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Computable Isomorphisms, Degree Spectra of Relations, and Scott Families
 Ann. Pure Appl. Logic
, 1998
"... this paper we are interested in those structures in which the basic computations can be performed by Turing machines. ..."
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this paper we are interested in those structures in which the basic computations can be performed by Turing machines.
Computable categoricity of trees of finite height
 Journal of Symbolic Logic
"... We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ0 3conditi ..."
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Cited by 13 (6 self)
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We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ0 3condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n ≥ 1 in ω, there exists a computable tree of finite height which is ∆0 n+1categorical but not ∆0 ncategorical.
Feasibly categorical models
 in Logic and Computational Complexity, editorD.Leivant, Lecture Notes in Comp. Science 960
, 1995
"... Abstract. We define a notion of a Scott family of formulas for a feasible model and give various conditions on a Scott family which imply that two models with the same family are feasibly isomorphic. For example, if A and B possess a common strongly ptime Scott family and both have universe (1}*, t ..."
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Abstract. We define a notion of a Scott family of formulas for a feasible model and give various conditions on a Scott family which imply that two models with the same family are feasibly isomorphic. For example, if A and B possess a common strongly ptime Scott family and both have universe (1}*, then they are ptime isomorphic. These results are applied to the study of permutation structures, linear orderings, equivalence relations, and Abelian groups. For example, conditions on two permutation structures (A, f) and (B, g) are given which imply that (A, f) and (B, g) are ptime isomorphic.
Computable categoricity for algebraic fields with splitting algorithms, submitted for publication
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A computably stable structure with no Scott family of finitary formulas
 Arch. Math. Logic
"... One of the goals of computability theory is to find syntactic equivalences for computational properties. The Limit Lemma is a classic example of this type of equivalence: X ⊆ ω is computable from 0 ′ if and only if it is arithmetically definable by a ∆ 0 2 formula. A more relevant example for this p ..."
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One of the goals of computability theory is to find syntactic equivalences for computational properties. The Limit Lemma is a classic example of this type of equivalence: X ⊆ ω is computable from 0 ′ if and only if it is arithmetically definable by a ∆ 0 2 formula. A more relevant example for this paper was proved independently by Ash, Knight, Manasse and Slaman
Complexity and Categoricity
, 2002
"... We define a notion of a feasible Scott family of formulas for a feasible model and give various conditions on a Scott family which imply that two models with the same family are feasibly isomorphic. For example, if A and B possess a common strongly ptime Scott family and both have universe f1g \La ..."
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We define a notion of a feasible Scott family of formulas for a feasible model and give various conditions on a Scott family which imply that two models with the same family are feasibly isomorphic. For example, if A and B possess a common strongly ptime Scott family and both have universe f1g \Lambda, then they are ptime isomorphic. These