Updating extended possibility‐based fuzzy relational databases

Two kinds of fuzziness in attribute values of the fuzzy relational databases can be distinguished: one is that attribute values are possibility distributions and the other is that there are resemblance relations in attribute domains. The fuzzy relational databases containing these two kinds of fuzziness simultaneously are called extended possibility‐based fuzzy relational databases. In this article, we focus on such fuzzy relational databases and investigate three update operations for the fuzzy relational databases, which are Insertion, Deletion, and Modification, respectively. We develop the strategies and implementation algorithms of these operations. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 237–258, 2007.     

Semantic measure of fuzzy data in extended possibility‐based fuzzy relational databases

In this paper, we propose notions of equivalence and inclusion of fuzzy data in relational databases for measuring their semantic relationship. The fuzziness of data appears in attribute values in forms of possibility distribution as well as resemblance relations in attribute domain elements. An approach for evaluating semantic measures is presented. With the proposal, one can remove fuzzy data redundancy and define fuzzy functional dependency. © 2000 John Wiley & Sons, Inc.     

Fuzzy relational algebra for possibility-distribution-fuzzy-relational model of fuzzy data

In the real world, there exist a lot of fuzzy data which cannot or need not be precisely defined. We distinguish two types of fuzziness: one in an attribute value itself and the other in an association of them. For such fuzzy data, we propose a possibility-distribution-fuzzy-relational model, in which fuzzy data are represented by fuzzy relations whose grades of membership and attribute values are possibility distributions. In this model, the former fuzziness is represented by a possibility distribution and the latter by a grade of membership. Relational algebra for the ordinary relational database as defined by Codd includes the traditional set operations and the special relational operations. These operations are classified into the primitive operations, namely, union, difference, extended Cartesian product, selection and projection, and the additional operations, namely, intersection, join, and division. We define the relational algebra for the possibility-distribution-fuzzy-relational model of fuzzy databases.     

Data dependencies in extended possibility‐based fuzzy relational databases

Based on the semantic equivalence degree the formal definitions of fuzzy functional dependencies (FFDs) and fuzzy multivalued dependencies (FMVDs) are first introduced to the fuzzy relational databases, where fuzziness of data appears in attribute values in the form of possibility attributions, as well as resemblance relations in attribute domain elements, called extended possibility‐based fuzzy relational databases. A set of inference rules for FFDs and FMVDs is then proposed. It is shown that FFDs and FMVDs are consistent and the inference rules are sound and complete, just as Armstrong's axioms for classic cases. © 2002 Wiley Periodicals, Inc.     

Algebraic operations in fuzzy object-oriented databases

The fuzzy object-oriented databases have been proposed to meet the need of dealing with fuzzy as well as complex objects. In this paper, we present a formal fuzzy object-oriented database model. Based on the semantic measure of fuzzy data, we first identify two kinds of fuzzy object redundancies, which are inclusion redundancy and equivalence redundancy, and then define three kinds of merging operation for redundancy removal. On the basis, we define some fuzzy algebraic operations for fuzzy classes and fuzzy objects. Finally, in the paper, we discuss fuzzy querying strategies and give the form of SQL-like fuzzy querying for the fuzzy object-oriented databases.     

Querying fuzzy relational databases through fuzzy domain calculus

In this paper we present a definition of a domain relational calculus for fuzzy relational databases using the GEFRED model as a starting point. It is possible to define an equivalent fuzzy tuple relational calculus and consequently we achieve the two query language levels that Codd designed for relational databases but these are extended to fuzzy relational databases: Fuzzy relational algebra (defined in the GEFRED model) and the fuzzy relational calculus which is put forward in this paper. The expressive power of this fuzzy relational calculus is demonstrated through the use of a method to translate any algebraic expression into an equivalent expression in fuzzy domain relational calculus. Furthermore, we include a useful system so that the degree to which each value has satisfied the query condition can be measured. Some examples are also included in order to clarify the definition. ©1999 John Wiley & Sons, Inc.     

