Preface Volume 66, Issue 5

AbstractThis part of the volume contains the papers accepted for presentation at the workshop on Unification in Non-Classical Logics (UNCL), co-located with ICALP 2002, which took place on July 12, 2002 in M\'alaga, Spain.The workshop was concerned with one of the most promising areas of research on non-classical logics and its applications. Unification in non-classical logics, with various approaches to handling generalised terms, has drawn more and more attention in recent years. So far, most popular lines of research include fuzzy unification of (conventional) databases and the use of fuzzy concepts in information retrieval.This workshop was conceived as a forum for the exchange of ideas relevant for the concept of unification in non-classical logics, including, but not limited to, the topics of: •Unification in multiple-valued and fuzzy logic programming.•Unification based on similarities and fuzzy equivalence relations.•Categorical unification.•Practical use of non-classical unification, e.g. in expert systems and information retrieval.The program committee selected six papers after a reviewing process in which each submitted paper received at least two reviews. Considerable effort was devoted for the evaluation of the submissions and to providing the authors with helpful feedback. The criteria for selection were originality, quality, and relevance to the topic of the workshop.Alsinet et al reviewed and compared two models which extend first order possibilistic logic in order to enable fuzzy unification. The extension considers mainly fuzzy constants, and in form of restrictions on existential quantifiers.Banerjee and Bujosa presented a non-classical interpretion of classical unification in terms of geometrical constructions over a suitable R-module M. The main result is that unification of two terms can be seen as the intersection of their corresponding affine varieties on M. This paves the way of using methods from computer algebra in the field of unification.In Eklund et al, substitutions and unifiers appear as constructs in Kleisli categories related to particular composed powerset term monads. It is shown that an often used similarity-based approach to fuzzy unification is compatible with the categorical approach, and can be adequately extended.Kutsia presented a unification procedure for a theory with individual and sequence variables, free fixed and flexible arity function symbols and patterns. These theories have been used in different contexts such as databases, rewriting, programming languages, or theorem proving.Medina et al introduced a formal model for similarity-based fuzzy unification in multi-adjoint logic programs. On this computational model, a similarity-based unification approach which provides a semantic framework for logic programming with different notions of similarity was constructed.Virtanen introduced unification in similarity-based logic programming. One of the crucial points is the definition of similarity degrees between sets, giving rise to [lambda]-interpretations. The selection of so called most significant terms again is one of the cornerstones of the paper.We would like to thank all those who submitted papers for consideration, the authors of accepted papers for their interesting discussions during the workshop, the additional referees for their careful work, and Inma Fortes from the local organising committee for her assistance

Relation-based Galois connections: towards the residual of a relation

Inma P. Cabrera, Pablo Cordero, Manuel Ojeda-Aciego, Relation-based Galois connections: towards the residual of a relation, CMMSE 2017: Proceedings of the 17th International Conference on Mathematical Methods in Science and Engineering ( ISBN: 978-84-617-8694-7) , pp. 469--475We explore a suitable generalization of the notion of Galois connection in which their components are binary relations. Many different approaches are possible depending both on the (pre-)order relation between subsets in the underlying powerdomain and the chosen type of relational composition.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

The f -index of inclusion as optimal adjoint pair for fuzzy modus ponens

We continue studying the properties of the f -index of inclusion and show that, given a fixed pair of fuzzy sets, their f -index of inclusion can be linked to a fuzzy conjunction which is part of an adjoint pair. We also show that, when this pair is used as the underlying structure to provide a fuzzy interpretation of the modus ponens inference rule, it provides the maximum possible truth-value in the conclusion among all those values obtained by fuzzy modus ponens using any other possible adjoint pair.Partially supported by the Spanish Ministry of Science, Innovation and Universities (MCIU), State Agency of Research (AEI), Junta de Andalucía (JA), Universidad de Málaga (UMA) and European Regional Development Fund (FEDER) through the projects PGC2018-095869-B-I00 (MCIU/AEI/FEDER) and UMA2018-FEDERJA-001 (JA/UMA/FEDER). Funding for open access charge: Universidad de Málaga / CBU

On the notion of fuzzy adjunctions between fuzzy orders

Las adjunciones (también denominadas conexiones de Galois isótonas) entre dos estructuras matemáticas proporcionan una manera de conectar ambas teorías que permite compartir las ventajas de ambas. Hay varios resultados en la literatura previa acerca de las condiciones necesarias y suficientes para la existencia de conexiones de Galois entre dos conjuntos parcialmente ordenados. En otros trabajos anteriores, los autores han estudiado la existencia y construcción del adjunto por la derecha de una aplicación dada entre conjuntos preordenados o dotados de un orden difuso, entendido éste como una relación binaria difusa satisfaciendo reflexividad, transitividad y antisimetría. Se entiende el término adjunción difusa como una pareja de aplicaciones entre dos conjuntos crisp que están dotados de órdenes difusos para las culaes se verifica la condición ρ(a,g(b)) = ρ(f(a),b). Esta definición no está suficientemente justificada en un contexto difuso puesto que las aplicaciones que se consideran se dan entre conjuntos clásicos. En este trabajo se explica la forma en la que la citada definición está relacionada con las funciones difusas, mostrando así que la definición sí es adecuada en ambiente difuso también. La extensión natural de la noción difusa de adjunción contempla dos posibilidades, según si uno considera igualdades difusas o equivalencias difusas asociadas al orden difuso o no.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Ordering objects via attribute preferences

We apply recent results on the construction of suitable orderings for the existence of right adjoint to the analysis of the following problem: given a preference ordering on the set of attributes of a given context, we seek an induced preference among the objects which is compatible with the information provided by the context.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Using the Chu Construction for generalizing formal concept analysis

L. Antoni, I. P. Cabrera, S. Krajči, O. Krídlo, and M. Ojeda-Aciego. Using the Chu construction for generalizing formal concept analysis. In CLA 2015, pp. 147–158, Blaise Pascal University, LIMOS laboratory, Clermont-Ferrand, 2015El objetivo de este artículo es mostrar la conexión entre generalizaciones de Análisis de Conceptos Formales y la construcción de Chu sobre la categoría ChuCors de contextos formales y correspondencias de Chu. Todas las propiedades categóricas necesarias para la comprensión de los resultados de este trabajo como producto categórico, producto tensorial o propiedades de su bifuntor se presentan y demuestran. Finalmente, la generalización de Análisis de Conceptos Formales de segundo orden se representa por una categoría construida en términos de la Construcción de Chu

A neural implementation of multi-adjoint logic programs via sf-homogenization

A generalization of the homogenization process needed for the neural im- plementation of multi-adjoint logic programming (a unifying theory to deal with uncertainty, imprecise data or incomplete information) is presented here. The idea is to allow to represent a more general family of adjoint pairs, but maintaining the advantage of the existing implementation recently introduced in [6]. The soundness of the transformation is proved and its complexity is analysed. In addition, the corresponding generalization of the neural-like implementation of the fixed point semantics of multi-adjoint is presented

Relational Galois connections between transitive fuzzy digraphs

Fuzzy-directed graphs are often chosen as the data structure to model and implement solutions to several problems in the applied sciences. Galois connections have also shown to be useful both in theoretical and in practical problems. In this paper, the notion of relational Galois connection is extended to be applied between transitive fuzzy directed graphs. In this framework, the components of the connection are crisp relations satisfying certain reasonable properties given in terms of the so-called full powering