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28
Degree Spectra of Relations on Computable Structures
 J. Symbolic Logic
, 1999
"... Abstract We give some new examples of possible degree spectra of invariant relations on \Delta 02categorical computable structures that demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the kinds of sets of degrees that can ..."
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Abstract We give some new examples of possible degree spectra of invariant relations on \Delta 02categorical computable structures that demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the kinds of sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable relation on a \Delta 02categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses [22] to establish the same result for computable relations on computable linear orderings.
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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Cited by 8 (0 self)
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Degree spectra of relations on Boolean algebras
, 2002
"... Abstract We show that every computable relation on a computable Boolean algebra B is either definable by a quantifierfree formula with constants from B (in which case it is obviously intrinsically computable) or has infinite degree spectrum. Computable mathematics has been the focus of a large amou ..."
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Abstract We show that every computable relation on a computable Boolean algebra B is either definable by a quantifierfree formula with constants from B (in which case it is obviously intrinsically computable) or has infinite degree spectrum. Computable mathematics has been the focus of a large amount of research in the past few decades. Computable model theory in particular has seen vigorous and varied activity, leading to the discovery and intensive investigation of a number of central recurring themes. Among these is the study of the computabilitytheoretic properties of the images of a relation on a structure in different computable copies of the structure. In this paper, we investigate computable relations on Boolean algebras from this point of view. Boolean algebras are very interesting to computable model theorists because, like linear orderings, they are a natural, nontrivial, and wellstudied class of structures that exhibits much more structure than is present in the general case. Thus, studying computable Boolean algebras can give us insight into the nature of computation under constraints. We will define the relevant concepts from computable model theory below. A valuable recent reference covering a wide range of topics in computable mathematics is The first and second authors ' research was partially supported by the Marsden Fund of New Zealand.
The isomorphism problem for computable abelian pgroups of bounded length
"... Abstract. Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are nonclassifiable in general, but are classifiable when we consider only countable members. This paper explores such a ..."
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Cited by 6 (3 self)
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Abstract. Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are nonclassifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from nonclassifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian pgroups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new backandforth relations on such groups are calculated. 1.
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
The isomorphism problem for classes of computable fields
 Arch. Math. Logic
, 2003
"... Abstract. Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are nonclassifiable in general, but are classifiable when we consider only countable members. This paper explores such a ..."
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Cited by 5 (3 self)
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Abstract. Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are nonclassifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out several examples. One motivation is to see whether some classes whose set of countable members is very complex become classifiable when we consider only computable members. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from nonclassifiable. For arbitrary fields — even real closed fields — we show that the isomorphism problem is Σ1 1 complete (the maximum possible), and for others we show that it is of relatively low complexity. We show that the isomorphism problem for algebraically closed fields, Archimedean real closed fields, or vector spaces is Π0 3 complete. 1.
Comparing classes of finite structures
 Algebra and Logic
"... In many branches of mathematics, there is work classifying a collection of objects, up to isomorphism or other important equivalence, in terms of nice invariants. In descriptive set theory, there is a body of work using a notion of ..."
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Cited by 5 (3 self)
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In many branches of mathematics, there is work classifying a collection of objects, up to isomorphism or other important equivalence, in terms of nice invariants. In descriptive set theory, there is a body of work using a notion of
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.
A computably categorical structure whose expansion by a constant has infinite computable dimension
 Journal of Symbolic Logic
"... Cholak, Goncharov, Khoussainov, and Shore [J. Symbolic Logic 64 (1999) 13– 37] showed that for each k> 0 there is a computably categorical structure whose expansion by a constant has computable dimension k. We show that the same is true with k replaced by ω. Our proof uses a version of Goncharov’s m ..."
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Cited by 3 (2 self)
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Cholak, Goncharov, Khoussainov, and Shore [J. Symbolic Logic 64 (1999) 13– 37] showed that for each k> 0 there is a computably categorical structure whose expansion by a constant has computable dimension k. We show that the same is true with k replaced by ω. Our proof uses a version of Goncharov’s method of left and right operations. ∗Partially supported by an Alfred P. Sloan Doctoral Dissertation Fellowship and NSF Grant DMS0200465. ∗∗Partially supported by NSF Grants DMS0100035 and INT9602579. Thanks also to the Mathe
Realizing levels of the hyperarithmetic hierarchy as degree spectra of relations on computable structures
, 2003
"... We construct a class of relations on computable structures whose degree spectra form natural classes of degrees. Given any computable ordinal ff and reducibility r stronger than or equal to mreducibility, we show how to construct a structure with an intrinsically \Sigma ff invariant relation whose ..."
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We construct a class of relations on computable structures whose degree spectra form natural classes of degrees. Given any computable ordinal ff and reducibility r stronger than or equal to mreducibility, we show how to construct a structure with an intrinsically \Sigma ff invariant relation whose degree spectrum consists of all nontrivial \Sigma ff rdegrees. We extend this construction to show that \Sigma ff can be replaced by either \Pi ff or \Delta ff.