Complementary Relations and Their Concept Lattices in Relational Databases

The storage of data is a key issue of information systems, which is an important foundation for data query and data mining. Relational database model has been proven to be a very useful data-storage technique. As information is stored as data in relational databases, the induction of concepts from data is a pivotal topic in the data mining field. Formal concept analysis (FCA) turns out to be a perfect instrument for a meaningful and conceptual exploration of the stored data. In FCA, conceptual scaling provides a complete framework for transforming any many-valued context (i.e., relation/table) into a context (called a derived context), in which each many valued attribute is given a scale. The attributes in a scale basically describe meaningful features of the values of the initial attribute. From the logical point of view, complement operation plays a very important role in relational databases and data query systems. In this paper, we provide the connections between the concepts of binary relations and those of complementary binary relations, and propose an approach toward normalizing (complementary) scales, i.e., each (complementary)scale can be represented by a set of statements. One advantage of normalizing scales is to avoid generating huge derived relations, and hence this approach reduces storage cost. By the normalization, the concept lattice of the complement of a derived relation is reduced to a combination of the concept lattice of the derived relation and a set of statements.