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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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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.
Classification from a computable viewpoint
 The Bulletin of Symbolic Logic
"... Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism ..."
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Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism
Index sets for computable structures
 Algebra and Logic
"... The index set of a computable structure A is the set of indices for computable copies of A. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, Qvector spaces, Archimedean real closed ordered fields, reduced Abelian ..."
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The index set of a computable structure A is the set of indices for computable copies of A. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, Qvector spaces, Archimedean real closed ordered fields, reduced Abelian pgroups of length less than ω 2, and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be mcomplete Π 0 n, dΣ 0 n,orΣ 0 n, for various n. In each case, the calculation involves finding an “optimal ” sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable Πn, dΣn, or Σn) yields a bound on the complexity of the index set. When we show mcompleteness of the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey theory.
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.