Model-theoretic Point Of View Research Articles

A compositional semantics for logic programs

This paper considers open logic programs originally as a tool to build an OR-compositional semantics of logic programs. We extend the original semantic definitions in the framework of the general approach to the semantics of logic programs described by Gabbrielli and Levi (1991). We first define an operational semantics O Ω(P) which models computed answer substitutions and which is compositional w.r.t. the union of programs. Next, we consider the semantic domain of Ω- denotations, which are sets of clauses with a suitable equivalence relation. The fixpoint semantics F Ω(P) given by Bossi and Menegus (1991) is proved equivalent to the operational semantics. From the model-theoretic viewpoint, an Ω-denotation is mapped onto a set of Herbrand interpretations, thus leading to a definition of an Ω-model based on the classical notion of truth. Moreover, we consider a particular kind of Ω-models (compositional models), and we show that O Ω(P) is a (nonminimal) compositional Ω-model. A suitable abstraction of O Ω(P) is compositional and fully abstract w.r.t. the equivalence induced by successful derivations. Finally, some applications of our semantics are discussed. In particular, O Ω(P) can be viewed as the foundations of modular program analysis.

Propositional knowledge base revision and minimal change

The semantics of revising knowledge bases represented by sets of propositional sentences is analyzed from a model-theoretic point of view. A characterization of all revision schemes that satisfy the Gärdenfors rationality postulates is given in terms of minimal change with respect to an ordering among interpretations. Revision methods proposed by various authors are surveyed and analyzed in this framework. The correspondences between Gärdenfors-like rationality postulates and minimal changes with respect to other orderings are also investigated.

Homogeneous models and almost decidability

AbstractCountable homogeneous models are ‘simple’ objects from a model theoretic point of view. From a recursion theoretic point of view they can be complex. For instance the elementary theory of such a model might be undecidable, or the set of complete types might be recursively complex. Unfortunately even if neither of these conditions holds, such a model still can be undecidable. This paper investigates countable homogeneous models with respect to a weaker notion of decidability called almost decidable. It is shown that for theories that have only countably many type spectra, any countable homogeneous model of such a theory that has a Σ2 type spectrum is almost decidable.

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

On the development of the model-theoretic viewpoint in logical theory

On the development of the model-theoretic viewpoint in logical theory

Logic Programming, Substitutions and Finite Computability

<p>Apt and van Emden have studied the semantics of logic programming by means of fixed point methods. From a model theoretic point of view, their formalisation is very nice. Least and greatest fixed points correspond to least and greatest Herbrand-models respectively.</p><p>Viewed operationally, there is an ugly asymmetry. The least fixed point expresses finite computability, but the greatest fixed point denotes negation by <em>trans</em>-finite failure, i.e. the underlying operator is not omega-continuous for decreasing chains in general.</p><p>We use the notion of finite computability inherent in Scott domains to build a domainlike construction (the cd-domain) that offers omega-continuity for increasing and decreasing chains equally. On this basis negation by finite failure is expressed in terms of a fixed point.</p><p>The fixed point semantics of Apt and van Emden is very abstract concerning the concept of substitution, although it is fundamental for any implementation. Hence it becomes quite tedious to prove the correctness of a concrete resolution algorithm. The fixed point semantics of this paper offers an intermediate step in this respect. Any commitments to specific resolution strategies are avoided, and the semantics may be the basis of sequential and parallel implementations equally. Simultaneously the set of substitution dataobjects is structured by a Scott information theoretic partial order, namely the cd-domain.</p>

Model Theoretic Approaches to Definability

When approaching the problems of mathematical logic from a model-theoretic point of view, one has at his disposal a variety of ‘methods of attack.’ We have in mind, for example, methods based on the compactness theorem and utilized by Robinson, Svenonius and others (see [18] and [20]) or methods based on the ultraproduct construction (perhaps more precisely, the Łos theorem [16]) and its generalizations and employed by Kochen, Keisler and others (see [8], [11], [12], [13], [14]). Some of the more important of these methods are considered here with a view toward revealing the underlying uniformity. The analysis is made in the context of the problems of definability. Particularly in Section 3, an attempt is made to indicate basis similarities between the various methods considered while addressing each approach to a proof of the Beth theorem [1]. These different proofs of the Beth theorem invoke Craig’s lemma (see [6], or [7]). Although much of Section 3 is expository in nature, we feel that the material should be available in the unified presentation.