The correspondence between the concepts in description logics for contexts and formal concept analysis

Abstract

Formal concept analysis (FCA) and description logic (DL) are meant to be formalizations of concepts. A formal concept in the former consists of its intent and extent, where the intent is the set of all the attributes shared by each object in the extent of the concept, and the extent is the set of all the objects sharing each property in the intent of the concept. A concept in the latter formalization is simply a concept name, the interpretation of which is a subset of a universe. To consider the correspondence between concepts in both formalizations, a multi-valued formal context must be represented both as a knowledge base and as a model of the DL for contexts, where concepts are decomposed into tuple concepts C, interpreted as a set of tuples and value concepts V, interpreted as a set of attribute-value pairs. We show that there is a difference between the interpretation of concepts ∀R.V/∀R−.C and the Galois connection between the extent/intent of formal concepts in FCA. According to the Galois connection, there should be concepts of the form +∀R.V and +∀R−.C interpreted in FCA, and hence the logical language L for DL is extended to be L+ together with +∀ as a constructor so that +∀R.V and +∀R−.C are well-defined concepts. Conversely, according to the interpretation in DL there should be pseudo concepts in FCA so that the interpretation of concepts ∀R.V/∀R−.C is the extent/intent of pseudo concepts. The correspondence between formal concepts and concepts in L+, and between pseudo concepts and concepts in L are presented in this paper.