Spectra of theories and structures

Andrews, Uri; Miller, Joseph S.

doi:10.1090/s0002-9939-2014-12283-0

Cited by 11 publications

(21 citation statements)

References 18 publications

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“…Andrews and Miller showed that the family

{d : {boldd}^{(ω + 1)} \geq {bold0}^{(ω \cdot 2 + 2)}}

is not the theory spectrum of a structure . Rossegger's result therefore gives an example of a bi‐embeddability spectrum which can not be a theory spectrum.…”

Section: Be Trivialitymentioning

confidence: 99%

“…At every stage s check if any i < s enters W A e,s and if so build a component isomorphic to F i using elements bigger than s not yet used during the construction. 2 As the construction is A-computable and tr(A) = tr(G), the constructed structure is an A-computable copy of S G .…”

Section: Strongly Locally Finite Graphsmentioning

confidence: 99%

“…Two structures are

Σ_{n}

equivalent,

A \equiv_{n} B

, if every first order

Σ_{n}

sentence true of

scriptA

is true of

scriptB

and vice versa. Andrews and Miller studied spectra of theories, the family of degrees of models of a complete theory T . In terms of the above definition, the theory spectrum of T is the spectrum of a model

scriptA

of T under elementary equivalence,

{prefixDgSp}_{\equiv} false(scriptA false)

.…”

Section: Introductionmentioning

confidence: 99%

“…Knight studied degree spectra of structures, the set of degrees of isomorphic copies of a given structure [16]. Since then the question which sets of degrees can be realized by degree spectra has been widely studied, cf., e.g., [1,2,14,15,25,28].…”

Section: Introductionmentioning

confidence: 99%

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Bi‐embeddability spectra and bases of spectra

Fokina

Rossegger

Mauro

2019

Mathematical Logic Qtrly

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We study degree spectra of structures with respect to the bi‐embeddability relation. The bi‐embeddability spectrum of a structure is the family of Turing degrees of its bi‐embeddable copies. To facilitate our study we introduce the notions of bi‐embeddable triviality and basis of a spectrum. Using bi‐embeddable triviality we show that several known families of degrees are bi‐embeddability spectra of structures. We then characterize the bi‐embeddability spectra of linear orderings and study bases of bi‐embeddability spectra of strongly locally finite graphs.

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“…Andrews and Miller showed that the family

{d : {boldd}^{(ω + 1)} \geq {bold0}^{(ω \cdot 2 + 2)}}

is not the theory spectrum of a structure . Rossegger's result therefore gives an example of a bi‐embeddability spectrum which can not be a theory spectrum.…”

Section: Be Trivialitymentioning

confidence: 99%

Section: Strongly Locally Finite Graphsmentioning

confidence: 99%

“…Two structures are

Σ_{n}

equivalent,

A \equiv_{n} B

, if every first order

Σ_{n}

sentence true of

scriptA

is true of

scriptB

scriptA

of T under elementary equivalence,

{prefixDgSp}_{\equiv} false(scriptA false)

.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bi‐embeddability spectra and bases of spectra

Fokina

Rossegger

Mauro

2019

Mathematical Logic Qtrly

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“…Define B i to be the result of applying Marker's (∀∃)-extension n times. It follows from [17] or [18] that if A i is computable then B i is n-decidable. And properties of the Marker's extensions proved in [19] imply that A i is computably categorical iff B i is categorical relative to n-decidable presentations.…”

Section: Complexity Of Index Setsmentioning

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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