Notions of locality and their logical characterizations over finite models

1999 ◽  
Vol 64 (4) ◽  
pp. 1751-1773 ◽  
Author(s):  
Lauri Hella ◽  
Leonid Libkin ◽  
Juha Nurmonen
Keyword(s):  
Small Degree ◽  
Order Logic ◽  
First Order Logic ◽  
Bounded Degree ◽  
First Order ◽  
Finite Models ◽  
Order Case ◽  
The Relationship

AbstractMany known tools for proving expressibility bounds for first-ordér logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressibility bounds. These results apply beyond the first-order case. We use them to derive expressibility bounds for first-order logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree.

Relational Chase Procedures Interpreted as Resolution with Paramodulation1

1991 ◽  
Vol 15 (2) ◽  
pp. 123-138
Author(s):  
Joachim Biskup ◽  
Bernhard Convent
Keyword(s):  
Alternative Interpretation ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Problem ◽  
First Order ◽  
Inference Problems ◽  
The One ◽  
The Relationship ◽  
Theoretical Foundations ◽  
Insight Into

In this paper the relationship between dependency theory and first-order logic is explored in order to show how relational chase procedures (i.e., algorithms to decide inference problems for dependencies) can be interpreted as clever implementations of well known refutation procedures of first-order logic with resolution and paramodulation. On the one hand this alternative interpretation provides a deeper insight into the theoretical foundations of chase procedures, whereas on the other hand it makes available an already well established theory with a great amount of known results and techniques to be used for further investigations of the inference problem for dependencies. Our presentation is a detailed and careful elaboration of an idea formerly outlined by Grant and Jacobs which up to now seems to be disregarded by the database community although it definitely deserves more attention.

On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem Löwenheim theorems and compactness of related quantifiers

1980 ◽  
Vol 45 (2) ◽  
pp. 265-283 ◽  
Author(s):  
Matatyahu Rubin ◽  
Saharon Shelah
Keyword(s):  
Boolean Algebra ◽  
Automorphism Groups ◽  
Order Logic ◽  
Boolean Algebras ◽  
First Order Logic ◽  
Elementary Equivalence ◽  
First Order ◽  
Finite Models ◽  
Countably Compact

AbstractTheorem 1. (◊ℵ1,) If B is an infinite Boolean algebra (BA), then there is B1, such that ∣ Aut (B1) ≤∣B1∣ = ℵ1 and 〈B1, Aut (B1)〉 ≡ 〈B, Aut(B)〉.Theorem 2. (◊ℵ1) There is a countably compact logic stronger than first-order logic even on finite models.This partially answers a question of H. Friedman. These theorems appear in §§1 and 2.Theorem 3. (a) (◊ℵ1) If B is an atomic ℵ-saturated infinite BA, Ψ Є Lω1ω and 〈B, Aut (B)〉 ⊨Ψ then there is B1, Such that ∣Aut(B1)∣ ≤ ∣B1∣ =ℵ1, and 〈B1, Aut(B1)〉⊨Ψ. In particular if B is 1-homogeneous so is B1. (b) (a) holds for B = P(ω) even if we assume only CH.

scholarly journals Successor-Invariant First-Order Logic on Classes of Bounded Degree (Extended Abstract)

2021 ◽  
Author(s):  
Julien Grange
Keyword(s):  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Bounded Degree ◽  
First Order ◽  
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 a successor relation on the vertices of the graph is allowed, as long as the validity of formulas is independent on the choice of a particular successor. We show that when the degree is bounded, successor-invariant first-order logic is no more expressive than first-order logic.

scholarly journals A dichotomy in classifying quantifiers for finite models

2005 ◽  
Vol 70 (4) ◽  
pp. 1297-1324
Author(s):  
Saharon Shelah ◽  
Mor Doron
Keyword(s):  
Partial Order ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Natural Partial Order ◽  
Existential Quantifier ◽  
First Order ◽  
Finite Models ◽  
One To One

AbstractWe consider a family of finite universes. The second order existential quantifier Qℜ means for each U Є quantifying over a set of n(ℜ)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called interpretability. We show that for every Qℜ, either Qℜ is interpretable by quantifying over subsets of U and one to one functions on U both of bounded order, or the logic L(Qℜ) (first order logic plus the quantifier Qℜ) is undecidable.

A NORMAL FORM FOR FIRST-ORDER LOGIC OVER DOUBLY-LINKED DATA STRUCTURES

2008 ◽  
Vol 19 (01) ◽  
pp. 205-217 ◽  
Author(s):  
STEVEN LINDELL
Keyword(s):  
Normal Form ◽  
Data Structures ◽  
Linked Data ◽  
Linear Time ◽  
Order Logic ◽  
Information Storage ◽  
First Order Logic ◽  
Total Size ◽  
Bounded Degree ◽  
First Order

We use singulary vocabularies to analyze first-order definability over doubly-linked data structures. Singulary vocabularies contain only monadic predicate and monadic function symbols. A class of mathematical structures in any vocabulary can be elementarily interpreted in a singulary vocabulary, while preserving notions of total size and degree. Doubly-linked data structures are a special case of bounded-degree finite structures in which there are reciprocal connections between elements, corresponding closely with physically feasible models of information storage. They can be associated with logical models involving unary relations and bijective functions in what we call an invertible singulary vocabulary. Over classes of these models, there is a normal form for first-order logic which eliminates all quantification of dependent variables. The paper provides a syntactically based proof using counting quantifiers. It also makes precise the notion of implicit calculability for arbitrary arity first-order formulas. Linear-time evaluation of first-order logic over doubly-linked data structures becomes a direct corollary. Included is a discussion of why these special data structures are appropriate for physically realizable models of information.

