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Degree spectra and computable dimension in algebraic structures
 Annals of Pure and Applied Logic 115 (2002
, 2002
"... \Lambda \Lambda ..."
Effective model theory: the number of models and their complexity
 MODELS AND COMPUTABILITY
, 1999
"... Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems i ..."
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Cited by 18 (6 self)
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Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems is computable) and decidable structures (ones with decidable theories). If the objects of initial interest are typical mathematical structures, then the starting point is computable structures. We present an introduction to both of these aspects of effective model theory organized roughly around the themes of the number and types of models of theories with particular attention to categoricity (as either a hypothesis or a conclusion) and the analysis of various computability issues in families of models.
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
On the nbackandforth types of Boolean algebras
 In preparation
"... Abstract. The objective of this paper is to uncover the structure of the backandforth equivalence classes at the finite levels for the class of Boolean algebras. As an application, we obtain bounds on the computational complexity of determining the backandforth equivalence classes of a Boolean al ..."
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Cited by 5 (1 self)
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Abstract. The objective of this paper is to uncover the structure of the backandforth equivalence classes at the finite levels for the class of Boolean algebras. As an application, we obtain bounds on the computational complexity of determining the backandforth equivalence classes of a Boolean algebra for finite levels. This result has implications for characterizing the relatively intrinsically Σ 0 n relations of Boolean algebras as existential formulas over a finite set of relations. 1.
Relations on computable structures
 Contemporary Mathematics (University of Belgrade
"... Gődel’s incompleteness theorem from 1931 is an astonishing early result of computable mathematics. Gődel showed that “there are in fact relatively simple problems in the theory of ordinary whole numbers which cannot be decided from the axioms. ” The work of Gődel, Turing, Kleene, Church, ..."
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Cited by 5 (4 self)
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Gődel’s incompleteness theorem from 1931 is an astonishing early result of computable mathematics. Gődel showed that “there are in fact relatively simple problems in the theory of ordinary whole numbers which cannot be decided from the axioms. ” The work of Gődel, Turing, Kleene, Church,
EFFECTIVELY CATEGORICAL ABELIAN GROUPS
"... We study effective categoricity of computable abelian groups of the form ⊕ i∈ω H, where H is a subgroup of (Q, +). Such groups are called homogeneous completely decomposable. It is wellknown that a homogeneous completely decomposable group is computably categorical if and only if its rank is finit ..."
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We study effective categoricity of computable abelian groups of the form ⊕ i∈ω H, where H is a subgroup of (Q, +). Such groups are called homogeneous completely decomposable. It is wellknown that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite. We study ∆0 ncategoricity in this class of groups, for n> 1. We introduce a new algebraic concept of Sindependence which is a generalization of the wellknown notion of pindependence. We develop the theory of Pindependent sets. We apply these techniques to show that every homogeneous completely decomposable group is ∆0 3categorical. We prove that a homogeneous completely decomposable group of infinite rank is ∆0 2categorical if and only if it is isomorphic to the free module over the localization of Z by a computably enumerable set of primes P with the semilow complement (within the set of all primes). Finally, we apply these results and techniques to study the complexity of generating bases of computable free modules over localizations of integers, including the free abelian group.
Computable Structures: Presentations Matter
 IN PROCEEDINGS OF THE INTL. CONG. LMPS
, 1999
"... The computability properties of a relation R not included in the language of a computable structure A can vary from one computable presentation to another. We describe some classic results giving conditions on A or R that restrict the possible variations in the computable dimension of A (i.e. the ..."
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Cited by 1 (1 self)
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The computability properties of a relation R not included in the language of a computable structure A can vary from one computable presentation to another. We describe some classic results giving conditions on A or R that restrict the possible variations in the computable dimension of A (i.e. the number of isomorphic copies of A up to computable isomorphism) and the computational complexity of R. For example, what conditions guarantee that A is computably categorical (i.e. of dimension 1) or that R is intrinsically computable (i.e. computable in every presentation). In the absence of such conditions, we discuss the possible computable dimensions of A and variations (in terms of Turing degree) of R in different presentations (the degree spectrum of R). In particular, various classic theorems and more recent ones of the author, B. Khoussainov, D. Hirschfeldt and others about the possible degree spectra of computable relations on computable structures and the connections with ...
COUNTING THE BACKANDFORTH TYPES
"... Abstract. Given a class of structures K and n ∈ ω, we study the dichotomy between there being countably many nbackandforth equivalence classes and there being continuum many. In the latter case we show that, relative to some oracle, every set can be weakly coded in the (n − 1)st jump of some stru ..."
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Abstract. Given a class of structures K and n ∈ ω, we study the dichotomy between there being countably many nbackandforth equivalence classes and there being continuum many. In the latter case we show that, relative to some oracle, every set can be weakly coded in the (n − 1)st jump of some structure in K. In the former case we show that there is a countable set of infinitary Πn relations that captures all of the Πn information about the structures in K. In most cases where there are countably many nbackandforth equivalence classes, there is a computable description of them. We will show how to use this computable description to get a complete set of computably infinitary Πn formulas. This will allow us to completely characterize the relatively intrinsically Σ 0 n+1 relations in the computable structures of K, and to prove that no Turing degree can be coded by the (n − 1)st jump of any structure in K unless that degree is already below 0 (n−1). 1.
THE CLASSIFICATION PROBLEM FOR COMPACT COMPUTABLE METRIC SPACES
"... Abstract. We adjust methods of computable model theory to effective analysis. We use index sets and infinitary logic to obtain classificationtype results for compact computable metric spaces. We show that every compact computable metric space can be uniquely described, up to an isomorphism, by a com ..."
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Abstract. We adjust methods of computable model theory to effective analysis. We use index sets and infinitary logic to obtain classificationtype results for compact computable metric spaces. We show that every compact computable metric space can be uniquely described, up to an isomorphism, by a computable Π3 formula, and that orbits of elements are uniformly given by computable Π2 formulas. We show that the index set for such spaces is Π 0 3complete, and the isomorphism problem is Π 0 2complete within its index set. We also give further classification results for special classes of compact spaces, and for other related classes of Polish spaces. Finally, as our main result we show that each compact computable metric space is ∆ 0 3categorical, and there exists a compact computable Polish space which is not ∆ 0 2categorical. 1.