Degrees of autostability relative to strong constructivizations

Гончаров, С. С.

doi:10.1134/s0081543811060071

Cited by 23 publications

(10 citation statements)

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“…He showed that every cardinal from 1 through ω can be the computable dimension of a computable structure. (See [11] and [12] for these and related results.) However, by far the most common computable dimensions are 1 (which is equivalent to computable categoricity) and ω, and for many classes of structures, these are the only possible computable dimensions: linear orders, Boolean algebras, and trees, for example.…”

Section: Conclusion and Questionsmentioning

confidence: 89%

Categoricity properties for computable algebraic fields

Hirschfeldt

Kramer

Miller

et al. 2014

Trans. Amer. Math. Soc.

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Section: Conclusion and Questionsmentioning

confidence: 89%

Categoricity properties for computable algebraic fields

Hirschfeldt

Kramer

Miller

et al. 2014

Trans. Amer. Math. Soc.

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“…On the other hand, the index set of relatively computably categorical structures is Σ 0 3 -complete [2]. More recently, Goncharov started to study the complexity of the notion of categoricity restricted to decidable structures [9][10][11].…”

Section: Definitionmentioning

confidence: 99%

Index Sets for n-Decidable Structures Categorical Relative to m-Decidable Presentations

et al. 2015

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Keywords: index set, structure categorical relative to n-decidable presentations, n-decidable structure categorical relative to m-decidable presentations.We say that a structure is categorical relative to n-decidable presentations (or autostable relative to n-constructivizations) if any two n-decidable copies of the structure are computably isomorphic. For n = 0, we have the classical definition of a computably categorical (autostable) structure. Downey, Kach, Lempp, Lewis, Montalbán, and Turetsky proved that there is no simple syntactic characterization of computable categoricity. More formally, they showed that the index set of computably categorical structures is Π 1 1 -complete. Here we study index sets of n-decidable structures that are categorical relative to m-decidable presentations, for various m, n ∈ ω. If m ≥ n ≥ 0, then the index set is again Π 1 1 -complete, i.e., there is no nice description of the class of n-decidable structures that are categorical relative to m-decidable presentations. In the case m = n − 1 ≥ 0, the index set is Π 0 4 -complete, while if 0 ≤ m ≤ n − 2, the index set is Σ 0 3 -complete. * Supported by Austrian Science Fund FWF (projects V 206 and I 1238).

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“…Proof. Goncharov showed in [13] that if two computable structures are isomorphic via a 0 -computable isomorphism but not via any computable isomorphism, then the isomorphism type of those structures has computable dimension ω. By Corollary 5.4, this result applies to all computable fields with low splitting sets which are not computably categorical, and by Theorem 3.4, such a field exists.…”

Section: D-computable Categoricitymentioning

confidence: 99%

“…On the other hand, the known structural characterization of computably categorical trees requires a description by recursion on the heights of finite trees. The question has been studied for a number of other theories as well, and results along these lines may be found in [12], [13], [14], [15], [16], [21], [24], [27], [30], and [31].…”

Section: Introductionmentioning

confidence: 99%