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Computing Formal Concepts by Attribute Sorting

Krajca P., Outrata J., Vychodil V.

Fundamenta Informaticae

2012

Vol. 115, nr 4

395-417

We present a novel approach to compute formal concepts of formal context. In terms of operations with Boolean matrices, the presented algorithm computes all maximal rectangles of the input Boolean matrix which are full of 1s. The algorithm combines basic ideas of previous approaches with our recent observations on the influence of attribute permutations and attribute sorting on the number of formal concepts which are computed multiple times. As a result, we present algorithm which computes formal concepts by successive context reduction and attribute sorting. We prove its soundness, discuss its complexity and efficiency, and show that it outperforms other algorithms from the CbO family in terms of substantially lower numbers of formal concepts which are computed multiple times.

Attribute Reduction in Formal Contexts: A Covering Rough Set Approach

Li T-J., Wu W. Z.

Fundamenta Informaticae

2011

Vol. 111, nr 1

15-32

This paper proposes an approach to attribute reduction in formal contexts via a covering rough set theory. The notions of reducible attributes and irreducible attributes of a formal context are first introduced and their properties are examined. Judgment theorems for determining all attribute reducts in the formal context are then obtained. According to the attribute reducts, all attributes of the formal context are further classified into three types and the characteristic of each type is characterized by the properties of irreducible classes of the formal context. Finally, by using the discernibility attribute sets, a method of distinguishing the reducible attributes and the irreducible attributes in formal contexts is presented.

Relational Contexts and Relational Concepts

Jiang F., Sui Y., Cao C.

Fundamenta Informaticae

2010

Vol. 99, nr 3

293-314

Formal concept analysis (FCA) is a mathematical description and theory of concepts implied in formal contexts. And the current formal contexts of FCA aim to model the binary relations between individuals (objects) and attributes in the real world. In the real world we usually describe each individual by some attributes, which induces the relations between individuals and attributes. But there also exist many relations between individuals, for instance, the parent-children relation in a family. In this paper, to model the relations between individuals in the real world, we propose a new context - relational context for FCA, which contains a set U of objects and a binary relation r on U. Corresponding to the formal concepts in formal contexts, we present different kinds of relational concepts in relational contexts, which are the pairs of sets of objects. First we define the standard relational concepts in relational contexts. Moreover, we discuss the indirect relational concepts and negative relational concepts in relational contexts, which aim to concern the indirection and negativity of the relations in relational contexts, respectively. Finally, we define the hybrid relational concepts in relational contexts, which are the combinations of any two different kinds of relational concepts. In addition, we also discuss the application of relational contexts and relational concepts in the supply chain management field.