Knowledge reduction in formal decision contexts based on an order-preserving mapping

Knowledge reduction is one of the basic issues in knowledge presentation and data mining. In this study, an order-preserving mapping between the set of all the extensions of the conditional concept lattice and that of the decision concept lattice is defined to classify formal decision contexts into consistent and inconsistent categories. Then, methods of knowledge reduction for both the consistent and the inconsistent formal decision contexts are formulated by constructing proper discernibility matrices and their associated Boolean functions. For the consistent formal decision contexts, the proposed reduction method can avoid redundancy subject to maintaining consistency, while for the inconsistent formal decision contexts, the reduction method can make the set of all the compact non-redundant decision rules complete in the initial formal decision context.