114,750 research outputs found
Some characteristics of matroids through rough sets
At present, practical application and theoretical discussion of rough sets
are two hot problems in computer science. The core concepts of rough set theory
are upper and lower approximation operators based on equivalence relations.
Matroid, as a branch of mathematics, is a structure that generalizes linear
independence in vector spaces. Further, matroid theory borrows extensively from
the terminology of linear algebra and graph theory. We can combine rough set
theory with matroid theory through using rough sets to study some
characteristics of matroids. In this paper, we apply rough sets to matroids
through defining a family of sets which are constructed from the upper
approximation operator with respect to an equivalence relation. First, we prove
the family of sets satisfies the support set axioms of matroids, and then we
obtain a matroid. We say the matroids induced by the equivalence relation and a
type of matroid, namely support matroid, is induced. Second, through rough
sets, some characteristics of matroids such as independent sets, support sets,
bases, hyperplanes and closed sets are investigated.Comment: 13 page
Two-sorted Modal Logic for Formal and Rough Concepts
In this paper, we propose two-sorted modal logics for the representation and
reasoning of concepts arising from rough set theory (RST) and formal concept
analysis (FCA). These logics are interpreted in two-sorted bidirectional
frames, which are essentially formal contexts with converse relations. On one
hand, the logic contains ordinary necessity and possibility
modalities and can represent rough set-based concepts. On the other hand, the
logic has window modality that can represent formal concepts. We
study the relationship between \textbf{KB} and \textbf{KF} by proving a
correspondence theorem. It is then shown that, using the formulae with modal
operators in \textbf{KB} and \textbf{KF}, we can capture formal concepts based
on RST and FCA and their lattice structures
Topological and algebraic characterization of coverings sets obtained in rough sets discretization and attribute reduction algorithms
Abstract. A systematic study on approximation operators in covering based rough sets and some relations with relation based rough sets are presented. Two different frameworks of approximation operators in covering based rough sets were unified in a general framework of dual pairs. This work establishes some relationships between the most important generalization of rough set theory: Covering based and relation based rough sets. A structured genetic algorithm to discretize, to find reducts and to select approximation operators for classification problems is presented.Se presenta un estudio sistemático de los diferentes operadores de aproximación en conjuntos aproximados basados en cubrimientos y operadores de aproximación basados en relaciones binarias. Se unifican dos marcos de referencia sobre operadores de aproximación basados en cubrimientos en un único marco de referencia con pares duales. Se establecen algunas relaciones entre operadores de aproximación de dos de las más importantes generalizaciones de la teoría de conjuntos aproximados. Finalmente, se presenta un algoritmo genético estructurado, para discretizar, reducir atributos y seleccionar operadores de aproximación, en problemas de clasificación.Doctorad
Decision Analysis via Granulation Based on General Binary Relation
Decision theory considers how best to make decisions in the light of uncertainty about data. There are several methodologies that may be used to determine the best decision. In rough set theory, the classification of objects according to approximation operators can be fitted into the Bayesian decision-theoretic model, with respect to three regions (positive, negative, and boundary region). Granulation using equivalence classes is a restriction that limits the decision makers. In this paper, we introduce a generalization and modification of decision-theoretic rough set model by using granular computing on general binary relations. We obtain two new types of approximation that enable us to classify the objects into five regions instead of three regions. The classification of decision region into five areas will enlarge the range of choice for decision makers
The structure of oppositions in rough set theory and formal concept analysis - Toward a new bridge between the two settings
Rough set theory (RST) and formal concept analysis (FCA) are two formal settings in information management, which have found applications in learning and in data mining. Both rely on a binary relation. FCA starts with a formal context, which is a relation linking a set of objects with their properties. Besides, a rough set is a pair of lower and upper approximations of a set of objects induced by an indistinguishability relation; in the simplest case, this relation expresses that two objects are indistinguishable because their known properties are exactly the same. It has been recently noticed, with different concerns, that any binary relation on a Cartesian product of two possibly equal sets induces a cube of oppositions, which extends the classical Aristotelian square of oppositions structure, and has remarkable properties. Indeed, a relation applied to a given subset gives birth to four subsets, and to their complements, that can be organized into a cube. These four subsets are nothing but the usual image of the subset by the relation, together with similar expressions where the subset and / or the relation are replaced by their complements. The eight subsets corresponding to the vertices of the cube can receive remarkable interpretations, both in the RST and the FCA settings. One facet of the cube corresponds to the core of RST, while basic FCA operators are found on another facet. The proposed approach both provides an extended view of RST and FCA, and suggests a unified view of both of them. © 2014 Springer International Publishing
A non-distributive logic for semiconcepts of a context and its modal extension with semantics based on Kripke contexts
A non-distributive two-sorted hypersequent calculus \textbf{PDBL} and its
modal extension \textbf{MPDBL} are proposed for the classes of pure double
Boolean algebras and pure double Boolean algebras with operators respectively.
A relational semantics for \textbf{PDBL} is next proposed, where any formula is
interpreted as a semiconcept of a context. For \textbf{MPDBL}, the relational
semantics is based on Kripke contexts, and a formula is interpreted as a
semiconcept of the underlying context. The systems are shown to be sound and
complete with respect to the relational semantics. Adding appropriate sequents
to \textbf{MPDBL} results in logics with semantics based on reflexive,
symmetric or transitive Kripke contexts. One of these systems is a logic for
topological pure double Boolean algebras. It is demonstrated that, using
\textbf{PDBL}, the basic notions and relations of conceptual knowledge can be
expressed and inferences involving negations can be obtained. Further, drawing
a connection with rough set theory, lower and upper approximations of
semiconcepts of a context are defined. It is then shown that, using the
formulae and sequents involving modal operators in \textbf{MPDBL}, these
approximation operators and their properties can be captured
Knowledge structure, knowledge granulation and knowledge distance in a knowledge base
AbstractOne of the strengths of rough set theory is the fact that an unknown target concept can be approximately characterized by existing knowledge structures in a knowledge base. Knowledge structures in knowledge bases have two categories: complete and incomplete. In this paper, through uniformly expressing these two kinds of knowledge structures, we first address four operators on a knowledge base, which are adequate for generating new knowledge structures through using known knowledge structures. Then, an axiom definition of knowledge granulation in knowledge bases is presented, under which some existing knowledge granulations become its special forms. Finally, we introduce the concept of a knowledge distance for calculating the difference between two knowledge structures in the same knowledge base. Noting that the knowledge distance satisfies the three properties of a distance space on all knowledge structures induced by a given universe. These results will be very helpful for knowledge discovery from knowledge bases and significant for establishing a framework of granular computing in knowledge bases
Matroidal structure of generalized rough sets based on symmetric and transitive relations
Rough sets are efficient for data pre-process in data mining. Lower and upper
approximations are two core concepts of rough sets. This paper studies
generalized rough sets based on symmetric and transitive relations from the
operator-oriented view by matroidal approaches. We firstly construct a
matroidal structure of generalized rough sets based on symmetric and transitive
relations, and provide an approach to study the matroid induced by a symmetric
and transitive relation. Secondly, this paper establishes a close relationship
between matroids and generalized rough sets. Approximation quality and
roughness of generalized rough sets can be computed by the circuit of matroid
theory. At last, a symmetric and transitive relation can be constructed by a
matroid with some special properties.Comment: 5 page
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