Concept Reduction Preserving Binary Relations

Formal concept analysis(FCA)and three-way concept analysis(3WCA)are tools for knowledge discovery and decision making which are based on formal contexts.The two theories have both achieved abundant results,and have been applied to diverse fields,such as artificial intelligence,information retrieval.Concept,a basic cognition unit,can be mathematically described by FCA and 3WCA.There is an one-to-one mapping between a formal context and a formal concept lattice(three-way concept lattice),which is composed by all the formal concepts(three-way concepts)generated from this context.However,there is no necessity to get all the formal concepts(three-way concepts)in some cases.Inspired by the thought of using formal concepts to solve Boolean factorization,and the thought of attribute reduction,we propose the idea of concept reduction preserving binary relations.The main contents of this paper are as follows.(1)The definition of formal concept reduction is given,and the sufficiency conditions of consistent sets(reduct)preserving binary relations are presented.(2)According to the roles of formal concepts played in concept reduction preserving binary relations,formal concepts are classified into three types,that is,core concepts,relative necessary concepts and unnecessary concepts.And then,based on binary relations,the characteristics of the three types of formal concepts are discussed.Further,the characteristics of object concepts and attribute concepts are discussed based on operators and boolean matrix.Based on the above work,the approaches to computing formal concept reduction are presented.(3)Based on the connections between three-way concepts and formal concepts,three-way concept reductions are discussed similarly.