Attribute combination reduction in formal concept analysis: A theoretical characterization

Formal concept analysis is an approach relying on hierarchies of formal concepts to acquire knowledge from formal contexts, and is developed on the foundation of lattice theory. In formal concept analysis, reduction theory mainly consists of two types: attribute reduction and concept reduction. The former achieves data reduction but inevitably leads to the loss of the original information in the formal context. The latter involves the deletion of formal concepts while preserving the original information. This paper proposes a new type of reduction called attribute combination reduction to preserve object intents, which leverages the strengths of both attribute reduction and concept reduction while avoiding the limitations of attribute reduction. First, the definition of attribute combination reducts and the judgment theorem of attribute combination consistent sets are given. Then, the relationship between attribute combination reducts and concept reducts is investigated, and the properties of attribute combination reducts are explored. In addition, to narrow the scope for searching attribute combination reducts, a lower bound and an upper bound for their cardinality are provided from the perspectives of the set dimension and the Ferrers dimension of a formal context.