Implication functions in interval-valued fuzzy set theory

Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory

Constructing interval-valued fuzzy material implication functions derived from general interval-valued grouping functions

Grouping functions and their dual counterpart, overlap functions, have drawn the attention of many authors, mainly because they constitute a richer class of operators compared to other types of aggregation functions. Grouping functions are a useful theoretical tool to be applied in various problems, like decision making based on fuzzy preference relations. In pairwise comparisons, for instance, those functions allow one to convey the measure of the amount of evidence in favor of either of two given alternatives. Recently, some generalizations of grouping functions were proposed, such as (i) the n-dimensional grouping functions and the more flexible general grouping functions, which allowed their application in n-dimensional problems, and (ii) n-dimensional and general interval-valued grouping functions, in order to handle uncertainty on the definition of the membership functions in real-life problems. Taking into account the importance of interval-valued fuzzy implication functions in several application problems under uncertainty, such as fuzzy inference mechanisms, this paper aims at introducing a new class of interval-valued fuzzy material implication functions. We study their properties, characterizations, construction methods and provide examples.upported by CNPq (301618/2019-4, 311429/2020-3), FAPERGS (19/2551-0001660-3), UFERSA, the Spanish Ministry of Science and Technology (TIN2016-77356-P, PID2019-108392GB I00 (MCIN/AEI/10.13039/501100011033)) and Navarra de Servicios y Tecnologías, S.A. (NASERTIC)

Fuzzy Sets, Fuzzy Logic and Their Applications 2020

The present book contains the 24 total articles accepted and published in the Special Issue “Fuzzy Sets, Fuzzy Logic and Their Applications, 2020” of the MDPI Mathematics journal, which covers a wide range of topics connected to the theory and applications of fuzzy sets and systems of fuzzy logic and their extensions/generalizations. These topics include, among others, elements from fuzzy graphs; fuzzy numbers; fuzzy equations; fuzzy linear spaces; intuitionistic fuzzy sets; soft sets; type-2 fuzzy sets, bipolar fuzzy sets, plithogenic sets, fuzzy decision making, fuzzy governance, fuzzy models in mathematics of finance, a philosophical treatise on the connection of the scientific reasoning with fuzzy logic, etc. It is hoped that the book will be interesting and useful for those working in the area of fuzzy sets, fuzzy systems and fuzzy logic, as well as for those with the proper mathematical background and willing to become familiar with recent advances in fuzzy mathematics, which has become prevalent in almost all sectors of the human life and activity

Preference Modelling

This paper provides the reader with a presentation of preference modelling fundamental notions as well as some recent results in this field. Preference modelling is an inevitable step in a variety of fields: economy, sociology, psychology, mathematical programming, even medicine, archaeology, and obviously decision analysis. Our notation and some basic definitions, such as those of binary relation, properties and ordered sets, are presented at the beginning of the paper. We start by discussing different reasons for constructing a model or preference. We then go through a number of issues that influence the construction of preference models. Different formalisations besides classical logic such as fuzzy sets and non-classical logics become necessary. We then present different types of preference structures reflecting the behavior of a decision-maker: classical, extended and valued ones. It is relevant to have a numerical representation of preferences: functional representations, value functions. The concepts of thresholds and minimal representation are also introduced in this section. In section 7, we briefly explore the concept of deontic logic (logic of preference) and other formalisms associated with "compact representation of preferences" introduced for special purpoes. We end the paper with some concluding remarks

Semantic Similarity of Spatial Scenes

The formalization of similarity in spatial information systems can unleash their functionality and contribute technology not only useful, but also desirable by broad groups of users. As a paradigm for information retrieval, similarity supersedes tedious querying techniques and unveils novel ways for user-system interaction by naturally supporting modalities such as speech and sketching. As a tool within the scope of a broader objective, it can facilitate such diverse tasks as data integration, landmark determination, and prediction making. This potential motivated the development of several similarity models within the geospatial and computer science communities. Despite the merit of these studies, their cognitive plausibility can be limited due to neglect of well-established psychological principles about properties and behaviors of similarity. Moreover, such approaches are typically guided by experience, intuition, and observation, thereby often relying on more narrow perspectives or restrictive assumptions that produce inflexible and incompatible measures. This thesis consolidates such fragmentary efforts and integrates them along with novel formalisms into a scalable, comprehensive, and cognitively-sensitive framework for similarity queries in spatial information systems. Three conceptually different similarity queries at the levels of attributes, objects, and scenes are distinguished. An analysis of the relationship between similarity and change provides a unifying basis for the approach and a theoretical foundation for measures satisfying important similarity properties such as asymmetry and context dependence. The classification of attributes into categories with common structural and cognitive characteristics drives the implementation of a small core of generic functions, able to perform any type of attribute value assessment. Appropriate techniques combine such atomic assessments to compute similarities at the object level and to handle more complex inquiries with multiple constraints. These techniques, along with a solid graph-theoretical methodology adapted to the particularities of the geospatial domain, provide the foundation for reasoning about scene similarity queries. Provisions are made so that all methods comply with major psychological findings about people’s perceptions of similarity. An experimental evaluation supplies the main result of this thesis, which separates psychological findings with a major impact on the results from those that can be safely incorporated into the framework through computationally simpler alternatives