Possibility theory and formal concept analysis: Characterizing independent sub-contexts

Formal concept analysis is a lattice-theoretic framework devised for the extraction of knowledge from Boolean data tables. A possibility-theoretic view of formal concept analysis has been recently introduced, and in particular set-valued counterparts of the four set-functions, respectively, evaluating potential or actual, possibility or necessity, that underlie bipolar possibility theory. It enables us to retrieve an enlarged perspective for formal concept analysis, already laid bare by some researchers like Dünsch and Gediga, or Georgescu and Popescu. The usual (Galois) connection that defines the notion of a formal concept as the pair of its extent and its intent is based on the actual (or guaranteed) possibility function, where each object in a concept has all properties of its intent, and each property is possessed by all objects of its extent. Noticing the formal similarity between the operator underlying classical formal concept analysis and the notion of division in relational algebra, we briefly indicate how to define approximate concepts by relaxing the universal quantifier in the definition of intent and extent as already done for relational divisions. The main thrust of the paper is the detailed study of another connection based on the counterpart to necessity measures. We show that it leads to partition a formal context into disjoint subsets of objects having distinct properties, and to split a data table into independent sub-tables.