Equality-free saturated models

Saturated models are a powerful tool in model theory. The properties of universality and homogeneity of the saturated models of a theory are useful for proving facts about this theory. They are used in the proof of interpolation and preservation theorems and also as work-spaces. Sometimes we work with models which are saturated only for some sets of formulas, for example, recursively saturated models, in the study of models of arithmetic or atomic compact, in model theory of modules. In this article we introduce the notion of equality-free saturated model, that is, roughly speaking, a model which is saturated for the set of equality-free formulas. Our aim is to understand better the role that identity plays in classical model theory, in particular with regard to this process of saturation. Given an infinite cardinal κ, we say that a model is equality-free κsaturated if it satisfies all the 1-types over sets of parameters of power less than κ, with all the formulas in the type that are equality-free. We compare this notion with the usual notion of κ-saturated model. We prove the existence of infinite models A, which are L −-|A | +-saturated. Fro

Contribucions a la teoria de models de la lògica sense identitat

[spa] La tesis doctoral es un estudio de la teoria de modelos de la lógica sin identidad. Se estudia el fragmento de la lógica de primer orden compuesto por las fórmulas que no tienen el símbolo de identidad. Los conceptos fundamentales estudiados son el de "Congruencia de Leibniz" y el de "Relación de parentesco (Relative Relation)". El interés actual de estas nociones procede de los trabajos de W. Blok y de D. Pigozzi. Hemos estudiado esta lógica desde el punto de vista de la teoria de modelos clásica, desarrollando técnicas usuales en teoria de modelos: Metodo de los diagramas, sistemas de Back-and-Forth, etc. con el fin de obtener caracterizaciones algebraicas de la equivalencia elemental en esta lógica y teoremas de preservación. Una de las contribuciones más importantes de este trabajo es la caracterización de los enunciados de primer orden que son lógicamente equivalentes a un enunciado sin identidad. Hemos introducido las nociones de modelo saturado, universal y homogéneo sin identidad. Hemos estudiado sus propiedades y las hemos comparado con las de las nociones análogas en lógica de primer orden con identidad. Finalmente hemos estudiado el fragmento universal de Horn sin identidad de los lenguajes infinitarios, con Y cardinales infinitos regulares. Hemos obtenido resultados de caracterización y de preservación. Usando estos resultados hemos demostrado teoremas de interpolación y definibilidad para este fragmento

On elementary equivalence in fuzzy predicate logics

Our work is a contribution to the model theory of fuzzy predicate logics. In this paper we characterize elementary equivalence between models of fuzzy predicate logic using elementary mappings. Refining the method of diagrams we give a solution to an open problem of Hájek and Cintula (J Symb Log 71(3):863-880, 2006, Conjectures 1 and 2). We investigate also the properties of elementary extensions in witnessed and quasi-witnessed theories, generalizing some results of Section 7 of Hájek and Cintula (J Symb Log 71(3):863-880, 2006) and of Section 4 of Cerami and Esteva (Arch Math Log 50(5/6):625-641, 2011) to non-exhaustive model

On the existence of free models in fuzzy universal Horn classes

This paper is a contribution to the study of the universal Horn fragment of predicate fuzzy logics, focusing on some relevant notions in logic programming. We introduce the notion of term structure associated to a set of formulas in the fuzzy context and we show the existence of free models in fuzzy universal Horn classes. We prove that every equality-free consistent universal Horn fuzzy theory has a Herbrand model

Extending possibilistic logic over Gödel logic

In this paper we present several fuzzy logics trying to capture different notions of necessity (in the sense of possibility theory) for Gödel logic formulas. Based on different characterizations of necessity measures on fuzzy sets, a group of logics with Kripke style semantics is built over a restricted language, namely, a two-level language composed of non-modal and modal formulas, the latter, moreover, not allowing for nested applications of the modal operator N. Completeness and some computational complexity results are shown

On similarity in fuzzy description logics

This paper is a contribution to the study of similarity relations between objects represented as attribute-value pairs in Fuzzy Description Logics . For this purpose we use concrete domains in the fuzzy description logic IALCEF(D)IALCEF(D) associated either with a left-continuous or with a finite t-norm. We propose to expand this fuzzy description logic by adding a Similarity Box (SBox) including axioms expressing properties of fuzzy equalities. We also define a global similarity between objects from similarities between the values of each object attribute (local similarities) and we prove that the global similarity defined using a t-norm inherits the usual properties of the local similarities (reflexivity, symmetry or transitivity). We also prove a result relative to global similarities expressing that, in the context of the logic MTL∀, similar objects have similar properties, being these properties expressed by predicate formulas evaluated in these object

Back-and-forth systems for fuzzy first-order models

This paper continues the study of model theory for fuzzy logics by addressing the fundamental issue of classifying models according to their first-order theory. Three different definitions of elementary equivalence for fuzzy first-order models are introduced and separated by suitable counterexamples. We propose several back-and-forth conditions, based both on classical two-sorted structures and on non-classical structures, that are useful to obtain elementary equivalence in particular cases as we illustrate with several example

Perspectives innovadores en el disseny d'algoritmes intel·ligents per a la classificació d'obres d'art : eines per a una millor integració humà-màquina

La present investigació explora una perspectiva innovadora en el disseny d'algoritmes capaços de classificar pintures segons el seu estil artístic. Evitant alguns inconvenients d'integració humà-màquina que presenta el disseny amb aprenentatge automàtic, ℓ-SHE és un algoritme basat en sistemes de representació lògica i conceptes qualitatius que, a més de reconèixer i classificar amb èxit l'estil artístic d'una pintura, treballa amb informació semàntica, genera explicacions del perquè dels seus resultats, i pot ser entrenat fàcilment i amb bancs de dades relativament petits.La presente investigación explora una perspectiva innovadora en el diseño de algoritmos capaces de clasificar pinturas según su estilo artístico. Evitando algunos inconvenientes de integración humano-máquina que presenta el diseño con aprendizaje automático, ℓ-SHE es un algoritmo basado en sistemas de representación lógica y conceptos cualitativos que, además de reconocer y clasificar con éxito el estilo artístico de una pintura, trabaja con información semántica, genera explicaciones del por qué de sus resultados, y puede ser entrenado fácilmente y con bancos de datos relativamente pequeños.The present research explores an innovative perspective in the design of algorithms for classifying pictorial artworks according to their art style. Avoiding some of the human-machine integration issues associated with machine learning algorithm design, ℓ-SHE is an algorithm based on logical representation and qualitative concepts which in addition to successfully recognizing and classifying the art style of paintings, it works with semantic information, generates explanations of which reasonings have been followed for the classification results, and can easily be trained using relatively small databases