Rules acquisition of formal decision contexts based on three-way concept lattices

Abstract Three-way concept lattices can supply much more information than classical concept lattices since they contain the positive information and negative information between objects and attributes simultaneously. Taking advantage of this, the rules acquisition for formal decision contexts from the perspective of three-way concept lattices are discussed, and the results are compared with the common decision rules based on classical concept lattices. Firstly, the definition of object-induced three-way consistence of a formal decision context is presented. Then, positive decision rules and negative decision rules are proposed for an object-induced three-way consistent formal decision context. Furthermore, we give semantic explanation for these rules, and compare them with common rules obtained from a strongly consistent lattice. In parallel, the similar issues are investigated from the perspective of attributes, including the definition of attribute-induced three-way consistence of a formal decision context, rules acquisition based on an attribute-induced three-way consistent formal decision context, and rules comparison. Finally, the relationships among the three types of consistence, namely, object-induced three-way consistence, attribute-induced three-way consistence, and existing strongly consistence are discussed to make the contents compact. Some theoretical examples are given to illustrate the main results of this paper.