scholarly journals Ordered completion for first-order logic programs on finite structures

2012 ◽  
Vol 177-179 ◽  
pp. 1-24 ◽  
Author(s):  
Vernon Asuncion ◽  
Fangzhen Lin ◽  
Yan Zhang ◽  
Yi Zhou
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Logic Programs ◽  
First Order ◽  
Finite Structures

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 The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context.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

scholarly journals Learning Relational Representations with Auto-encoding Logic Programs

2019 ◽  
Author(s):  
Sebastijan Dumancic ◽  
Tias Guns ◽  
Wannes Meert ◽  
Hendrik Blockeel
Keyword(s):  
Deep Learning ◽  
Relational Learning ◽  
Order Logic ◽  
First Order Logic ◽  
Relational Data ◽  
Logic Programs ◽  
Learning Methods ◽  
First Order ◽  
Relational Structures ◽  
Representation Language

Deep learning methods capable of handling relational data have proliferated over the past years. In contrast to traditional relational learning methods that leverage first-order logic for representing such data, these methods aim at re-representing symbolic relational data in Euclidean space. They offer better scalability, but can only approximate rich relational structures and are less flexible in terms of reasoning. This paper introduces a novel framework for relational representation learning that combines the best of both worlds. This framework, inspired by the auto-encoding principle, uses first-order logic as a data representation language, and the mapping between the the original and latent representation is done by means of logic programs instead of neural networks. We show how learning can be cast as a constraint optimisation problem for which existing solvers can be used. The use of logic as a representation language makes the proposed framework more accurate (as the representation is exact, rather than approximate), more flexible, and more interpretable than deep learning methods. We experimentally show that these latent representations are indeed beneficial in relational learning tasks.

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 Verification of Imperative Programs through Transformation of Constraint Logic Programs

10.29007/163x ◽  
2018 ◽  
Author(s):  
Emanuele De Angelis ◽  
Fabio Fioravanti ◽  
Alberto Pettorossi ◽  
Maurizio Proietti
Keyword(s):  
Program Transformation ◽  
Program Verification ◽  
Order Logic ◽  
Initial Configuration ◽  
First Order Logic ◽  
Logic Programs ◽  
Partial Correctness ◽  
First Order ◽  
Transformation Rules ◽  
Constraint Logic Programs

We present a method for verifying partial correctness properties of imperative programs by using techniques based on the transformation of constraint logic programs (CLP). We consider: (i) imperative programs that manipulate integers and arrays, and (ii) first order logic properties that define <i>configurations</i> of program executions. We use CLP as a metalanguage for representing imperative programs, their executions, and their properties. First, we encode the correctness of an imperative program, say Prog, as the negation of a predicate 'incorrect' defined by a CLP program T. By construction, 'incorrect' holds in the least model of T if and only if the execution of Prog from an initial configuration eventually halts in an error configuration. Then, we apply to program T a sequence of transformations that preserve its least model semantics. These transformations are based on well-known transformation rules, such as unfolding and folding, guided by suitable transformation strategies, such as specialization and generalization. The objective of the transformations is to derive a new CLP program TransfT where the predicate 'incorrect' is defined either by (i) the fact `incorrect.' (and in this case Prog is incorrect), or by (ii) the empty set of clauses (and in this case Prog is correct). In the case where we derive a CLP program such that neither (i) nor (ii) holds, we iterate the transformation. Since the problem is undecidable, this process may not terminate. We show through examples that our method can be applied in a rather systematic way, and is amenable to automation by transferring to the field of program verification many techniques developed in the field of program transformation.

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 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.

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