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14
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
Computable trees, prime models, and relative decidability
 PROC. AMER. MATH. SOC
, 2005
"... We show that for every computable tree T with no dead ends and all paths computable, and every D>T ∅, there is a Dcomputable listing of the isolated paths of T. It follows that for every complete decidable theory T such that all the types of T are computable and every D>T ∅, there is a Ddecidable ..."
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We show that for every computable tree T with no dead ends and all paths computable, and every D>T ∅, there is a Dcomputable listing of the isolated paths of T. It follows that for every complete decidable theory T such that all the types of T are computable and every D>T ∅, there is a Ddecidable prime model of T. This result extends a theorem of Csima and yields a stronger version of the theorem, due independently to Slaman and Wehner, that there is a structure with presentations of every nonzero degree but no computable presentation.
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.
Enumerations in computable structure theory
 Ann. Pure Appl. Logic
"... Goncharov, Harizanov, Knight, Miller, and Solomon gratefully acknowledge NSF support under binational ..."
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Goncharov, Harizanov, Knight, Miller, and Solomon gratefully acknowledge NSF support under binational
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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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
Model Theoretic Complexity of Automatic Structures
 PROC. TAMC ’08, LNCS 4978
, 2008
"... We study the complexity of automatic structures via wellestablished concepts from both logic and model theory, including ordinal heights (of wellfounded relations), Scott ranks of structures, and CantorBendixson ranks (of trees). We prove the following results: 1) The ordinal height of any autom ..."
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We study the complexity of automatic structures via wellestablished concepts from both logic and model theory, including ordinal heights (of wellfounded relations), Scott ranks of structures, and CantorBendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic wellfounded partial order is bounded by ωω; 2) The ordinal heights of automatic wellfounded relations are unbounded below ωCK 1, the first noncomputable ordinal; 3) For any computable ordinal α, there is an automatic structure of Scott rank at least α. Moreover, there are automatic structures of Scott rank ωCK 1, ωCK 1 +1; 4) For any computable ordinal α, there is an automatic successor tree of CantorBendixson rank α.
Trivial, strongly minimal theories are model complete after naming constants
 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
, 2002
"... We prove that if M is any model of a trivial, strongly minimal theory, then the elementary diagram Th(MM) is a model complete LMtheory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are 0 ′ ′decidable, and that the spectrum of compu ..."
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We prove that if M is any model of a trivial, strongly minimal theory, then the elementary diagram Th(MM) is a model complete LMtheory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are 0 ′ ′decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is Σ 0 5.
Soare, Bounding homogeneous models
"... A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a ddecidable homogeneous model A, i.e., the elementary diagram D e (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithm ..."
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A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a ddecidable homogeneous model A, i.e., the elementary diagram D e (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogeneous model of T has a PA degree. 1