Tsau Young Lin

Generalization of Rough Sets using Modal Logics

ABSTRACTThe theory of rough sets is an extension of set theory with two additional unary set-theoretic operators defined based on a binary relation on the universe. These two operators are related to the modal operators in modal logics. By exploring the relationship between rough sets and modal logics, this paper proposes and examines a number of extended rough set models. By the properties satisfied by a binary relation, such as serial, reflexive, symmetric, transitive, and Euclidean, various classes of algebraic rough set models can be derived. They correspond to different modal logic systems. With respect to graded and probabilistic modal logics, graded and probabilistic rough set models are also discussed.

Topological and Fuzzy Rough Sets

The approximation theory is studied via rough sets, fuzzy sets and topological spaces (more precisely, Frechet spaces). Rough set theory is a set theory via knowledge bases. This set theory is extended to fuzzy sets and Frechet topological spaces. By these results one can show that the classification preserves the approximation. We also showed that within the approximation theory, fuzzy set and Frechet topology are intrinsically equivalent notions. Finally, we show that even though approximation is a compromised solution, the three theories allow one to draw an exact solution whenever there are adequate approximations. This implies that these three approaches are good approximation theories.

Neighborhood systems and relational databases

Queries in database can be classified roughly into two types: specific targets and fuzzy targets. Many queries are in effect fuzzy targets, however, because of lacking the supports, the user has been emulating them with specific targets by retiring a query repeatedly with minor changes. In this paper, we augment the relational database, with neighborhood systems, so the database can answer a fuzzy query. There have been many works to combine relational databases and fuzzy theory. Bucklles and Petry replaced attributes values by sets of values. Zemankova-Leech, Kandel, and Zviell used fuzzy logic. The formalism of present work is quite general, it allows numerical or nonnumerical measurements of fuzziness in relational databases. The fuzzy theory present here is quite different from the usual theory. Our basic assumption here is that: the data are not fuzzy, the queries are.Motro [Motr86] introduced the notion of distance into the relational databases. From that he can, then, define the notion of “close-ness” and develop goal queries. Though “distance” is a useful concept, yet very often the quantification of it is meaningless or extremely difficult. For example, “very close”, “very far” are meaningful concept of distance, yet there is no practical way to quantity them for all occasions. Our approach here is more direct, we define directly the meaning of “very close neighborhood”. Using the concept of neighborhoods is not very original, in fact, in the theory of topological spaces [Dugu66], mathematician has been using the “neighborhood system” to study the phenomena of “close-ness”. In the territory of fuzzy queries, the notion of “neighborhood” captures the essence of the qualitative information of “close-ness” better than the brute-force-quantified information (distance). A “fuzzy” neighborhood is a qualitative measure of fuzziness. On the surface, it seems a very complicated procedure to define a neighborhood for each value in the attribute. In fact, if we use the characteristic function (membership function) to define a subset, then the defining procedure is merely another type of distance function (non-measure distance or symbolic distance). Now, to define the neighborhood system one can simply re-entered the third column of the relation with linguistic values: “very close”, “close”, “far”. Note that there is a “greater than” relation among these linguistic values. In mathematical terms, they forms a lattice [Jaco60]. For technical reason, we require the values in third column be elements of a lattice. Note that real number is a lattice, so we get Motros results back.

Rough Approximate Operators: Axiomatic Rough Set Theory

In rough set theory, the upper and lower approximations are defined in terms of equivalence relation. In this paper, the reverse problem is considered. Let H and L are two abstract operators acting on the power set of U, the universe of discourse. If the two operators satisfy six axioms, then there is an equivalence relation defined on U such that H(X) and L(X) are precisely the upper and lower approximations. The six axioms are adopted from the axioms of Kuratowski’s closure operator. The proof is an easy application of point set topology. Similar results (five axioms) are also obtained for neighborhood systems (a generalized rough set theory) which are based on Frechet (V)spaces. The results can be viewed as a beginning of an axiomatic rough set theory.

