4 research outputs found
Fuzzy algebras of concepts
Preconcepts are basic units of knowledge that form the basis of formal concepts in formal concept analysis (FCA). This paper investigates the relations among different kinds of preconcepts, such as protoconcepts, meet and join-semiconcepts and formal concepts. The first contribution of this paper, is to present a fuzzy powerset lattice gradation, that coincides with the preconcept lattice at its 1-cut. The second and more significant contribution, is to introduce a preconcept algebra gradation that yields different algebras for protoconcepts, semiconcepts, and concepts at different cuts. This result reveals new insights into the structure and properties of the different categories of preconcepts.Partial funding for open access charge: Universidad de Málag
A non-distributive logic for semiconcepts of a context and its modal extension with semantics based on Kripke contexts
A non-distributive two-sorted hypersequent calculus \textbf{PDBL} and its
modal extension \textbf{MPDBL} are proposed for the classes of pure double
Boolean algebras and pure double Boolean algebras with operators respectively.
A relational semantics for \textbf{PDBL} is next proposed, where any formula is
interpreted as a semiconcept of a context. For \textbf{MPDBL}, the relational
semantics is based on Kripke contexts, and a formula is interpreted as a
semiconcept of the underlying context. The systems are shown to be sound and
complete with respect to the relational semantics. Adding appropriate sequents
to \textbf{MPDBL} results in logics with semantics based on reflexive,
symmetric or transitive Kripke contexts. One of these systems is a logic for
topological pure double Boolean algebras. It is demonstrated that, using
\textbf{PDBL}, the basic notions and relations of conceptual knowledge can be
expressed and inferences involving negations can be obtained. Further, drawing
a connection with rough set theory, lower and upper approximations of
semiconcepts of a context are defined. It is then shown that, using the
formulae and sequents involving modal operators in \textbf{MPDBL}, these
approximation operators and their properties can be captured