Learning Fuzzy Dual-Modal Concepts and Constructing Concept LatticesUsing Modal Operators in Formal Concept Analysis

Abstract

Formal Concept Analysis (FCA) is a mathematically founded method for extracting formal concepts froma binary relation between a set of objects and a set of properties (attributes), referred to as a crisp formalcontext. A formal concept is a pair hobjects; propertiesi induced by the sufficiency operator and organizedinto a lattice structure. Since real-world data may involve fuzzy properties, several fuzzy extensions of FCAhave been proposed. These approaches aim to extract fuzzy formal concepts using a fuzzy generalization ofthe sufficiency operator, which often leads to a combinatorial explosion in the number of induced concepts,limiting their practical applicability. Beyond sufficiency-based reasoning, modal FCA introduces necessityand possibility operators to extract modal concepts, yet it remains largely confined to crisp contexts.Extending such modalities to fuzzy settings raises the challenge of constructing concept lattices of manageablesize while preserving computational feasibility. To address this issue, we propose a modal extension ofFCA for fuzzy formal contexts based on necessity and possibility operators, referred to as FM-operators,which induce fuzzy dual-modal concepts (FDM-concepts). We establish a fundamental theorem characterizingthe algebraic and lattice structure of these FDM-concepts. To support practical computation, wedevelop two algorithms for extracting FDM-concepts: a powerset-based algorithm and an incrementalalgorithm. Both algorithms have exponential worst-case complexity O(2m n m), where n and m denotethe numbers of objects and properties, while the incremental algorithm is output-polynomial in the numberof extracted concepts. Experiments on synthetic and real-world datasets demonstrate that the proposedFDMframework significantly reduces the number of induced concepts compared to classical fuzzy FCA.We further illustrate its applicability to information retrieval, disjunctive attribute implications, and adaptivenonlinear control systems.