On The Characterization And Generation Of Three-way Concept Lattices

The era of big data not only provides a lot of data and information for human,but also challenges how to effectively mine useful knowledge,analyze and process these massive and dynamic complex data.Formal concept analysis,rough set,fuzzy set,three-way decision and granular computing are all effective methods in the field of data analysis and knowledge discovery.Among them,the three-way concept analysis the-ory is a new mathematical tool for information processing and knowledge discovery by combining the three-way decision with formal concept analysis.The core content of three-way concept analysis theory includes the characterization of three-way con-cept lattices and the acquisition of three-way concepts.The number of the three-way concepts is huge,especially in the face of dynamic formal context,each recalculation concept,to some extent,leads to its low data processing efficiency,which also hinders the rapid development and wide application of the theory.This paper focuses on the characterization of three-way concept lattices and the acquisition of three-way concept-s under dynamic formal context.These studies have enriched the theoretical basis of three-way concept lattices and will play a positive role in promoting the application of three-way concept lattices.The main research contents of this paper are as follows:(1)Characteristics of three concept lattices.In the classical formal concept analysis proposed by Wille and Ganter et al.,there is a basic theorem:given a formal background,its corresponding formal concept lattice is a complete lattice;conversely,if given a com-plete lattice,there must be a concept lattice isomorphic with it.This conclusion holds in both object-oriented and attribute-oriented concept lattices.Is this conclusion true in three-way concept lattices?Firstly,a counter example is given to verify that this is not true in general.Secondly,we discuss some properties of special elements such as atoms and irreducible elements,and complements of three-way concept lattices.Finally,we establish an isomorphic mapping between complete lattices with these special properties and three-way concept lattices,so as to obtain the characteristic of three-way concept lattices.The characterization theorem of three-way rough concept lattices is obtained by a similar method.(2)The construction of three-way concept lattices when object changes.We de-fine dynamic formal context for several different cases.In the context of dynamic form,when adding or deleting objects,by analyzing the relationship between the new three-way concepts and the original three-way concepts,the necessary and sufficient condi-tions for obtaining the new three-way concepts from the original three-way concepts are obtained,and the incremental algorithm for acquiring three-way concepts is given.Through numerical experiments,the difference of computing time between incremen-tal algorithm and non incremental algorithm is compared,and the effectiveness of the algorithm is verified.(3)The construction of three-way concept lattices when attribute changes.When adding or deleting attributes,by analyzing the relationship between the new three-way concepts and the original three-way concepts,the necessary and sufficient conditions for obtaining the new three-way concepts from the original three-way concepts are obtained,and the algorithm for acquiring the new three-way concepts is given.The validity of the algorithm is verified by numerical experiments.