A two cardinal theorem for sets of formulas in a stable theory

1975 ◽  
Vol 21 (1) ◽  
pp. 7-23 ◽  
Author(s):  
Victor Harnik
Keyword(s):  
Stable Theory

scholarly journals Multisensor Fault Identification Scheme Based on Decentralized Sliding Mode Observers Applied to Reconfigurable Manipulators

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Bo Zhao ◽  
Yuanchun Li
Keyword(s):  
Sliding Mode ◽  
Sliding Mode Observer ◽  
Fault Identification ◽  
Stable Theory ◽  
Sensor Fault ◽  
Identification Scheme ◽  
Neural Network Controller ◽  
Network Controller ◽  
Sliding Mode Observers ◽  
Reconfigurable Manipulators

This paper concerns with a fault identification scheme in a class of nonlinear interconnected systems. The decentralized sliding mode observer is recruited for the investigation of position sensor fault or velocity sensor fault. First, a decentralized neural network controller is proposed for the system under fault-free state. The diffeomorphism theory is utilized to construct a nonlinear transformation for subsystem structure. A simple filter is implemented to convert the sensor fault into pseudo-actuator fault scenario. The decentralized sliding mode observer is then presented for multisensor fault identification of reconfigurable manipulators based on Lyapunov stable theory. Finally, two 2-DOF reconfigurable manipulators with different configurations are employed to verify the effectiveness of the proposed scheme in numerical simulation. The results demonstrate that one joint’s fault does not affect other joints and the sensor fault can be identified precisely by the proposed decentralized sliding mode observer.

Indiscernibles and decidable models

1983 ◽  
Vol 48 (1) ◽  
pp. 21-32 ◽  
Author(s):  
H. A. Kierstead ◽  
J. B. Remmel
Keyword(s):  
Order Theory ◽  
Stable Theory ◽  
Categorical Theory ◽  
Large Classes ◽  
Decidable Theory ◽  
First Order Theory ◽  
Decidable Model ◽  
Decidable Theories ◽  
Basic Facts ◽  
Infinite Set

Ehrenfeucht and Mostowski [3] introduced the notion of indiscernibles and proved that every first order theory has a model with an infinite set of order indiscernibles. Since their work, techniques involving indiscernibles have proved to be extremely useful for constructing models with various specialized properties. In this paper and in a sequel [5], we investigate the effective content of Ehrenfeucht's and Mostowski's result. In this paper we consider the question of which decidable theories have decidable models with infinite recursive sets of indiscernibles. In §1, using some basic facts from stability theory, we show that certain large classes of decidable theories have decidable models with infinite recursive sets of indiscernibles. For example, we show that every ω-stable decidable theory and every stable theory which possesses a certain strong decidability property called ∃Q-decidability have such models. In §2 we construct several examples of decidable theories which have no decidable models with infinite recursive sets of indiscernibles. These examples show that our hypothesis for our positive results in §1 are necessary. Finally in §3 we give two applications of our results. First as an easy application of our results in §1, we show that every ω-stable decidable theory has uncountable models which realize only recursive types. Also our counterexamples in §2 allow us to answer negatively two questions of Baldwin and Kueker [1] concerning the effectiveness of their elimination of Ramsey quantifiers for certain theories.In [5], we show that in general the problem of finding an infinite set of indiscernibles in a decidable model is recursively equivalent to finding a path through a recursive infinite branching tree. Similarly, we show that the problem of finding an co-type of a set of indiscernibles in a decidable ω-categorical theory is recursively equivalent to finding a path through a highly recursive finitely branching tree.

scholarly journals COUNTABLE MODELS OF THE THEORIES OF BALDWIN–SHI HYPERGRAPHS AND THEIR REGULAR TYPES

2019 ◽  
Vol 84 (3) ◽  
pp. 1007-1019
Author(s):  
DANUL K. GUNATILLEKA
Keyword(s):  
Large Body ◽  
Countable Model ◽  
Stable Theory ◽  
Generic Structure ◽  
Elementary Chain ◽  
Regular Type ◽  
Image Position ◽  
Original Class ◽  
Countable Models

