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29
Degree spectra and computable dimension in algebraic structures
 Annals of Pure and Applied Logic 115 (2002
, 2002
"... \Lambda \Lambda ..."
Computable Isomorphisms, Degree Spectra of Relations, and Scott Families
 Ann. Pure Appl. Logic
, 1998
"... this paper we are interested in those structures in which the basic computations can be performed by Turing machines. ..."
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Cited by 25 (12 self)
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this paper we are interested in those structures in which the basic computations can be performed by Turing machines.
On Presentations of Algebraic Structures
 in Complexity, Logic and Recursion Theory
, 1995
"... This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicat ..."
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Cited by 17 (6 self)
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This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995
Degree spectra of prime models
 J. Symbolic Logic
, 2004
"... 2.1 Notation from model theory................... 4 2.2 F ..."
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2.1 Notation from model theory................... 4 2.2 F
Questions in Computable Algebra and Combinatorics
, 1999
"... this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these ..."
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Cited by 5 (0 self)
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this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25
Turing degrees of the isomorphism types of algebraic objects
 the Journal of the London Mathematical Society
"... Abstract. The Turing degree spectrum of a countable structure A is the set of all Turing degrees of isomorphic copies of A. The Turing degree of the isomorphism type of A, if it exists, is the least Turing degree in its degree spectrum. We show there are countable fields, rings, and torsionfree abe ..."
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Abstract. The Turing degree spectrum of a countable structure A is the set of all Turing degrees of isomorphic copies of A. The Turing degree of the isomorphism type of A, if it exists, is the least Turing degree in its degree spectrum. We show there are countable fields, rings, and torsionfree abelian groups of arbitrary rank, whose isomorphism types have arbitrary Turing degrees. We also show that there are structures in each of these classes whose isomorphism types do not have Turing degrees. 1.
Ordercomputable sets
"... We give a straightforward computablemodeltheoretic definition of a property of ∆0 2 sets, called ordercomputability. We then prove various results about these sets which suggest that, simple though the definition is, the property defies any characterization in pure computability theory. The most ..."
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Cited by 4 (0 self)
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We give a straightforward computablemodeltheoretic definition of a property of ∆0 2 sets, called ordercomputability. We then prove various results about these sets which suggest that, simple though the definition is, the property defies any characterization in pure computability theory. The most striking example is the construction of two computably isomorphic c.e. sets, one of which is ordercomputable and the other not. 1
Constructive models of uncountably categorical theories
 PROC. AMER. MATH. SOC
, 1999
"... We construct a strongly minimal (and thus uncountably categorical) but not totally categorical theory in a finite language of binary predicates whose only constructive (or recursive) model is the prime model. ..."
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Cited by 3 (2 self)
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We construct a strongly minimal (and thus uncountably categorical) but not totally categorical theory in a finite language of binary predicates whose only constructive (or recursive) model is the prime model.
On the complexity of the successivity relation in computable linear orderings, in preparation
"... Abstract. In this paper, we solve a longstanding open question (see, e.g., Downey [6, §7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering L has infinitely many successivities, then the spectrum ..."
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Abstract. In this paper, we solve a longstanding open question (see, e.g., Downey [6, §7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering L has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we use a new method of constructing ∆ 0 3isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is of independent interest. It would seem to promise many further applications.
Minimal pairs and quasiminimal degrees for the joint spectra of structures
 Lecture
"... Abstract. Two properties of the CoSpectrum of the Joint Spectrum of finitely many abstract structures are presented a Minimal Pair type theorem and the existence of a QuasiMinimal degree with respect to the Joint Spectrum of the structures. 1 ..."
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Abstract. Two properties of the CoSpectrum of the Joint Spectrum of finitely many abstract structures are presented a Minimal Pair type theorem and the existence of a QuasiMinimal degree with respect to the Joint Spectrum of the structures. 1