an undecidable theory. This solves an open problem of Maass and Stob [9]. We develop a method to prove undecidability by interpreting ideal lattices, which can also be applied to degree s ..."
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Cited by 6 (3 self)
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We prove that each interval of the lattice E of c.e. sets under inclusion is either a boolean algebra or has an undecidable theory. This solves an open problem of Maass and Stob [9]. We develop a method to prove undecidability by interpreting ideal lattices, which can also be applied to degree structures from complexity theory. We also answer a question left open in [6] by giving an example of a nondefinable subclass of E which has an arithmetical index set and is invariant under automorphisms. 1 Introduction Intervals play an important role in the study of the lattice E of computably enumerable (c.e.) sets under inclusion. Several interesting properties of a c.e. set can be given alternative definitions in terms of the structure of L(A), the lattice of c.e. supersets of A. For instance, hyperhypersimplicity of a coinfinite c.e. set A is equivalent to L(A) being a boolean algebra, and A is rmaximal if and only if L(A) has no nontrivial complemented elements. A further typ...
Slender classes
, 2006
"... Abstract. A Π 0 1 class P is called thin if, given a subclass P ′ of P there is a clopen C with P ′ = P ∩ C. Cholak, Coles, Downey and Herrmann [7] proved that a Π 0 1 class P is thin if and only if its lattice of subclasses forms a Boolean algebra. Those authors also proved that if this boolean al ..."
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Cited by 2 (1 self)
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Abstract. A Π 0 1 class P is called thin if, given a subclass P ′ of P there is a clopen C with P ′ = P ∩ C. Cholak, Coles, Downey and Herrmann [7] proved that a Π 0 1 class P is thin if and only if its lattice of subclasses forms a Boolean algebra. Those authors also proved that if this boolean algebra is the free Boolean algebra, then all such think classes are automorphic in the lattice of Π 0 1 classes under inclusion. From this it follows that if the boolean algebra has a finite number n of atoms then the resulting classes are all automorphic. We prove a conjecture of Cholak and Downey [8] by showing that this is the only time the Boolean algebra determines the automorphism type of a thin class. 1.
Effectively and noneffectively nowhere simple sets
 Mathematical Logic Quarterly 42
, 1996
"... Abstract. R. Shore proved that every recursively enumerable (r.e.) set can be split into two (disjoint) nowhere simple sets. Splitting theorems play an important role in recursion theory since they provide information about the lattice E of all r.e. sets. Nowhere simple sets were further studied by ..."
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Cited by 2 (2 self)
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Abstract. R. Shore proved that every recursively enumerable (r.e.) set can be split into two (disjoint) nowhere simple sets. Splitting theorems play an important role in recursion theory since they provide information about the lattice E of all r.e. sets. Nowhere simple sets were further studied by D. Miller and J. Remmel, and we generalize some of their results. We characterize r.e. sets which can be split into two (non)effectively nowhere simple sets, and r.e. sets which can be split into two r.e. nonnowhere simple sets. We show that every r.e. set is either the disjoint union of two effectivelynowheresimplesets or two noneffectively nowhere simple sets. We characterize r.e. sets whose every nontrivial splitting is into nowhere simple sets, and r.e. sets whose every nontrivial splitting is into effectively nowhere simple sets. R. Shore proved that for every effectively nowhere simple set A, thelattice L ∗ (A) is effectively isomorphic to E ∗ , and that there is a nowhere simple set A such that L ∗ (A) is not effectively isomorphic to E ∗. We prove that every nonzero r.e. Turing degree contains a noneffectively nowhere simple set A with the lattice L ∗ (A) effectively isomorphic to E ∗. 1. Introduction and
Index sets for computable differential equations
, 2004
"... Key words Index set, computable analysis. ..."
Invariance And Noninvariance In The Lattice Of Pi Classes
, 2006
"... This paper continues the study of the lattice of \Pi ..."
INVARIANCE AND NONINVARIANCE IN THE LATTICE OF � 0 1 CLASSES
"... Abstract. We prove that there are two minimal �0 1 classes that are not automorphic. ..."
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Abstract. We prove that there are two minimal �0 1 classes that are not automorphic.