The hierarchy theorem for generalized quantifiers

1996 ◽  
Vol 61 (3) ◽  
pp. 802-817 ◽  
Author(s):  
Lauri Hella ◽  
Kerkko Luosto ◽  
Jouko Väänänen
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Generalized Quantifiers ◽  
Similarity Type ◽  
First Order ◽  
Generalized Quantifier ◽  
Finite Structures

AbstractThe concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindström [17] with a counting argument. We extend his method to arbitrary similarity types.

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 The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context.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.

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 Undecidable relativizations of algebras of relations

1999 ◽  
Vol 64 (2) ◽  
pp. 747-760 ◽  
Author(s):  
Szabolcs Mikulás ◽  
Maarten Marx
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Similarity Type ◽  
First Order ◽  
Equational Theories ◽  
Algebras Of Relations ◽  
Guarded Fragment ◽  
Cylindric Set Algebras

AbstractIn this paper we show that relativized versions of relation set algebras and cylindric set algebras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic.

On generalized quantifiers in arithmetic

1982 ◽  
Vol 47 (1) ◽  
pp. 187-190 ◽  
Author(s):  
Carl Morgenstern
Keyword(s):  
Axiom System ◽  
Peano Arithmetic ◽  
Order Logic ◽  
First Order Logic ◽  
Generalized Quantifiers ◽  
First Order ◽  
Order Language ◽  
Truth Definition ◽  
Order Arithmetic ◽  
Infinite Set

In this note we investigate an extension of Peano arithmetic which arises from adjoining generalized quantifiers to first-order logic. Markwald [2] first studied the definability properties of L1, the language of first-order arithmetic, L, with the additional quantifer Ux which denotes “there are infinitely many x such that…. Note that Ux is the same thing as the Keisler quantifier Qx in the ℵ0 interpretation.We consider L2, which is L together with the ℵ0 interpretation of the Magidor-Malitz quantifier Q2xy which denotes “there is an infinite set X such that for distinct x, y ∈ X …”. In [1] Magidor and Malitz presented an axiom system for languages which arise from adding Q2 to a first-order language. They proved that the axioms are valid in every regular interpretation, and, assuming ◊ω1, that the axioms are complete in the ℵ1 interpretation.If we let denote Peano arithmetic in L2 with induction for L2 formulas and the Magidor-Malitz axioms as logical axioms, we show that in we can give a truth definition for first-order Peano arithmetic, . Consequently we can prove in that is Πn sound for every n, thus in we can prove the Paris-Harrington combinatorial principle and the higher-order analogues due to Schlipf.

Successor-invariant first-order logic on finite structures

2007 ◽  
Vol 72 (2) ◽  
pp. 601-618 ◽  
Author(s):  
Benjamin Rossman
Keyword(s):  
Binary Relation ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Arbitrary Choice ◽  
Finite Structures ◽  
Successor Relation ◽  
Order Invariant

AbstractWe consider successor-invariant first-order logic (FO + succ)inv, consisting of sentences Φ involving an “auxiliary” binary relation S such that (, S1) ⊨ Φ ⇔ (, S2) ⊨ Φ for all finite structures and successor relations S1, S2 on . A successor-invariant sentence Φ has a well-defined semantics on finite structures with no given successor relation: one simply evaluates Φ on (, S) for an arbitrary choice of successor relation S. In this article, we prove that (FO + succ)inv is more expressive on finite structures than first-order logic without a successor relation. This extends similar results for order-invariant logic [8] and epsilon-invariant logic [10].

Epsilon-logic is more expressive than first-order logic over finite structures

2000 ◽  
Vol 65 (4) ◽  
pp. 1749-1757 ◽  
Author(s):  
Martin Otto
Keyword(s):  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Database Theory ◽  
Finite Structures

AbstractThere are properties of finite structures that are expressible with the use of Hilbert's ∈-operator in a manner that does not depend on the actual interpretation for ∈-terms. but not expressible in plain first-order. This observation strengthens a corresponding result of Gurevich, concerning the invariant use of an auxiliary ordering in first-order logic over finite structures. The present result also implies that certain non-deterministic choice constructs, which have been considered in database theory, properly enhance the expressive power of first-order logic even as far as deterministic queries are concerned, thereby answering a question raised by Abiteboul and Vianu.

Finite Variable Logics in Descriptive Complexity Theory

1998 ◽  
Vol 4 (4) ◽  
pp. 345-398 ◽  
Author(s):  
Martin Grohe
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Complexity Classes ◽  
Finite Model ◽  
Polynomial Space ◽  
Finite Model Theory ◽  
First Order ◽  
First Order Theory ◽  
Finite Structures

Throughout the development of finite model theory, the fragments of first-order logic with only finitely many variables have played a central role. This survey gives an introduction to the theory of finite variable logics and reports on recent progress in the area.For each k ≥ 1 we let Lk be the fragment of first-order logic consisting of all formulas with at most k (free or bound) variables. The logics Lk are the simplest finite-variable logics. Later, we are going to consider infinitary variants and extensions by so-called counting quantifiers.Finite variable logics have mostly been studied on finite structures. Like the whole area of finite model theory, they have interesting model theoretic, complexity theoretic, and combinatorial aspects. For finite structures, first-order logic is often too expressive, since each finite structure can be characterized up to isomorphism by a single first-order sentence, and each class of finite structures that is closed under isomorphism can be characterized by a first-order theory. The finite variable fragments seem to be promising candidates with the right balance between expressive power and weakness for a model theory of finite structures. This may have motivated Poizat [67] to collect some basic model theoretic properties of the Lk. Around the same time Immerman [45] showed that important complexity classes such as polynomial time (PTIME) or polynomial space (PSPACE) can be characterized as collections of all classes of (ordered) finite structures definable by uniform sequences of first-order formulas with a fixed number of variables and varying quantifier-depth.

scholarly journals Successor-Invariant First-Order Logic on Classes of Bounded Degree

2021 ◽  
Vol Volume 17, Issue 3 ◽  
Author(s):  
Julien Grange
Keyword(s):  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Bounded Degree ◽  
First Order ◽  
Finite Structures ◽  
Successor Relation

We study the expressive power of successor-invariant first-order logic, which is an extension of first-order logic where the usage of an additional successor relation on the structure is allowed, as long as the validity of formulas is independent of the choice of a particular successor on finite structures. We show that when the degree is bounded, successor-invariant first-order logic is no more expressive than first-order logic.

The hierarchy theorem for second order generalized quantifiers

2006 ◽  
Vol 71 (1) ◽  
pp. 188-202 ◽  
Author(s):  
Juha Kontinen
Keyword(s):  
Order Type ◽  
Second Order ◽  
Order Logic ◽  
Generalized Quantifiers ◽  
Generalized Quantifier ◽  
Finite Structures ◽  
Second Order Logic

AbstractWe study definability of second order generalized quantifiers on finite structures. Our main result says that for every second order type t there exists a second order generalized quantifier of type t which is not definable in the extension of second order logic by all second order generalized quantifiers of types lower than t.

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