AbstractWe continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } \right\}$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $ if and only if the dimension of $\mathfrak{M}_\beta $ is at most the dimension of $\mathfrak{M}_\gamma $ and that each countable model is isomorphic to some $\mathfrak{M}_\beta $. We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω-stable theory with a nonlocally modular regular type, answering a question of Pillay in [11].

scholarly journals GROUPOIDS AND RELATIVE INTERNALITY

2019 ◽  
Vol 84 (3) ◽  
pp. 987-1006
Author(s):  
LÉO JIMENEZ
Keyword(s):  
Stable Theory ◽  
Sufficient Condition ◽  
Natural Type ◽  
Image Position ◽  
Stationary Type

AbstractIn a stable theory, a stationary type $q \in S\left( A \right)$ internal to a family of partial types ${\cal P}$ over A gives rise to a type-definable group, called its binding group. This group is isomorphic to the group $Aut\left( {q/{\cal P},A} \right)$ of permutations of the set of realizations of q, induced by automorphisms of the monster model, fixing ${\cal P}\,\mathop \cup \nolimits \,A$ pointwise. In this article, we investigate families of internal types varying uniformly, what we will call relative internality. We prove that the binding groups also vary uniformly, and are the isotropy groups of a natural type-definable groupoid (and even more). We then investigate how properties of this groupoid are related to properties of the type. In particular, we obtain internality criteria for certain 2-analysable types, and a sufficient condition for a type to preserve internality.

scholarly journals Constructing strongly equivalent nonisomorphic models for unsuperstable theories, Part C

1999 ◽  
Vol 64 (2) ◽  
pp. 634-642
Author(s):  
Tapani Hyttinen ◽  
Saharon Shelah
Keyword(s):  
Stable Theory ◽  
Part C ◽  
Saturated Models

AbstractIn this paper we prove a strong nonstructure theorem for κ(T)-saturated models of a stable theory T with dop. This paper continues the work started in [1].

Pseudo-finite homogeneity and saturation

1999 ◽  
Vol 64 (4) ◽  
pp. 1689-1699 ◽  
Author(s):  
Jörg Flum ◽  
Martin Ziegler
Keyword(s):  
Expressive Power ◽  
Query Languages ◽  
Stable Theory ◽  
Finite Cover ◽  
Database Query ◽  
Homogeneity Property ◽  
Finite States

AbstractWhen analyzing database query languages a roperty, of theories, the pseudo-finite homogeneity property, has been introduced and applied (cf. [3]). We show that a stable theory has the pseudo-finite homogeneity property just in case its expressive power for finite states is bounded. Moreover, we introduce the corresponding pseudo-finite saturation property and show that a theory fails to have the finite cover property if and only if it has the pseudo-finite saturation property.

Expansions of models of ω-stable theories

1984 ◽  
Vol 49 (2) ◽  
pp. 470-477
Author(s):  
Steven Buechler
Keyword(s):  
Large Class ◽  
Stable Theory ◽  
Universal Model

AbstractWe prove that every relation-universal model of an ω-stable theory is saturated. We also show there is a large class of ω-stable theories for which every resplendent model is homogeneous.

Expansions of geometries

2003 ◽  
Vol 68 (3) ◽  
pp. 803-827 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Stable Theory ◽  
Successor Relation ◽  
Edge Probability ◽  
Limit Probability

AbstractFor n < ω, expand the structure (n, S, I, F) (with S the successor relation, I, F as the initial and final element) by forming graphs with edge probability n−α for irrational α, with 0 < α < 1. The sentences in the expanded language, which have limit probability 1, form a complete and stable theory.

scholarly journals Definability of groups in ℵ0-stable metric structures

2010 ◽  
Vol 75 (3) ◽  
pp. 817-840 ◽  
Author(s):  
Itaï Ben Yaacov
Keyword(s):  
Group Action ◽  
Stable Theory ◽  
Translation Invariant ◽  
Definable Groups ◽  
Metric Structures

AbstractWe prove that in a continuous ℵ0-stable theory every type-definable group is definable. The two main ingredients in the proof are:(i) Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from [Ben08], allowing us to prove the theorem in case the metric is invariant under the group action; and(ii) Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones.

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