Multirretículos y reducción de atributos en retículos de conceptos multiadjuntos

Abstract

Since its introduction in the eighties by B. Ganter and R. Wille, Formal Concept Analysis has become an appealing research topic. It is a theory of data analysis which identifies conceptual structures among data sets. Specifically, it is a tool for extracting pieces of information from databases that contain a set of attributes A and a set of objects B together with a relationship between them. These pieces of information are called concepts and they can be hierarchized to obtain concept lattices.

Attribute reduction is a very important part in Formal Concept Analysis because the difficulty in building the concept lattice increases exponentially when the number of objects and attributes increases. Therefore, one of the most important goals in this theory is the reduction of the context, removing the irrelevant information. Moreover, real databases usually give rise to complex concept lattices, from which extracting conclusions can be a really difficult task. Consequently, another important issue is the reduction of the size of the original concept lattice.

This thesis has been focused on both these goals. Firstly, it introduces several results in order to classify the set of attributes. From this classification a mechanism to reduce the context on a fuzzy environment is obtained, which generalizes the current existing procedures. The most innovative aspect related to this contribution is that it maintains all the knowledge of the relational system.

In addition, two procedures to reduce the size of a multi-adjoint concept lattice are presented. One of them considers thresholds in the concept-forming operators and this reduction method generalizes existing mechanisms based on this philosophy. Another procedure introduced shows a reduction from the irreducible elements of the lattice. This one provides an interesting property, that is, the reduced concept lattice is a sublattice of the original one and, consequently, the use of this mechanism does not involve the loss or modification of the original information.

Lastly, the thesis concludes by demonstrating an extension of the theory of Formal Concept Analysis based on multilattices. As a consequence, the range of applications of Formal Concept Analysis has been increased.