Effective Model Theory: The Number of Models and Their Complexity

Khoussainov, Bakhadyr; Shore, Richard A.

doi:10.1017/cbo9780511565670.009

Cited by 21 publications

(12 citation statements)

References 47 publications

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“…We give an example. If all types of an Ehrenfeucht theory T are computable then T must have at least three strongly constructive models (a proof of this can, for example, be found in [25]). Therefore for any decidable Ehrenfeucht theory T with exactly three models, the saturated model of T has a strong constructivization if and only if all models of T have strong constructivizations.…”

Section: Theorem 313 [34] [36] For Each N ≥ 3 There Exists a Decidamentioning

confidence: 99%

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Open problems in the theory of constructive algebraic systems

Goncharov¹,

Khoussainov²

2000

Contemporary Mathematics

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Abstract. In this paper we concentrate on open problems in two directions in the development of the theory of constructive algebraic systems. The first direction deals with universal algebras whose positive open diagrams can be computably enumerated. These algebras are called positive algebras. Here we emphasize the interplay between universal algebra and computability theory. We propose a systematic study of positive algebras as a new direction in the development of the theory of constructive algebraic systems. The second direction concerns the traditional topics in constructive model theory. First we propose the study of constructive models of theories with few models such as countably categorical theories, uncountably categorical theories, and Ehrenfeucht theories. Next, we propose the study of computable isomorphisms and computable dimensions of such models. We also discuss issues related to the computability-theoretic complexity of relations in constructive algebraic systems.

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Section: Theorem 313 [34] [36] For Each N ≥ 3 There Exists a Decidamentioning

confidence: 99%

“…By the theorem of Nurtazin this will give the desired result. The theory T is basically the one constructed in [25] (page 204). The language of T consists of infinitely many unary predicates R i .…”

Section: Isomorphisms Of Uncountably Categorical Modelsmentioning

confidence: 99%

Open problems in the theory of constructive algebraic systems

Goncharov¹,

Khoussainov²

2000

Contemporary Mathematics

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“…For example, the authors in [9] and in [8] deal with countable, computable models of uncountably categorical theories. For further related results we refer the reader to [10] and to [6]; they also contain a rather complete list of references. For more recent related investigations we refer to [5] and [1].…”

Section: Introductionmentioning

confidence: 99%

Non-computable models of certain first order theories

Sági

Horvath²

2017

2017 IEEE 14th International Scientific Conference on Informatics

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Abstract-Let D be a complexity class. A countable first order structure is defined to be D-presented iff all of its basic relations and functions are in D. We show, that if T is a first order theory with at least one uncountable Stone space then T has a countable model not isomorphic to any D-presented one. We also show that there is a countable ℵ0-categorical structure in a finite language which is not isomorphic to any D-presented structure; in addition, there exists a consistent first order theory in a finite language that does not have D-presented models, at all. Our proofs utilize model theoretic methods and do not involve any nontrivial recursion theoretic notion or construction.

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“…This is not the case, however, in computable model theory, since two isomorphic computable structures might have quite different computability-theoretic properties. (See [10] for examples of this phenomenon, as well as background relevant to this paper.) This leads us to study computable structures up to computable isomorphism, a point of view reflected in the following definition.…”

Section: Introductionmentioning

confidence: 99%