Approximate Description of Indefinable Granules Based on Classical and Three-Way Concept Lattices

Granule description is a fundamental problem in granular computing. However, how to describe indefinable granules is still an open, interesting, and important problem. The main objective of this paper is to give a preliminary solution to this problem. Before proceeding, the framework of approximate description is introduced. That is, any indefinable granule is characterized by an ordered pair of formulas, which form an interval set, where the first formula is the beta-prior approximate optimal description and the second formula is the alpha-prior approximate optimal description. More concretely, given an indefinable granule, by exploring the description of its lower approximate granule, its beta-prior approximate optimal description is obtained. Likewise, by consulting the description of its upper approximate granule, its alpha-prior approximate optimal description can also be derived. Following this idea, the descriptions of indefinable granules are investigated. Firstly, boolean AND-approximate descriptions of indefinable granules are investigated based on the classical concept lattice, and (boolean AND,boolean OR)-approximate descriptions of indefinable granules are given via object pictorial diagrams. And then, it is revealed from some examples that the classical concept lattice is no longer effective and negative attributes must be taken into consideration. Therefore, a three-way concept lattice is adopted instead of the classical concept lattice to study (boolean AND,<not sign>)-approximate descriptions and (boolean AND,boolean OR,<not sign>)-approximate descriptions of indefinable granules. Finally, some discussions are presented to show the differences and similarities between our study and existing ones.