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.

Second-order languages and mathematical practice

1985 ◽  
Vol 50 (3) ◽  
pp. 714-742 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Mathematical Practice ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Infinite Cardinal ◽  
First Order ◽  
Order Case ◽  
Skolem Theorem ◽  
Infinite Model ◽  
Second Order Logic

There are well-known theorems in mathematical logic that indicate rather profound differences between the logic of first-order languages and the logic of second-order languages. In the first-order case, for example, there is Gödel's completeness theorem: every consistent set of sentences (vis-à-vis a standard axiomatization) has a model. As a corollary, first-order logic is compact: if a set of formulas is not satisfiable, then it has a finite subset which also is not satisfiable. The downward Löwenheim-Skolem theorem is that every set of satisfiable first-order sentences has a model whose cardinality is at most countable (or the cardinality of the set of sentences, whichever is greater), and the upward Löwenheim-Skolem theorem is that if a set of first-order sentences has, for each natural number n, a model whose cardinality is at least n, then it has, for each infinite cardinal κ (greater than or equal to the cardinality of the set of sentences), a model of cardinality κ. It follows, of course, that no set of first-order sentences that has an infinite model can be categorical. Second-order logic, on the other hand, is inherently incomplete in the sense that no recursive, sound axiomatization of it is complete. It is not compact, and there are many well-known categorical sets of second-order sentences (with infinite models). Thus, there are no straightforward analogues to the Löwenheim-Skolem theorems for second-order languages and logic.There has been some controversy in recent years as to whether “second-order logic” should be considered a part of logic, but this boundary issue does not concern me directly, at least not here.

Compactly expandable models and stability

1995 ◽  
Vol 60 (2) ◽  
pp. 673-683 ◽  
Author(s):  
Enrique Casanovas
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Unary Predicate ◽  
Similarity Type ◽  
First Order ◽  
Related Notion ◽  
Special Part ◽  
Consistent Extension ◽  
The Relationship ◽  
Close Parallelism

In analogy to ω-logic, one defines M-logic for an arbitrary structure M (see [5],[6]). In M-logic only those structures are considered in which a special part, determined by a fixed unary predicate, is isomorphic to M. Let L be the similarity type of M and T its complete theory. We say that M-logic is κ-compact if it satisfies the compactness theorem for sets of < κ sentences. In this paper we introduce the related notion of compactness for expandability: a model M is κ-compactly expandable if for every extension T′ ⊇ T of cardinality < κ, if every finite subset of T′ can be satisfied in an expansion of M, then T′ can also be satisfied in an expansion of M. Moreover, M is compactly expandable if it is ∥M∥+-compactly expandable. It turns out that M-logic is κ-compact iff M is κ-compactly expandable.Whereas for first-order logic consistency and finite satisfiability are the same, consistency with T and finite satisfiability in M are, in general, no longer the same thing. We call the model Mκ-expandable if every consistent extension T′ ⊇ T of cardinality < κ can be satisfied in an expansion of M. We say that M is expandable if it is ∥M∥+-expandable. Here we study the relationship between saturation, expandability and compactness for expandability. There is a close parallelism between our results about compactly expandable models and some theorems of S. Shelah about expandable models, which are in fact expressed in terms of categoricity of PC-classes (see [7, Th. VI.5.3, VI.5.4 and VI.5.5]). Our results could be obtained directly from these theorems of Shelah if expandability and compactness for expandability were the same notion.

First-order and counting theories of ω-automatic structures

2008 ◽  
Vol 73 (1) ◽  
pp. 129-150 ◽  
Author(s):  
Dietrich Kuske ◽  
Markus Lohrey
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Bounded Degree ◽  
First Order ◽  
Automatic Structures ◽  
Decidable Theories ◽  
Infinite Cardinality ◽  
Elementary Algorithm

AbstractThe logic extends first-order logic by a generalized form of counting quantifiers (“the number of elements satisfying … belongs to the set C”). This logic is investigated for structures with an injectively ω-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable [6]. It is shown that, as in the case of automatic structures [21], also modulo-counting quantifiers as well as infinite cardinality quantifiers (“there are many elements satisfying …”) lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of that contains only effective quantifiers is shown to be decidable and an elementary algorithm for this decision is presented. Both assumptions (ω-automaticity and bounded degree) are necessary for this result to hold.

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