Index sets of decidable models

Fokina, Ekaterina B.

doi:10.1007/s11202-007-0097-y

Cited by 15 publications

(5 citation statements)

References 14 publications

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“…As a systematic framework, this strategy of looking at index sets of computable structures originated in [29] and has been used in many other applications, see [8,21,22,27,28,48].…”

Section: Decision Problemmentioning

confidence: 99%

Automatic and Polynomial-Time Algebraic Structures

Bazhenov¹,

Harrison-Trainor²,

Kalimullin³

et al. 2019

J. symb. log.

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A structure is automatic if its domain, functions, and relations are all regular languages. Using the fact that every automatic structure is decidable, in the literature many decision problems have been solved by giving an automatic presentation of a particular structure. Khoussainov and Nerode asked whether there is some way to tell whether a structure has, or does not have, an automatic presentation. We answer this question by showing that the set of Turing machines that represent automata-presentable structures is ${\rm{\Sigma }}_1^1 $-complete. We also use similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel.

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“…As a systematic framework, this strategy of looking at index sets of computable structures originated in [29] and has been used in many other applications, see [8,21,22,27,28,48].…”

Section: Decision Problemmentioning

confidence: 99%

Automatic and Polynomial-Time Algebraic Structures

Bazhenov¹,

Harrison-Trainor²,

Kalimullin³

et al. 2019

J. symb. log.

View full text Add to dashboard Cite

show abstract

“…In [6][7][8], it was shown that the given constructions allow of preserving the property of being prime and, as it is not hard to see, of being almost prime. The properties of being d-decidable for constructed models and of being d-autostable relative to d-strong constructivizations are also inherited.…”

Section: Reducing An Arbitrary Finite Signature To a Signature For Grmentioning

confidence: 99%

“…PROPOSITION 1.1 [6][7][8]. For any finite relational signature σ, there exists a signature σ 1 that consists of a single predicate P and is such that for any model M of the signature σ, there exists a model M of the signature σ 1 such that: [6][7][8].…”

Section: Reducing An Arbitrary Finite Signature To a Signature For Grmentioning

confidence: 99%

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Index Sets of Constructive Models of Bounded Signature that are Autostable Relative to Strong Constructivizations

Гончаров

Marchuk

2015

Algebra Logic

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“…Take some class K of models of σ closed under isomorphism. Many articles deal with index sets [5][6][7][8][9][10][11][12], as well as some estimate of the index sets of classes of models [13].…”

mentioning

confidence: 99%

Estimation of the algorithmic complexity of classes of computable models

Pavlovskiï

2008

Sib Math J

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We estimate the algorithmic complexity of the index set of some natural classes of computable models: finite computable models (Σ 0 2 -complete), computable models with ω-categorical theories (Δ 0 ω -complex Π 0 ω+2 -set), prime models (Δ 0 ω -complex Π 0 ω+2 -set), models with ω 1 -categorical theories (Δ 0 ω -complex Σ 0 ω+1 -set). We obtain a universal lower bound for the model-theoretic properties preserved by Marker's extensions (Δ 0 ω ).While studying the structural properties of theories and the algebraic properties of models we are especially interested in the complexity of the definitions of these properties from an algorithmic viewpoint. The characterization of classes, and hence, the abstract properties of models in this case can be obtained through bounds for index sets in the universal numbering of computable models [1, § 4]. Fix some computable signature σ with no function symbols. A model M of this signature is called computable if the universe of M and underlying relations are uniformly computable. Without loss of generality we will consider the models based on nonnegative integers.It is shown in [2] that given some finite signature with no function symbols there exists a universal computable numbering of computable models of this signature; i.e., some computable numbering of all computable models of the fixed signature such that every other computable numbering of computable models reduces to it. In the case of an infinite signature σ there exists a universal computable numbering of all computable models of σ together with all finite computable models of the finite parts of this signature [3, § 1.4; 4, § 1.3]. Both of these numberings are principal, i.e., every computable sequence of computable models of σ reduces to these numberings.The existence of this type of universal numberings of models allows us to obtain a characterization of classes of computable models by relating the index sets of concrete classes of models with a level in a suitable algorithmic hierarchy. The classes of algebraic systems in a signature with function symbols can be characterized using this approach via the representation of functions by their graphs which are predicates.Denote by ν σ = {M 0 , M 1 , . . . , M n , . . . } the universal computable numbering of models of the corresponding signature σ. Take some class K of models of σ closed under isomorphism.Definition 1. We call the index set I(K) of the class K of models the setMany articles deal with index sets [5][6][7][8][9][10][11][12], as well as some estimate of the index sets of classes of models [13].In this article we construct bounds in the corresponding complexity hierarchies for the following index sets of the available classes of computable models considered with various signatures:Fin σ = {i | M i is finite}, Cat σ = {i | M i has an ℵ 0 -categorical theory}, UCat σ = {i | M i has an ℵ 1 -categorical theory}, Prim σ = {i | M i is a prime model}.

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Index sets of decidable models

Cited by 15 publications

References 14 publications

Automatic and Polynomial-Time Algebraic Structures

Automatic and Polynomial-Time Algebraic Structures

Index Sets of Constructive Models of Bounded Signature that are Autostable Relative to Strong Constructivizations

Estimation of the algorithmic complexity of classes of computable models

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