SEPARABLE MODELS OF RANDOMIZATIONS

2015 ◽  
Vol 80 (4) ◽  
pp. 1149-1181 ◽  
Author(s):  
URI ANDREWS ◽  
H. JEROME KEISLER
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Order Theory ◽  
Prime Model ◽  
First Order ◽  
First Order Theory ◽  
Separable Models ◽  
Homogeneous Models ◽  
Countable Models

AbstractEvery complete first order theory has a corresponding complete theory in continuous logic, called the randomization theory. It has two sorts, a sort for random elements of models of the first order theory, and a sort for events. In this paper we establish connections between properties of countable models of a first order theory and corresponding properties of separable models of the randomization theory. We show that the randomization theory has a prime model if and only if the first order theory has a prime model. And the randomization theory has the same number of separable homogeneous models as the first order theory has countable homogeneous models. We also show that when T has at most countably many countable models, each separable model of TR is uniquely characterized by a probability density function on the set of isomorphism types of countable models of T. This yields an analogue for randomizations of the results of Baldwin and Lachlan on countable models of ω1-categorical first order theories.

Prime models and almost decidability

1986 ◽  
Vol 51 (2) ◽  
pp. 412-420 ◽  
Author(s):  
Terrence Millar
Keyword(s):  
Order Theory ◽  
The Other ◽  
Prime Model ◽  
First Order ◽  
Decidable Theory ◽  
Complete Extension ◽  
First Order Theory ◽  
Additional Constant ◽  
Open Question ◽  
Countable Models

This paper introduces and investigates a notion that approximates decidability with respect to countable structures. The paper demonstrates that there exists a decidable first order theory with a prime model that is not almost decidable. On the other hand it is proved that if a decidable complete first order theory has only countably many complete types, then it has a prime model that is almost decidable. It is not true that every decidable complete theory with only countably many complete types has a decidable prime model. It is not known whether a complete decidable theory with only countably many countable models up to isomorphism must have a decidable prime model. In [1] a weaker result was proven—if every complete extension, in finitely many additional constant symbols, of a theory T fails to have a decidable prime model, then T has 2ω nonisomorphic countable models. The corresponding statement for saturated models is false, even if all the complete types are recursive, as was shown in [2]. This paper investigates a variation of the open question via a different notion of effectiveness—almost decidable.A tree Tr will be a subset of ω<ω that is closed under predecessor. For elements f, g in ω<ω ∪ ωω, ƒ ⊲ g iffdf ∀i < lh(ƒ)[ƒ(i) = g(i)].

Countable models of trivial theories which admit finite coding

1996 ◽  
Vol 61 (4) ◽  
pp. 1279-1286 ◽  
Author(s):  
James Loveys ◽  
Predrag Tanović
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Superstable Theories ◽  
Countable Models

AbstractWe prove:Theorem. A complete first order theory in a countable language which is strictly stable, trivial and which admits finite coding hasnonisomorphic countable models.Combined with the corresponding result or superstable theories from [4] our result confirms the Vaught conjecture for trivial theories which admit finite coding.

scholarly journals Generalized prime models

1971 ◽  
Vol 36 (4) ◽  
pp. 593-606 ◽  
Author(s):  
Robert Fittler
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
Prime Model ◽  
First Order ◽  
First Order Theory ◽  
General Conditions

A prime model O of some complete theory T is a model which can be elementarily imbedded into any model of T (cf. Vaught [7, Introduction]). We are going to replace the assumption that T is complete and that the maps between the models of T are elementary imbeddings (elementary extensions) by more general conditions. T will always be a first order theory with identity and may have function symbols. The language L(T) of T will be denumerable. The maps between models will be so called F-maps, i.e. maps which preserve a certain set F of formulas of L(T) (cf. I.1, 2). Roughly speaking a generalized prime model of T is a denumerable model O which permits an F-map O→M into any model M of T. Furthermore O has to be “generated” by formulas which belong to a certain subset G of F.

On some distributional properties of a first-order nonnegative bilinear time series model

2001 ◽  
Vol 38 (3) ◽  
pp. 659-671 ◽  
Author(s):  
Zhiqiang Zhang ◽  
Howell Tong
Keyword(s):  
Time Series ◽  
Probability Density Function ◽  
Numerical Method ◽  
Probability Density ◽  
Density Function ◽  
Stationary Distribution ◽  
Time Series Model ◽  
First Order ◽  
Distributional Properties ◽  
Tail Behaviour

We study a simple first-order nonnegative bilinear time-series model and give conditions under which the model is stationary. The probability density function of the stationary distribution (when it exists) is found. We also discuss the tail behaviour of the stationary distribution and calculate the probability density function by a numerical method. Simulation is used to check the calculation.

Prime e.c. commutative rings in characteristic n ≥ 2

1999 ◽  
Vol 64 (2) ◽  
pp. 629-633
Author(s):  
Dan Saracino
Keyword(s):  
Commutative Rings ◽  
Order Theory ◽  
Prime Model ◽  
Finite Model ◽  
First Order ◽  
First Order Theory ◽  
Building Models ◽  
Open Questions ◽  
A Chain ◽  
Mathematics Department

Let CR denote the first-order theory of commutative rings with unity, formulated in the language L = 〈 +, •, 0, 1〉. Virtually everything that is known about existentially complete (e.c.) models of CR is contained in Cherlin's paper [2], where it is shown, in particular, that the e.c. models are not first-order axiomatizable. The purpose of this note is to show that, in analogy with the case of fields, there exists a unique prime e.c. model of CR in each characteristic n > 2. As a consequence we settle Problem 8 in the list of open questions at the end of Hodges' book Building models by games ([5], p. 278).By a “prime” e.c. model of characteristic n ≥ 2 we mean one that embeds in every e.c. model of characteristic n. (The embedding is not always elementary, since [2] not all e.c. models of characteristic n are elementarily equivalent.) The prime model is characterized by the fact that it is the union of a chain of finite subrings each of which is an amalgamation base for CR. In §1 we describe the finite amalgamation bases for CR and show that every finite model embeds in a finite amalgamation base; in §2 we use this information to obtain prime e.c. models and answer Hodges' question.Our results on prime e.c. models were obtained some years ago, during the fall term of 1982, while the author was a visitor at Wesleyan University. The author wishes to take this opportunity to thank the mathematics department at Wesleyan for its hospitality during that visit, and subsequent ones.

An Example of a Complete First-Order Theory with All Models Algorithmically Trivial but Without Locally Finite Models

1982 ◽  
Vol 5 (3-4) ◽  
pp. 313-318
Author(s):  
Paweł Urzyczyn
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
Prime Model ◽  
Program Schema ◽  
Locally Finite ◽  
First Order ◽  
Finite Models ◽  
First Order Theory

We show an example of a first-order complete theory T, with no locally finite models and such that every program schema, total over a model of T, is strongly equivalent in that model to a loop-free schema. For this purpose we consider the notion of an algorithmically prime model, what enables us to formulate an analogue to Ryll-Nardzewski Theorem.

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