A new spectrum of recursive models using an amalgamation construction

2011 ◽  
Vol 76 (3) ◽  
pp. 883-896 ◽  
Author(s):  
Uri Andrews
Keyword(s):  
Model Theory ◽  
Recursive Model ◽  
Recursive Models ◽  
Saturated Models ◽  
Minimal Theory ◽  
Finite Language ◽  
Amalgamation Construction

AbstractWe employ an infinite-signature Hrushovski amalgamation construction to yield two results in Recursive Model Theory. The first result, that there exists a strongly minimal theory whose only recursively presentable models are the prime and saturated models, adds a new spectrum to the list of known possible spectra. The second result, that there exists a strongly minimal theory in a finite language whose only recursively presentable model is saturated, gives the second non-trivial example of a spectrum produced in a finite language.

Effective model theory vs. recursive model theory

1990 ◽  
Vol 55 (3) ◽  
pp. 1168-1191 ◽  
Author(s):  
John Chisholm
Keyword(s):  
Model Theory ◽  
Classical Model ◽  
Effective Model ◽  
Regularity Conditions ◽  
Strong Regularity ◽  
Recursive Model ◽  
Recursive Models

Recursive model theory is supposed to be the study of the effectiveness of constructions and theorems in model theory. This often involves getting “effective” versions of various classical model-theoretic notions. The traditional way of doing this is to restrict attention to recursive models, and recursive isomorphisms between them, etc. Thus for example the following definition appears in the literature (in [3] and [1]).Definition. Given a recursive model A and an n Є ω, a subset R ⊆ An is called intrinsically r.e. provided that for every recursive model B ≈ A, the isomorphic image in B of R is an r.e. subset of Bn.It is clear that if R is definable by a (recursive, infinitary) Σ10 formula (with finitely many parameters from A), then R is intrinsically r.e. It seems natural for the converse to be true. Indeed, provided that (A, R) is sufficiently “regular” in a sense made precise in a theorem of Ash and Nerode (see [3]), the converse is true. However, if we drop the (rather strong) regularity conditions, there exist “pathological” examples of intrinsically r.e. relations which are not definable by a Σ10 formula (see [7]).In this paper, we suggest a rather different approach to studying the effectiveness of model theory, an approach we have dubbed “effective model theory”. The basic idea is to allow arbitrary nonrecursive models, but to require all notions to be relativized to the complexity of the models involved. (Much the same notion has been used in [2] under the name “relatively recursive model theory”.) Thus for example we have the following effective model theory version of the property of being intrinsically r.e.

The Complexity of intrinsically r.e. subsets of existentially decidable models

1990 ◽  
Vol 55 (3) ◽  
pp. 1213-1232 ◽  
Author(s):  
John Chisholm
Keyword(s):  
Model Theory ◽  
Recursion Theory ◽  
Natural Numbers ◽  
Recursive Model ◽  
Recursive Models ◽  
Pairing Function ◽  
Effective Pairing ◽  
Theory And Model

Recursive model theory involves the study of relationships between recursion theory and model theory. One direction this often takes is to study the effectiveness of various aspects of model theory. This paper examines such questions by examining some properties of recursive models; that is, models whose basic relations, functions, and constants are all uniformly recursive (and whose universe is the set of natural numbers). Somewhat more precisely:Let be a model whose universe is N, and let (θ())i be an effective enumeration of all quantifier-free formulas of the language of . Then is recursive if {〈, i〉: satisfies (θ())i, in } is a recursive subset of N. (Here and throughout the paper, 〈 〉 denotes an effective pairing function, or an effective coding of sequences, as required.) Similarly, let (ϕ())i be an effective enumeration of all existential formulas of the language of . Then is existentially decidable if {〈, i〉: satisfies (ϕ())i in } is a recursive subset of N.It is clear that if is recursive and is a model isomorphic to , then may lose many of the recursive properties of . In the simplest example, could easily fail to be a recursive model. But even if we require that be a recursive model, it could still fail to retain other recursive properties of . An example of the sort of property which can be studied in this vein is the following notion, introduced by Ash and Nerode in [2].

Theories with recursive models

1979 ◽  
Vol 44 (1) ◽  
pp. 59-76 ◽  
Author(s):  
Manuel Lerman ◽  
James H. Schmerl
Keyword(s):  
Existential Quantifier ◽  
Categorical Theory ◽  
Recursive Model ◽  
Recursive Models ◽  
Decidable Theory ◽  
Decidable Model

A structure is recursive if the set of quantifier-free sentences in the complete diagram ⊿() of is recursive. It has been known for some time that every decidable theory has a recursive model. In fact, every decidable theory has a decidable model (that is a model such that ⊿() is recursive). In this paper we find other conditions which imply that a theory have a recursive model.In §1 we study the relation between an ℵ0-categorical theory T having a recursive model and the complexity of the quantificational hierarchy of that theory. We let ∃0 denote the set of quantifier-free sentences, and let ∃n÷1 denote the set of sentences beginning with an existential quantifier and having n alternations of quantifiers. (∀n is defined analogously.) Then we show that if T is an arithmetical ℵ0-categorical theory such that T ⋂ ∃n÷2 is Σn÷10 for each n < ω, then T has a recursive model. We show that this is a best possible result by giving an example of a ⊿n÷20 ℵ0-categorical theory T such that T ⋂ ∃n÷1 is recursive yet T has no recursive model.In §2 we consider the theory of trees. Ershov [1] had proved that every Σ10 theory of trees has a recursive model. We show this to be best possible by giving an example of a ⊿20 theory of trees which has no recursive model.

