Interval-valued algebras and fuzzy logics

In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter

Preassociative aggregation functions

The classical property of associativity is very often considered in aggregation function theory and fuzzy logic. In this paper we provide axiomatizations of various classes of preassociative functions, where preassociativity is a generalization of associativity recently introduced by the authors. These axiomatizations are based on existing characterizations of some noteworthy classes of associative operations, such as the class of Acz\'elian semigroups and the class of t-norms.Comment: arXiv admin note: text overlap with arXiv:1309.730

Householder triangularization of a quasimatrix

A standard algorithm for computing the QR factorization of a matrix A is Householder triangularization. Here this idea is generalized to the situation in which A is a quasimatrix, that is, a “matrix” whose “columns” are functions defined on an interval [a,b]. Applications are mentioned to quasimatrix leastsquares fitting, singular value decomposition, and determination of ranks, norms, and condition numbers, and numerical illustrations are presented using the chebfun system

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

Multivalued logic systems for technical applications

Velmi často je vyžadováno, aby automatizovaná zařízení byla jistým způsobem "inteligentní", tedy aby jejich řídicí systémy uměly emulovat rozhodovací proces. Tato diplomová práce poskytuje obecný formální popis vícehodnotových logických systémů schopných zmíněné emulace a jejich souvislost s teorií fuzzy množin. Jsou uvedeny způsoby vytváření matematických modelů založených na lingvistických datech. Dále se práce zabývá znalostními bázemi a jejich vlastnostmi. Součástí této práce je také počítačový program sloužící k tvorbě slovních modelů.Automated devices are very often required to exhibit some kind of an intelligent behaviour, which means that their control systems must be able to emulate the reasoning process. This diploma thesis provides a general formal description of multivalued logic systems capable of such an emulation and their connection with the fuzzy set theory. Ways of constructing mathematical models based on linguistic data are described. Also, knowledge bases and their properties are discussed. A computer program serving as a linguistic model development tool is a part of this thesis.