Available online at www.sciencedirect.com Fuzzy Sets and Systems 196 (2012) 4 – 16 www.elsevier.com/locate/fss Possibility theory and formal concept analysis: Characterizing independent sub-contexts Didier Dubois ∗ , Henri Prade IRIT, UniversitéPaul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex 09, France Available online 20 February 2011 Abstract 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. © 2011 Elsevier B.V. All rights reserved. Keywords: Formal concept analysis; Possibility theory; Galois connection; Relational division 1. Introduction Formal concept analysis (FCA for short) has been introduced under this name by Wille [1] and developed by Ganter and Wille [2]. However, the original lattice-theoretic construction can be found in chapter V of a much earlier book (in French) by Barbut and Monjardet [3] published in 1970. FCA exploits the duality between objects and properties in a Boolean data table, and it has led to an original and practical view of the notion of a formal concept with application to data mining [4]. A concept is then a pair made of a set of objects and a set of properties possessed by all of them, that are in mutual correspondence, through an antitone Galois connection. The latter expresses that these two sets are the extent and the intent of the concept, respectively. In this framework, properties are binary, and complete information is assumed about the relation linking objects and properties. More recently, FCA has been extended to fuzzy data tables in the framework of residuated lattices, principally by Belohlavek [5,6]. We suggest another extension exploiting the close relationship existing between division in relational algebra and the operator underlying classical FCA. On this basis, ∗ Corresponding author. E-mail addresses: dubois@irit.fr (D. Dubois), prade@irit.fr (H. Prade). 0165-0114/$-see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2011.02.008