Radim Bělohlávek

Crisply generated fuzzy concepts

In formal concept analysis of data with fuzzy attributes, both the extent and the intent of a formal (fuzzy) concept may be fuzzy sets. In this paper we focus on so-called crisply generated formal concepts. A concept

Fuzzy Closure Operators with Truth Stressers

\langle{A,B}\rangle \in \mathcal{B}(X, Y, I)

Data tables with similarity relations: functional dependencies, complete rules and non-redundant bases

is crisply generated if A = D↓ (and so B = D↓↑) for some crisp (i.e., ordinary) set D ⊆ Y of attributes (generator). Considering only crisply generated concepts has two practical consequences. First, the number of crisply generated formal concepts is considerably less than the number of all formal fuzzy concepts. Second, since crisply generated concepts may be identified with a (ordinary, not fuzzy) set of attributes (the largest generator), they might be considered “the important ones” among all formal fuzzy concepts. We present basic properties of the set of all crisply generated concepts, an algorithm for listing all crisply generated concepts, a version of the main theorem of concept lattices for crisply generated concepts, and show that crisply generated concepts are just the fixed points of pairs of mappings resembling Galois connections. Furthermore, we show connections to other papers on formal concept analysis of data with fuzzy attributes. Also, we present examples demonstrating the reduction of the number of formal concepts and the speed-up of our algorithm (compared to listing of all formal concepts and testing whether a concept is crisply generated).

Algebras with fuzzy equalities

We study closure operators and closure structures in a fuzzy setting. Our main interest is the monotony condition of closure operators. In a fuzzy setting, the monotony condition may take several particular forms, all of them equivalent in the bivalent case. We study closure operators, called fuzzy closure operators with truth stresser, satisfying the monotony condition which can be linguistically described as “if it is (very) true that A is included in B then the closure of A is included in the closure of B.” We present examples of closure operators with truth stresser, investigate their basic properties and related structures.

Attribute implications in a fuzzy setting

We study rules

Fuzzy Horn logic II

A \Longrightarrow B

Formal concept analysis with constraints by closure operators

describing attribute dependencies in tables over domains with similarity relations.

Thresholds and shifted attributes in formal concept analysis of data with fuzzy attributes

A \Longrightarrow B

Birkhoff variety theorem and fuzzy logic

reads “for any two table rows: similar values of attributes from A imply similar values of attributes from B”. The rules generalize ordinary functional dependencies in that they allow for processing of similarity of attribute values. Similarity is modeled by reflexive and symmetric fuzzy relations. We show a system of Armstrong-like derivation rules and prove its completeness (two versions). Furthermore, we describe a non-redundant basis of all rules which are true in a data table and present an algorithm to compute bases.

Similarity relations and BK-relational products

An algebra with fuzzy equality is a set with operations on it that is equipped with similarity ~, i.e. a fuzzy equivalence relation, such that each operation f is compatible with ~. Described verbally, compatibility says that each f yields similar results if applied to pairwise similar arguments. On the one hand, algebras with fuzzy equalities are structures for the equational fragment of fuzzy logic and have been studied from this point of view before. On the other hand, they are the formal counterpart to the intuitive idea of having functions that are not allowed to map similar objects to dissimilar ones. The present paper aims at developing fundamental points of algebras with fuzzy equalities: we introduce the notion of an algebra with fuzzy equality, present natural examples, compare the notion with other approaches, and introduce and develop basic structural notions (subalgebras, morphisms, products, direct unions). In a follow-up paper [Vychodil, Direct limits and reduced products of algebras with fuzzy equalities, submitted for publication], we deal with advanced topics in algebras with fuzzy equalities (direct limits, reduced products).