scholarly journals A model-theoretic approach to descriptive general frames: the van Benthem characterization theorem

2020 ◽  
Vol 30 (7) ◽  
pp. 1331-1355
Author(s):  
Nick Bezhanishvili ◽  
Tim Henke
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Characterization Theorem ◽  
Interpolation Theorem ◽  
Theoretic Approach ◽  
Basic Model ◽  
Finite Structures ◽  
Beth Definability ◽  
Skolem Theorem ◽  
General Frames

Abstract The celebrated van Benthem characterization theorem states that on Kripke structures modal logic is the bisimulation-invariant fragment of first-order logic. In this paper, we prove an analogue of the van Benthem characterization theorem for models based on descriptive general frames. This is an important class of general frames for which every modal logic is complete. These frames can be represented as Stone spaces equipped with a ‘continuous’ binary relation. The proof of our theorem generalizes Rosen’s proof of the van Benthem theorem for finite frames and uses as an essential technique a new notion of descriptive unravelling. We also develop a basic model theory for descriptive general frames and show that in many ways it behaves like the model theory of finite structures. In particular, we prove the failure of the compactness theorem, of the Beth definability theorem, of the Craig interpolation theorem and of the upward Löwenheim–Skolem theorem.1

scholarly journals SOME MODEL THEORY OF GUARDED NEGATION

2018 ◽  
Vol 83 (04) ◽  
pp. 1307-1344
Author(s):  
VINCE BÁRÁNY ◽  
MICHAEL BENEDIKT ◽  
BALDER TEN CATE
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Description Logics ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Beth Definability ◽  
Order Translations ◽  
Certain Answers

AbstractThe Guarded Negation Fragment (GNFO) is a fragment of first-order logic that contains all positive existential formulas, can express the first-order translations of basic modal logic and of many description logics, along with many sentences that arise in databases. It has been shown that the syntax of GNFO is restrictive enough so that computational problems such as validity and satisfiability are still decidable. This suggests that, in spite of its expressive power, GNFO formulas are amenable to novel optimizations. In this article we study the model theory of GNFO formulas. Our results include effective preservation theorems for GNFO, effective Craig Interpolation and Beth Definability results, and the ability to express the certain answers of queries with respect to a large class of GNFO sentences within very restricted logics.

scholarly journals INFINITARY PROPOSITIONAL RELEVANT LANGUAGES WITH ABSURDITY

2017 ◽  
Vol 10 (4) ◽  
pp. 663-681
Author(s):  
GUILLERMO BADIA
Keyword(s):  
Expressive Power ◽  
Interpolation Theorem ◽  
Boolean Logic ◽  
Isomorphism Theorem ◽  
Strong Completeness ◽  
Lack Of Compactness ◽  
Relevant Logics ◽  
Beth Definability ◽  
Beth Definability Property

AbstractAnalogues of Scott’s isomorphism theorem, Karp’s theorem as well as results on lack of compactness and strong completeness are established for infinitary propositional relevant logics. An “interpolation theorem” (of a particular sort introduced by Barwise and van Benthem) for the infinitary quantificational boolean logic L∞ω holds. This yields a preservation result characterizing the expressive power of infinitary relevant languages with absurdity using the model-theoretic relation of relevant directed bisimulation as well as a Beth definability property.

A Course on Basic Model Theory

2017 ◽  
Author(s):  
Haimanti Sarbadhikari ◽  
Shashi Mohan Srivastava
Keyword(s):  
Model Theory ◽  
Basic Model

On the strong semantical completeness of the intuitionistic predicate calculus

1968 ◽  
Vol 33 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Richmond H. Thomason
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Predicate Calculus ◽  
First Order ◽  
Semantic Tableaux ◽  
Weak Completeness ◽  
Skolem Theorem ◽  
Completeness Theorems

In Kripke [8] the first-order intuitionjstic predicate calculus (without identity) is proved semantically complete with respect to a certain model theory, in the sense that every formula of this calculus is shown to be provable if and only if it is valid. Metatheorems of this sort are frequently called weak completeness theorems—the object of the present paper is to extend Kripke's result to obtain a strong completeness theorem for the intuitionistic predicate calculus of first order; i.e., we will show that a formula A of this calculus can be deduced from a set Γ of formulas if and only if Γ implies A. In notes 3 and 5, below, we will indicate how to account for identity, as well. Our proof of the completeness theorem employs techniques adapted from Henkin [6], and makes no use of semantic tableaux; this proof will also yield a Löwenheim-Skolem theorem for the modeling.

Some Aspects of Model Theory and Finite Structures

2002 ◽  
Vol 8 (3) ◽  
pp. 380-403 ◽  
Author(s):  
Eric Rosen
Keyword(s):  
Computer Science ◽  
Complexity Theory ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Finite Model ◽  
Finite Model Theory ◽  
First Order ◽  
Single Sentence ◽  
Finite Structures

Model theory is concerned mainly, although not exclusively, with infinite structures. In recent years, finite structures have risen to greater prominence, both within the context of mainstream model theory, e.g., in work of Lachlan, Cherlin, Hrushovski, and others, and with the advent of finite model theory, which incorporates elements of classical model theory, combinatorics, and complexity theory. The purpose of this survey is to provide an overview of what might be called the model theory of finite structures. Some topics in finite model theory have strong connections to theoretical computer science, especially descriptive complexity theory (see [26, 46]). In fact, it has been suggested that finite model theory really is, or should be, logic for computer science. These connections with computer science will, however, not be treated here.It is well-known that many classical results of ‘infinite model theory’ fail over the class of finite structures, including the compactness and completeness theorems, as well as many preservation and interpolation theorems (see [35, 26]). The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context. At present, there is no known example of a classical theorem that remains true over finite structures, yet must be proved by substantially different methods. It is generally concluded that first-order logic is ‘badly behaved’ over finite structures.From the perspective of expressive power, first-order logic also behaves badly: it is both too weak and too strong. Too weak because many natural properties, such as the size of a structure being even or a graph being connected, cannot be defined by a single sentence. Too strong, because every class of finite structures with a finite signature can be defined by an infinite set of sentences. Even worse, every finite structure is defined up to isomorphism by a single sentence. In fact, it is perhaps because of this last point more than anything else that model theorists have not been very interested in finite structures. Modern model theory is concerned largely with complete first-order theories, which are completely trivial here.

