Possibility theory and formal concept analysis in information systems

The setting of formal concept analysis presupposes the existence of a relation between objects and properties. Knowing that an unspecified object has a given property induces a formal possibility distribution that models the set of objects known to possess this property. This view expressed in a recent work by the authors of the present paper, has led to introduce the set-valued counterpart to the four set functions evaluating potential or actual, possibility or necessity that underlie bipolar possibility theory, and to study associated notions. This framework puts formal concept analysis in a new, enlarged perspective, further explored in this article. The "actual (or guaranteed) possibility" function induces the usual Galois connexion that defines the notion of a concept as the pair of its extent and its intent. A new Galois connexion, based on the necessity measure, partitions the relation in "orthogonal" subsets of objects having distinct properties. Besides, the formal similarity between the notion of division in relational algebra and the "actual possibility" function leads to define the fuzzy set of objects having most properties in a set, and other related notions induced by fuzzy extensions of division. Generally speaking, the possibilistic view of formal concept analysis still applies when properties are a matter of degree, as discussed in the paper. Lastly, cases where the object / property relation is incomplete due to missing information, or more generally pervaded with possibilistic uncertainty is also discussed.