Rough Set on Concept Lattice

A new type formulation of rough set theory can be developed on a binary relation by association on the elements of a universe of finite set of objects with the elements of another universe of finite set of properties.. This paper presents generalization of Pawlak rough set approximation operators on the concept lattices. The notion of rough set approximation is to approximate an undefinable set or concepts through two definable sets. We analyze these and from the results one can obtain a better understanding of data analysis using formal concept analysis and rough set theory. Key Words: Rough Set, Lattice, Formal Concept, Approximation operators

Concept Approximations: Approximative Notions for Concept Lattices

In this thesis, we present a lattice theoretical approach to the field of approximations. Given a pair consisting of a kernel system and a closure system on an underlying lattice, one receives a lattice of approximations. We describe the theory of these lattices of approximations. Furthermore, we put a special focus on the case of concept lattices. As it turns out, approximation of formal concepts can be interpreted as traces, which are preconcepts in a subcontext.:Preface 1. Preliminaries 2. Approximations in Complete Lattices 3. Concept Approximations 4. Rough Sets List of Symbols Index BibliographyIn der vorliegenden Arbeit beschreiben wir einen verbandstheoretischen Zugang zum Thema Approximieren. Ausgehend von einem Kern- und einem Hüllensystem auf einem vollständigen Verband erhält man einen Approximationsverband. Wir beschreiben die Theorie dieser Approximationsverbände. Des Weiteren liegt dabei ein Hauptaugenmerk auf dem Fall zugrundeliegender Begriffsverbände. Wie sich nämlich herausstellt, lassen sich Approximationen formaler Begriffe als Spuren auffassen, welche diese in einem vorgegebenen Teilkontext hinterlassen.:Preface 1. Preliminaries 2. Approximations in Complete Lattices 3. Concept Approximations 4. Rough Sets List of Symbols Index Bibliograph

Logics for Rough Concept Analysis

Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting point, in this paper we introduce some varieties of lattices expanded with normal modal operators which can be regarded as the natural rough algebra counterparts of certain subclasses of rough formal contexts, and introduce proper display calculi for the logics associated with these varieties which are sound, complete, conservative and with uniform cut elimination and subformula property. These calculi modularly extend the multi-type calculi for rough algebras to a ‘nondistributive’ (i.e. general lattice-based) setting.https://digitalcommons.chapman.edu/scs_books/1060/thumbnail.jp

Logics for Rough Concept Analysis

Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting point, in this paper we introduce some varieties of lattices expanded with normal modal operators which can be regarded as the natural rough algebra counterparts of certain subclasses of rough formal contexts, and introduce proper display calculi for the logics associated with these varieties which are sound, complete, conservative and with uniform cut elimination and subformula property. These calculi modularly extend the multi-type calculi for rough algebras to a `nondistributive' (i.e. general lattice-based) setting

Geometric lattice structure of covering and its application to attribute reduction through matroids

The reduction of covering decision systems is an important problem in data mining, and covering-based rough sets serve as an efficient technique to process the problem. Geometric lattices have been widely used in many fields, especially greedy algorithm design which plays an important role in the reduction problems. Therefore, it is meaningful to combine coverings with geometric lattices to solve the optimization problems. In this paper, we obtain geometric lattices from coverings through matroids and then apply them to the issue of attribute reduction. First, a geometric lattice structure of a covering is constructed through transversal matroids. Then its atoms are studied and used to describe the lattice. Second, considering that all the closed sets of a finite matroid form a geometric lattice, we propose a dependence space through matroids and study the attribute reduction issues of the space, which realizes the application of geometric lattices to attribute reduction. Furthermore, a special type of information system is taken as an example to illustrate the application. In a word, this work points out an interesting view, namely, geometric lattice to study the attribute reduction issues of information systems

Domains via approximation operators

In this paper, we tailor-make new approximation operators inspired by rough set theory and specially suited for domain theory. Our approximation operators offer a fresh perspective to existing concepts and results in domain theory, but also reveal ways to establishing novel domain-theoretic results. For instance, (1) the well-known interpolation property of the way-below relation on a continuous poset is equivalent to the idempotence of a certain set-operator; (2) the continuity of a poset can be characterized by the coincidence of the Scott closure operator and the upper approximation operator induced by the way below relation; (3) meet-continuity can be established from a certain property of the topological closure operator. Additionally, we show how, to each approximating relation, an associated order-compatible topology can be defined in such a way that for the case of a continuous poset the topology associated to the way-below relation is exactly the Scott topology. A preliminary investigation is carried out on this new topology.Comment: 17 pages; 1figure, Domains XII Worksho

Rough matroids based on coverings

The introduction of covering-based rough sets has made a substantial contribution to the classical rough sets. However, many vital problems in rough sets, including attribution reduction, are NP-hard and therefore the algorithms for solving them are usually greedy. Matroid, as a generalization of linear independence in vector spaces, it has a variety of applications in many fields such as algorithm design and combinatorial optimization. An excellent introduction to the topic of rough matroids is due to Zhu and Wang. On the basis of their work, we study the rough matroids based on coverings in this paper. First, we investigate some properties of the definable sets with respect to a covering. Specifically, it is interesting that the set of all definable sets with respect to a covering, equipped with the binary relation of inclusion $\subseteq$, constructs a lattice. Second, we propose the rough matroids based on coverings, which are a generalization of the rough matroids based on relations. Finally, some properties of rough matroids based on coverings are explored. Moreover, an equivalent formulation of rough matroids based on coverings is presented. These interesting and important results exhibit many potential connections between rough sets and matroids.Comment: 15page