Nonarithmetical ℵ0-categorical theories with recursive models

1994 ◽  
Vol 59 (1) ◽  
pp. 106-112 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
First Order ◽  
Recursive Model ◽  
Recursive Models ◽  
First Order Theory ◽  
Free Variables

In what follows, L is a recursive language. The structures to be considered are L-structures with universe named by constants from ω. A structure is recursive A if the open diagram D() is recursive. Lerman and Schmerl [L-S] proved the following result.Let T be an ℵ0-categorical elementary first-order theory. Suppose that for all n, , and T is arithmetical. Then T has a recursive model.The aim of this paper is to extend Theorem 0.1. Stating the extension requires some terminology. Consider finitary formulas with symbols from L and sometimes extra constants from ω. For each n ∈ ω, the Σn and Πn formulas are as usual. Then Bnformulas are Boolean combinations of Σn formulas. For an L-structure , Dn() denotes the set of Bn sentences in the complete diagram Dc(). A complete Σn theory is a maximal consistent set of ΣnL-sentences. We may write φ(x), or Γ(x), to indicate that the free variables of the formula φ, or the set Γ, are among those in x. A complete Bn type for x is a maximal consistent set Γ(x) of Bn formulas with just the free variables x.If T is ℵ0-categorical, then for each x only finitely many complete types Γ(x) are consistent with T. While Lerman and Schmerl stated their result just for ℵ0-categorical theories, essentially the same proof yields the following.Theorem 0.2. Let T be a consistent, complete theory such that for all n andx, only finitely many complete Bn types Γ(x) are consistent with T.

S. S. Goncharov. Autostability and computable families of constructivizations. Algebra and Logic, vol. 14 (1975), no. 6, pp. 392–409. - S. S. Goncharov. The quantity of nonautoequivalent constructivizations. Algebra and Logic, vol. 16 (1977), no. 3, pp. 169–185. - S. S. Goncharov and V. D. Dzgoev. Autostability of models. Algebra and Logic, vol. 19 (1980), no. 1, pp. 28–37. - J. B. Remmel. Recursively categorical linear orderings. Proceedings of the American Mathematical Society, vol. 83 (1981), no. 2, pp. 387–391. - Terrence Millar. Recursive categoricity and persistence. The Journal of Symbolic Logic, vol. 51 (1986), no. 2, pp. 430–434. - Peter Cholak, Segey Goncharov, Bakhadyr Khoussainov and Richard A. Shore. Computably categorical structures and expansions by constants. The Journal of Symbolic Logic, vol. 64 (1999), no. 1, pp. 13–137. - Peter Cholak, Richard A. Shore and Reed Solomon. A computably stable structure with no Scott family of finitary formulas. Archive for Mathematical Logic, vol. 45 (2006), no. 5, pp. 519–538. - Chris Ash, Julia Knight, Mark Manasse and Theodore Slaman. Generic copies of countable structures. Annals of Pure and Applied Logic, vol. 42 (1989), no. 3, pp. 195–205. - John Chisholm. Effective model theory vs. recursive model theory. The Journal of Symbolic Logic, vol. 55 (1990), no. 3, pp. 1168–1191.

2012 ◽  
Vol 18 (1) ◽  
pp. 131-134
Author(s):  
Daniel Turetsky
Keyword(s):  
Mathematical Logic ◽  
Model Theory ◽  
American Mathematical Society ◽  
Effective Model ◽  
Mathematical Society ◽  
Symbolic Logic ◽  
Applied Logic ◽  
Recursive Model ◽  
Linear Orderings ◽  
Scott Family

scholarly journals Foundations of recursive model theory

1978 ◽  
Vol 13 (1) ◽  
pp. 45-72 ◽  
Author(s):  
Terrence S. Millar
Keyword(s):  
Model Theory ◽  
Recursive Model

scholarly journals The Mediating Effect of Intellectual Capital Disclosure Between Firm Characteristics and Firm Value: Empirical Evidence From Indonesian Company With Non-recursive Model Analysis

2020 ◽  
Vol 11 (2) ◽  
pp. 14
Author(s):  
Toni Heryana ◽  
Sugeng Wahyudi ◽  
Wisnu Mawardi
Keyword(s):  
Firm Size ◽  
Intellectual Capital ◽  
Firm Value ◽  
Mediating Effect ◽  
Firm Characteristics ◽  
Recursive Model ◽  
Mediating Role ◽  
Recursive Models ◽  
Company Growth

Based on the signaling theory, this study seeks to explain the interaction of corporate value and the disclosure of intellectual capital in a framework of analysis of recursive models. Testing the recursive model also involves firm size and company growth as a characteristic of the company to clarify the mediating role of intellectual capital in mediating both of the firm's values. We find a positive relationship between firm size and growth on intellectual capital disclosure. The greater the size and growth of the company, the more it encourages companies to disclose intellectual capital in the company's annual report. Also, we find a non-recursive model between intellectual capital disclosure and firm value. This shows that the broader the disclosure of IC information by the company, the better the investor's perception of the company is reflected in the value of the company. Meanwhile, at different times the current condition of the company's value will encourage companies to disclose more complete IC information.

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