37 research outputs found
Fuzzy inequational logic
We present a logic for reasoning about graded inequalities which generalizes
the ordinary inequational logic used in universal algebra. The logic deals with
atomic predicate formulas of the form of inequalities between terms and
formalizes their semantic entailment and provability in graded setting which
allows to draw partially true conclusions from partially true assumptions. We
follow the Pavelka approach and define general degrees of semantic entailment
and provability using complete residuated lattices as structures of truth
degrees. We prove the logic is Pavelka-style complete. Furthermore, we present
a logic for reasoning about graded if-then rules which is obtained as
particular case of the general result
Parameterizing the semantics of fuzzy attribute implications by systems of isotone Galois connections
We study the semantics of fuzzy if-then rules called fuzzy attribute
implications parameterized by systems of isotone Galois connections. The rules
express dependencies between fuzzy attributes in object-attribute incidence
data. The proposed parameterizations are general and include as special cases
the parameterizations by linguistic hedges used in earlier approaches. We
formalize the general parameterizations, propose bivalent and graded notions of
semantic entailment of fuzzy attribute implications, show their
characterization in terms of least models and complete axiomatization, and
provide characterization of bases of fuzzy attribute implications derived from
data
Why most papers on filters are really trivial (including this one)
The aim of this note is to show that many papers on various kinds of filters (and related concepts) in (subreducts of) residuated structures are in fact easy consequences of more general results that have been known for a long time
Triadic fuzzy Galois connections as ordinary connections
Abstract-The paper presents results on representation of the basic structures related to ternary fuzzy relations by the structures related to ordinary ternary relations, such as Galois connections, closure operators, and trilattices (structures of maximal Cartesian subrelations). These structures appear as the fundamental structures in relational data analysis such as formal concept analysis or association rules. We prove several representation theorems that allow us to automatically transfer some of the known results from the ordinary case to fuzzy case. The transfer is demonstrated by examples. I. INTRODUCTION Relations play a fundamental role in mathematics, computer science, and their applications. Many results about ordinary relations have been generalized to the setting of fuzzy relations in the past. There has always been a fundamental question of how the various fuzzifications are related to the ordinary notions and results. Needless to say, this question is important both from a practical and theoretical point of view and is treated to some extent in textbooks, see e.g. In this paper we deal with basic structures associated to ternary relations that appear as fundamental ones in the methods of relational data analysis, namely the closure-like structures such as Galois connections, closure operators, structures of their fixpoints and the like. Such structures appear e.g. in formal concept analysis The most common way of looking at the relationship between ordinary notions and their fuzzy counterparts is in terms of a-cuts of fuzzy relations (see e.g. [15]) but there are additional possible views at the question as well. One of them, utilized in this paper, is provided in [3, Section 3.1.2]. Our paper is organized as follows. We first provide preliminaries in Section II. In Section III, we introduce the Galoi