A Boolean matrix approach for granular reduction in formal fuzzy contexts

Granular reduction is an important issue for knowledge representation and data analysis in formal concept analysis. Granular structure of crisp-fuzzy concepts with application in granular reduction in formal fuzzy contexts is examined in this paper. However, computing a minimal granular reduct of a formal fuzzy context by Boolean reasoning is an NP-hard problem. Therefore, it is natural to investigate a heuristic approach to deal with this problem. A new method based on Boolean matrix is proposed to search the granular reduction. Granular matrix representations for extensions and intensions are firstly proposed. Then, we develop a similar degree between attribute subsets to measure attribute significance. Subsequently, two heuristic algorithms for granular reduction in formal fuzzy contexts and formal fuzzy decision contexts are presented, respectively. We prove that the time complexities of the algorithms are polynomial. Finally, numerical experiments demonstrate the proposed algorithms are much more feasible and efficient. Our methods present a new framework for granular reduction in formal fuzzy contexts.