Catálogo de publicaciones - libros

This paper concerns the relationship between rough sets and flow graphs. It is shown that flow graph can be used both as formal language for computing approximations of sets in the sense of rough set theory, and as description tool for data structure. This description is employed next for finding patterns in data. To this end decision algorithm induced by the flow graph is defined and studied.

In this paper we present a modal logic for Pawlak’s information systems giving a modal characterization of 9 informational relations: strong indiscernibility, as well as weak and strong versions of forward and backward informational inclusion, as well as positive and negative similarities. extends the logic introduced in [4] by adding a modality corresponding to strong indiscernibility relation. The main problem in the modal treating of strong indiscernibility is that its definition is not modally definable. This requires special copying techniques, which in the presence of many interacting modalities presents complications. One of the main aims of the paper is to demonstrate such techniques and to present an information logic complete in the intended semantics and containing almost all natural information relations. It is proved that possesses finite model property and hence is decidable.

In this study, we discuss a concept of shadowed sets and elaborate on their applications. To establish some sound compromise between the qualitative Boolean (two-valued) description of data and quantitative membership grades, we introduce an interpretation framework of shadowed sets. Shadowed sets are discussed as three-valued constructs induced by fuzzy sets assuming three values (that could be interpreted as full membership, full exclusion, and uncertain). The algorithm of converting membership functions into this quantification is a result of a certain optimization problem guided by the principle of uncertainty localization. With the shadowed sets of clusters in place, discussed are various ideas of relational calculus on such constructs. We demonstrate how shadowed sets help in problems in data interpretation in fuzzy clustering by leading to the three-valued quantification of data structure that consists of core, shadowed, and uncertain structure.

We present a rough set approach to vague concept approximation within the adaptive learning framework. In particular, the role of extensions of approximation spaces in searching for concept approximation is emphasized. Boundary regions of approximated concepts within the adaptive learning framework are satisfying the higher order vagueness condition, i.e., the boundary regions of vague concepts are not crisp. There are important consequences of the presented framework for research on adaptive approximation of vague concepts and reasoning about approximated concepts. An illustrative example is included showing the application of Boolean reasoning in adaptive learning.

We introduce a pair of rough set approximations in formal concept analysis. The proposed approximation operators are defined based on both lattice-theoretic and set-theoretic operators. The properties of the approximation operators are examined. Algorithms for attribute reduction and object reduction in concept lattices are presented.

The rough approximations are considered in the context of multi-criteria classification problem where evaluations of objects on particular criteria and their assignments to decision classes are imprecise and given in the form of intervals of possible values. Within Dominance-based Rough Set Approach (DRSA), the lower and upper approximations reflect the inconsistencies with respect to dominance principle. In the considered case, also the interval assignments have to be taken into account. This requires a new formulation of the dominance principle. A possible solution to the problem consists in introducing the second-order rough approximations. The methodology based on these approximations preserves well-known properties of rough approximations, such as rough inclusion, complementarity, identity of boundaries and monotonicity.

The standard rough set theory has been introduced in 1982 [5]. In this paper we use a topological concepts to investigate a new definitions for the lower and upper approximation operators. This approach is a generalization for Pawlak approach and the generalizations in [2, 7, 10, 12, 13, 14, 15, 16]. Properties of the suggested concepts are obtained. Also comparison between our approach and previous approaches are given. In this case, we show that the generalized approximation space is a topological space for any reflexive relation.

Formal concept analysis and rough set theory are two different methods for knowledge representation and knowledge discovery, and both have been successfully applied to various fields. The basis of rough set theory is an equivalence relation on a universe of objects, and that of formal concept analysis is an ordered hierarchical structure — concept lattice. This paper discusses the basic connection between formal concept analysis and rough set theory, and also analyzes the relationship between a concept lattice and the power set of a partition. Finally, it is proved that a concept lattice can be transformed into a partition and vice versa, and transformation algorithms and examples are given.

We discuss characterizations of three important types of attribute sets in generalized approximation representation spaces, in which binary relations on the universe are reflexive. Many information tables, such as consistent or inconsistent decision tables, variable precision rough set models, consistent decision tables with ordered valued domains and with continuous valued domains, and decision tables with fuzzy decisions, can be unified to generalized approximation representation spaces. A general approach to knowledge reduction based on rough set theory is proposed.

[4] placed an approximation space (,≡ ) in a type-lowering retraction with its power set 2 such that the ≡ -exact subsets of comprise the kernel of the retraction, where ≡ is the equivalence relation of set-theoretic indiscernibility within the resulting universe of exact sets. Since a concept thus forms a set just in case it is ≡ -exact, set-theoretic comprehension in (,≡ ) is governed by the method of upper and lower approximations of Rough Set Theory. Some central features of this universe were informally axiomatized in [3] in terms of the notion of a Proximal Frege Structure and its associated modal Boolean algebra of exact sets. The present essay generalizes the axiomatic notion of a PFS to tolerance (reflexive, symmetric) relations, where the universe of exact sets forms a modal ortho-lattice. An example of this general notion is provided by the tolerance relation of “matching” over .