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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.
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
On the nbackandforth types of Boolean algebras
 IN PREPARATION
, 2010
"... 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 fo ..."
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Cited by 5 (1 self)
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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.
INDECOMPOSABLE LINEAR ORDERINGS AND HYPERARITHMETIC ANALYSIS
, 2006
"... A statement of hyperarithmetic analysis is a sentence of second order arithmetic S such that for every Y ⊆ ω, the minimum ωmodel containing Y of RCA0+S is HYP(Y), the ωmodel consisting of the sets hyperarithmetic in Y. We provide an example of a mathematical theorem which is a statement of hyper ..."
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A statement of hyperarithmetic analysis is a sentence of second order arithmetic S such that for every Y ⊆ ω, the minimum ωmodel containing Y of RCA0+S is HYP(Y), the ωmodel consisting of the sets hyperarithmetic in Y. We provide an example of a mathematical theorem which is a statement of hyperarithmetic analysis. This statement, that we call INDEC, is due to Jullien [Jul69]. To the author’s knowledge, no other already published, purely mathematical statement has been found with this property until now. We also prove that, over RCA0, INDEC is implied by ∆1 1CA0 and implies ACA0, but of course, neither ACA0, nor ACA0 + imply it. We introduce five other statements of hyperarithmetic analysis and study the relations among them. Four of them are related to finitelyterminating games. The fifth one, related to iterations of the Turing jump, is strictly weaker than all the other statements that we study in this paper, as we prove using Steel’s method of forcing with tagged trees.
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.
ON THE EQUIMORPHISM TYPES OF LINEAR ORDERINGS.
, 2006
"... A linear ordering (also known as total ordering) embeds into another linear ordering if it is isomorphic to a subset of it. Two linear orderings are said to be equimorphic if they can be embedded in each other. This is an equivalence relation, and we call the equivalence classes equimorphism types. ..."
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A linear ordering (also known as total ordering) embeds into another linear ordering if it is isomorphic to a subset of it. Two linear orderings are said to be equimorphic if they can be embedded in each other. This is an equivalence relation, and we call the equivalence classes equimorphism types. We
BEYOND THE ARITHMETIC
, 2005
"... Various results in different areas of Computability Theory are proved. First we work with the Turing degree structure, proving some embeddablity and decidability results. To cite a few: we show that every countable upper semilattice containing a jump operation can be embedded into the Turing degrees ..."
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Various results in different areas of Computability Theory are proved. First we work with the Turing degree structure, proving some embeddablity and decidability results. To cite a few: we show that every countable upper semilattice containing a jump operation can be embedded into the Turing degrees, of course, preserving jump and join; we show that every finite partial ordering labeled with the classes in the generalized high/low hierarchy can be embedded into the Turing degrees; we show that every generalized high degree has the complementation property; and we show that if a Turing degree a is either 1generic and ∆ 0 1, 2generic and arithmetic, nREA, or arithmetically generic, then the theory of the partial ordering of the Turing degrees below a is recursively isomorphic to true first order arithmetic. Second, we work with equimorphism types of linear orderings from the viewpoints of Computable Mathematics and Reverse Mathematics. (Two linear orderings are equimorphic if they can be embedded in each other.) Spector proved in 1955 that every hyperarithmetic ordinal is isomorphic to a recursive one. We extend his result and prove that every hyperarithmetic linear ordering is equimorphic to a recursive one. From the viewpoint of Reverse Mathematics, we look at the strength of Fraïssé’s conjecture. From our results, we deduce that Fraïssé’s conjecture is sufficient and necessary to develop a reasonable theory of equimorphism types of linear orderings. Other topics we include in this thesis are the following: we look at structures for which Arithmetic Transfinite Recursion is the natural system to study them; we study theories of hyperarithmetic analysis and present a new natural example; we look at the complexity of the elementary equivalence problem for Boolean algebras; and we prove that there is a minimal pair of Kolmogorovdegrees.