SMT-based verification of data-aware processes: a model-theoretic approach

2020 ◽  
Vol 30 (3) ◽  
pp. 271-313
Author(s):  
Diego Calvanese ◽  
Silvio Ghilardi ◽  
Alessandro Gianola ◽  
Marco Montali ◽  
Andrey Rivkin
Keyword(s):  
Model Theory ◽  
Satisfiability Modulo Theories ◽  
Theoretic Approach ◽  
Model Checker ◽  
Present Contribution ◽  
Relational Systems ◽  
Infinite State Systems ◽  
Algorithmic Techniques ◽  
The One ◽  
Infinite State

AbstractIn recent times, satisfiability modulo theories (SMT) techniques gained increasing attention and obtained remarkable success in model-checking infinite-state systems. Still, we believe that whenever more expressivity is needed in order to specify the systems to be verified, more and more support is needed from mathematical logic and model theory. This is the case of the applications considered in this paper: we study verification over a general model of relational, data-aware processes, to assess (parameterized) safety properties irrespectively of the initial database (DB) instance. Toward this goal, we take inspiration from array-based systems and tackle safety algorithmically via backward reachability. To enable the adoption of this technique in our rich setting, we make use of the model-theoretic machinery of model completion, which surprisingly turns out to be an effective tool for verification of relational systems and represents the main original contribution of this paper. In this way, we pursue a twofold purpose. On the one hand, we isolate three notable classes for which backward reachability terminates, in turn witnessing decidability. Two of such classes relate our approach to conditions singled out in the literature, whereas the third one is genuinely novel. On the other hand, we are able to exploit SMT technology in implementations, building on the well-known MCMT (Model Checker Modulo Theories) model checker for array-based systems and extending it to make all our foundational results fully operational. All in all, the present contribution is deeply rooted in the long-standing tradition of the application of model theory in computer science. In particular, this paper applies these ideas in an original mathematical context and shows how these techniques can be used for the first time to empower algorithmic techniques for the verification of infinite-state systems based on arrays, so as to make such techniques applicable to the timely, challenging settings of data-aware processes.

Barwise: Abstract Model Theory and Generalized Quantifiers

2004 ◽  
Vol 10 (1) ◽  
pp. 37-53 ◽  
Author(s):  
Jouko Väänänen
Keyword(s):  
Model Theory ◽  
Interpolation Property ◽  
Order Logic ◽  
First Order Logic ◽  
Generalized Quantifiers ◽  
Abstract Model ◽  
Famous Theorem ◽  
Abstract Model Theory ◽  
First Order ◽  
Skolem Theorem

§1. Introduction. After the pioneering work of Mostowski [29] and Lindström [23] it was Jon Barwise's papers [2] and [3] that brought abstract model theory and generalized quantifiers to the attention of logicians in the early seventies. These papers were greeted with enthusiasm at the prospect that model theory could be developed by introducing a multitude of extensions of first order logic, and by proving abstract results about relationships holding between properties of these logics. Examples of such properties areκ-compactness. Any set of sentences of cardinality ≤ κ, every finite subset of which has a model, has itself a model. Löwenheim-Skolem Theorem down to κ. If a sentence of the logic has a model, it has a model of cardinality at most κ. Interpolation Property. If ϕ and ψ are sentences such that ⊨ ϕ → Ψ, then there is θ such that ⊨ ϕ → θ, ⊨ θ → Ψ and the vocabulary of θ is the intersection of the vocabularies of ϕ and Ψ.Lindstrom's famous theorem characterized first order logic as the maximal ℵ0-compact logic with Downward Löwenheim-Skolem Theorem down to ℵ0. With his new concept of absolute logics Barwise was able to get similar characterizations of infinitary languages Lκω. But hopes were quickly frustrated by difficulties arising left and right, and other areas of model theory came into focus, mainly stability theory. No new characterizations of logics comparable to the early characterization of first order logic given by Lindström and of infinitary logic by Barwise emerged. What was first called soft model theory turned out to be as hard as hard model theory.

scholarly journals Linear Hahn–Banach extension operators

1989 ◽  
Vol 32 (1) ◽  
pp. 53-57 ◽  
Author(s):  
Brailey Sims ◽  
David Yost
Keyword(s):  
Banach Space ◽  
Model Theory ◽  
General Result ◽  
Extension Operator ◽  
Density Character ◽  
Finite Dimensional ◽  
Skolem Theorem

Given any subspace N of a Banach space X, there is a subspace M containing N and of the same density character as N, for which there exists a linear Hahn–Banach extension operator from M* to X*. This result was first proved by Heinrich and Mankiewicz [4, Proposition 3.4] using some of the deeper results of Model Theory. More precisely, they used the Banach space version of the Löwenheim–Skolem theorem due to Stern [11], which in turn relies on the Löwenheim–Skolem and Keisler–Shelah theorems from Model Theory. Previously Lindenstrauss [7], using a finite dimensional lemma and a compactness argument, obtained a version of this for reflexive spaces. We shall show that the same finite dimensional lemma leads directly to the general result, without any appeal to Model Theory.

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