The Constructions Of Classical Concept Lattices And Three-way Concept Lattices And Knowledge Acquisition

The formal concept analysis is a mathematical knowledge representation the-ory for philosophical concepts and concepts hierarchy structure. It was introduced by Wille, a German mathematician, in 1982, based on the discovery of formal concepts and the construction of concept hierarchy structures. Nowadays, this theory overlaps with many other disciplines and it has wide applications in many fields, such as computer networks, machine learning, analysis of traditional Chi-nese medicine, expert systems, etc.In the studying of formal concept analysis, construction of concept lattice is always an important and significant research direction. This paper focuses on the construction of concept lattices based on different Galois connections, i.e., the construction of classical concept lattice, the construction of three-way concept lat-tices and the construction of three-way object (property) oriented concept lattices. And we studied the relation between the classical concept lattices and three-way concept lattices, the relation between object (property) oriented concept lattices and three-way object (property) oriented concept lattices and the relation between three-way concept lattices and the three-way object (property) oriented concept lattices. Details are as follows:(1) We study the construction of classical concept lattices at both micro and macro levels. On the micro level, by using the viewpoint of granular, we divide object subsets into different levels, and then find the intents of each level of object subsets. We also explore relations of different levels of intents, and the cut-off condition of whether all intents are obtained. On the macro level, we divide the original formal context into several subcontexts and introduce its decompositions based on a minor minimum covering of the object set, a minor minimum covering of the attribute set and minor minimum coverings of two universes. We also prove that the above three decompositions exist and are unique. All the concepts of subcontexts in every decomposition constitute the concept lattice of the original formal context. And then we obtain concept lattice via the parallel algorithm.(2) We construct three-way concept lattices using the methods of construct-ing classical concept lattices. We define Type-? and Type-? combinatorial formal contexts through apposition and subposition of formal context and its complemen-tary context in which the objects and attributes are both distinguished. And then we prove that the concept lattice corresponding to lype-? (Type-?) combinatorial formal context is isomorphic to the object (attribute) induced three-way concept lattice.(3) We study the construction of three-way object (property) oriented con-cept lattices. By introducing locally completely possessing and locally completely no-possessing between an object set and an attribute, we firstly define three-way object (property) oriented operators, three-way object (property) oriented concept, three-way object (property) oriented concept extent and three-way object (prop-erty) oriented concept intent. Moreover, we prove that the set of all three-way ob-ject (property) oriented concepts is a complete lattice, i.e., three-way object (prop-erty) oriented concept lattice. And then we prove that the object (property) oriented concept lattice corresponding to Type-? (Type-?) combinatorial formal context is isomorphic to three-way object (property) oriented concept lattice.(4) We discuss the relations among classical concept lattice, three-way con-cept lattices and three-way object (property) oriented concept lattice. By the con-structions of three-way concept lattices and three-way object (property) oriented concept lattice, we prove that there exist a ? (?)-preserving order embedding of classical concept lattice in object (attribute) induced three-way concept lattice and a ? (?)-preserving order embedding of object (property) oriented concept lattice in three-way object (property) oriented concept lattice. We also prove that ob- ject (attribute) induced three-way concept lattice is anti-isomorphic (isomorphic) to three-way object (property) oriented concept lattice.