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Bounded Query Classes and the Difference Hierarchy
 Archive for Mathematical Logic
, 1995
"... Let A be any nonrecursive set. We define a hierarchy of sets (and a corresponding hierarchy of degrees) that are reducible to A based on bounding the number of queries to A that an oracle machine can make. When A is the halting problem K our hierarchy of sets interleaves with the difference hierarch ..."
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Cited by 15 (12 self)
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Let A be any nonrecursive set. We define a hierarchy of sets (and a corresponding hierarchy of degrees) that are reducible to A based on bounding the number of queries to A that an oracle machine can make. When A is the halting problem K our hierarchy of sets interleaves with the difference hierarchy Current address: Department of Computer Science, Yale University, 51 Prospect Street, P.O. Box 2158 Yale Station, New Haven, CT 06520. Supported in part by NSF grant CCR8808949. Part of this work was completed while this author was a student at Stanford University supported by fellowships from the National Science Foundation and from the Fannie and John Hertz Foundation. y Supported in part by NSF grant CCR8803641. z Part of this work was completed while this author was on sabbatical leave at the University of California, Berkeley. on the r.e. sets in a logarithmic way; this follows from a tradeoff between the number of parallel queries and the number of serial queries needed to...
On the Turing degrees of weakly computable real numbers
 Journal of Logic and Computation
, 1986
"... The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semicomputable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others ..."
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Cited by 6 (3 self)
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The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semicomputable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others we show that, there are two real numbers of c.e. binary expansions such that their difference does not have an ω.c.e. Turing degree. 1
On isolating r.e. and isolated d.r.e. degrees
 IN COMPUTABILITY, ENUMERABILITY, UNSOLVABILITY, VOLUME 224 OF LONDON MATH. SOC. LECTURE NOTE SER., 61–80
, 1996
"... ..."
Relative Enumerability in the Difference Hierarchy
 J. Symb. Logic
"... We show that the intersection of the class of 2REA degrees with that of the #r.e. degrees consists precisely of the class of d.r.e. degrees. We also include some applications and show that there is no natural generalization of this result to higher levels of the REA hierarchy. 1 Introduction The ..."
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Cited by 3 (1 self)
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We show that the intersection of the class of 2REA degrees with that of the #r.e. degrees consists precisely of the class of d.r.e. degrees. We also include some applications and show that there is no natural generalization of this result to higher levels of the REA hierarchy. 1 Introduction The # 0 2 degrees of unsolvability are basic objects of study in classical recursion theory, since they are the degrees of those sets whose characteristic functions are limits of recursive functions. A natural tool for understanding the Turing degrees is the introduction of hierarchies to classify various kinds of complexity. Because of its coarseness, the most common such hierarchy, the arithmetical hierarchy, is itself not of much use in the classification of the # 0 2 degrees. This fact leads naturally to the consideration of hierarchies based on finer distinctions than quantifier alternation. Two such hierarchies are by now well established. One, the REA hierarchy defined by Jockusch and S...
Strong minimal covers and a question of Yates: the story so far
 the proceedings of the ASL meeting
, 2006
"... Abstract. An old question of Yates as to whether all minimal degrees have a strong minimal cover remains one of the longstanding problems of degree theory, apparently largely impervious to present techniques. We survey existing results in this area, focussing especially on some recent progress. 1. ..."
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Cited by 2 (2 self)
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Abstract. An old question of Yates as to whether all minimal degrees have a strong minimal cover remains one of the longstanding problems of degree theory, apparently largely impervious to present techniques. We survey existing results in this area, focussing especially on some recent progress. 1.
Interpolating dr. e. and REA degrees between r. e. degrees
, 1995
"... This paper is a contribution to the investigation of the relationship between the ..."
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This paper is a contribution to the investigation of the relationship between the
Structural Properties of D.C.E. Degrees and Presentations of C.E. Reals
"... To my wife Caixia and my daughter Jiahui Abstract In this thesis, we are mainly concerned with the structural properties of the d.c.e. degrees and the distribution of the simple reals among the c.e. degrees. In chapters 2 and 3, we study the relationship between the isolation phenomenon and the jump ..."
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To my wife Caixia and my daughter Jiahui Abstract In this thesis, we are mainly concerned with the structural properties of the d.c.e. degrees and the distribution of the simple reals among the c.e. degrees. In chapters 2 and 3, we study the relationship between the isolation phenomenon and the jump operator. We prove in chapter 2 that there is a high d.c.e. degree d isolated by a low2 degree a. We improve this result in chapter 3 by showing that the isolating degree a can be low. Chapters 4 and 5 are devoted to the study of the pseudoisolation in the d.c.e. degrees. We prove that pseudoisolated d.c.e. degrees are dense in the c.e. degrees, and that there is a high d.c.e. degree pseudoisolated by a low d.c.e. degree.
The nr.e. degrees: undecidability and Σ1 substructures
, 2012
"... We study the global properties of Dn, the Turing degrees of the nr.e. sets. In Theorem 1.5, we show that the first order theory of Dn is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, Dn is not a Σ1substructure of Dm. 1 ..."
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We study the global properties of Dn, the Turing degrees of the nr.e. sets. In Theorem 1.5, we show that the first order theory of Dn is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, Dn is not a Σ1substructure of Dm. 1