Data Mining and Machine Oriented Modeling: A Granular Computing Approach

From the processing point of view, data mining is machine derivation of interesting properties (to human) from the stored data. Hence, the notion of machine oriented data modeling is explored: An attribute value, in a relational model, is a meaningful label (a property) of a set of entities (granule). A model using these granules themselves as attribute values (their bit patterns or lists of members) is called a machine oriented data model. The model provides a good database compaction and data mining environment. For moderate size databases, finding association rules, decision rules, and etc., can be reduced to easy computation of set theoretical operations of granules. In the second part, these notions are extended to real world objects, where the universe is granulated (clustered) into granules by binary relations. Data modeling and mining with such additional semantics are formulated and investigated. In such models, data mining is essentially a machine “calculus” of granules-granular computing.

A Review of Rough Set Models

Since introduction of the theory of rough set in early eighties, considerable work has been done on the development and application of this new theory. The paper provides a review of the Pawlak rough set model and its extensions, with emphasis on the formulation, characterization, and interpretation of various rough set models.

A new rough sets model based on database systems

Rough sets theory was proposed by Pawlak in the early 1980s and has been applied successfully in a lot of domains. One of the major limitations of the traditional rough sets model in the real applications is the inefficiency in the computation of core and reduct, because all the intensive computational operations are performed in flat files. In order to improve the efficiency of computing core attributes and reducts, many novel approaches have been developed, some of which attempt to integrate database technologies. In this paper, we propose a new rough sets model and redefine the core attributes and reducts based on relational algebra to take advantages of the very efficient set-oriented database operations. With this new model and our new definitions, we present two new algorithms to calculate core attributes and reducts for feature selections. Since relational algebra operations have been efficiently implemented in most widely-used database systems, the algorithms presented in this paper can be extensively applied to these database systems and adapted to a wide range of real-life applications with very large data sets. Compared with the traditional rough set models, our model is very efficient and scalable.

Granular Computing: Fuzzy Logic and Rough Sets

The primary goal of granular computing is to elevate the lower level data processing to a high level knowledge processing. Such an elevation is achieved by granulating the data space into a concept space. Each granule represents certain primitive concept, and the granulation as a whole represents a knowledge. In this paper, such an intuitive idea is formalized into a mathematical theory: Zadeh’s informal words are taken literally as a formal definition of granulation. Such a mathematical notion is a mild generalization of the “old” notion of crisp/fuzzy neighborhood systems of (pre-)topological spaces. A crisp/fuzzy neighborhood is a granule and is assigned a meaningful name to represent certain primitive concept or to summarize the information content. The set of all linear combinations of these names, called formal words, mathematically forms a vector space over real numbers. Each vector is intuitively an advanced concept represented by some “weighted averaged” of primitive concepts. In terms of these concepts, the universe can be represented by a formal word table; this is one form of Zadeh’s veristic constraints. Such a representation is useful; fuzzy logic designs can be formulated as series of table transformations. So table processing techniques of rough set theory may be used to simplify these tables and their transformations. Therefore the complexity of large scaled fuzzy systems may be reduced; details will be reported in future papers.

Granular computing: examples, intuitions and modeling

The notion of granular computing is examined. Obvious examples, such as fuzzy numbers, infinitesimal number and access control model, (pre-) topological spaces are examined. A general models are proposed; localized multi-level granulation can be modeled by generalized topological spaces, called neighborhood systems. For most general granulation are modeled by Tarski type relational structures.

Finding Association Rules Using Fast Bit Computation: Machine-Oriented Modeling

This paper continue the study of machine oriented models initiated by the second author. An attribute value is regarded as a name of the collection (called granule) of the entities that have the same property (specified by the attribute value). The relational model uses these granules (e.g., bit representation of subsets) as attribute values is called machine oriented data model. The model transforms data mining, particularly finding association rules, into Boolean operations. This paper show that this approach speed up data mining process tremendously; in the experiments, it is approximately 50 times faster, the pre-processing time was included).