Null values in fuzzy databases

Since in the real world, it often occurs that information is missing, database systems clearly need some facilities to deal with missing data. With respect to traditional database systems, the most commonly adopted approach to this problem is based on null values and three valued logic. This paper deals with the semantics and the use of null values in fuzzy databases. In dealing with missing information a distinction is made between incompleteness due to unavailability and incompleteness due to inapplicability. Both the database modelling and database querying aspects are described. With respect to attribute values, incompleteness due to unavailability is modelled by possibility distributions, which is a commonly used technique in the fuzzy databases. Domain specific null values, represented by a bottom symbol, are used to model incompleteness due to inapplicability. Extended possibilistic truth values are used to formalize the impact of data manipulation and (flexible) querying operations in the presence of these null values. The different cases of appearances of null values in the handling of selection conditions of flexible database queries are described in detail.     

Unacceptable components in fuzzy relational databases

The existence of unacceptable components, which consist of unacceptable tuples and elements in attribute values, is shown in fuzzy relational databases. The unacceptable components are created by update operations, insertion, deletion, and modification. An unacceptable tuple in a relation is a tuple such that the degree of its not belonging to that relation is greater than that of its belonging to. The unacceptable tuple can be easily eliminated from relations. There are three kinds of unacceptable elements. One case of unacceptable elements is a redundant element created by insertion and modification. Another is an element created by a possible tuple value not at all or partially satisfying integrity constraints in insertion and modification. The other is an element created by a possible tuple value completely or partially satisfying update conditions in deletion. The unacceptable elements can be eliminated from relations without loss of information. As a result, we can obtain fuzzy relational databases without unacceptable components by a reasonable way. © 1996 John Wiley & Sons, Inc.     

Storing and querying fuzzy XML data in relational databases

Information imprecision and uncertainty exist in many real-world applications and for this reason fuzzy data management has been extensively investigated in various database management systems. Currently, introducing native support for XML data in relational database management systems (RDBMs) has attracted considerable interest with a view to leveraging the powerful and reliable data management services provided by RDBMs. Although there is a rich literature on XML-to-relational storage, none of the existing solutions satisfactorily addresses the problem of storing fuzzy XML data in RDBMs. In this paper, we study the methodology of storing and querying fuzzy XML data in relational databases. In particular, we present an edge-based approach to shred fuzzy XML data into relational data. The unique feature of our approach is that no schema information is required for our data storage. On this basis, we present a generic approach to translate path expression queries into SQL for processing XML queries.     

Fuzzy relations and fuzzy relational databases

This paper deals with the connections existing between fuzzy set theory and fuzzy relational databases. Our new result dealing with fuzzy relations is how to calculate the greatest lower bound (glb) of two similarity relations. Our main contributions in fuzzy relational databases are establishing from fuzzy set theory what a fuzzy relational database should be (the result is both surprising and elegant), and making fuzzy relational databases even more robust.Our work in fuzzy relations and in fuzzy databases had led us into other interesting problems—two of which we mention in this paper. The first is primarily mathematical, and the second provides yet another connection between fuzzy set theory and artificial intelligence. In understanding similarity relations in terms of other fuzzy relations and in making fuzzy databases more robust, we work with closure and interior operators; we present some important properties of these operators. In establishing the connection between fuzzy set theory and artificial intelligence, we show that an abstraction on a set is in fact a partition on the set; that is, an abstraction defines an equivalence relation on the underlying set.     

Type-2 fuzzy variables and their arithmetic

This paper proposes an axiomatic framework from which we develop the theory of type-2 (T2) fuzziness, called fuzzy possibility theory. First, we introduce the concept of a fuzzy possibility measure in a fuzzy possibility space (FPS). The fuzzy possibility measure takes on regular fuzzy variable (RFV) values, so it generalizes the scalar possibility measure in the literature. One of the interesting consequences of the FPS is that it leads to a new definition of T2 fuzzy set on the Euclidean space $\Re^m,This paper proposes an axiomatic framework from which we develop the theory of type-2 (T2) fuzziness, called fuzzy possibility theory. First, we introduce the concept of a fuzzy possibility measure in a fuzzy possibility space (FPS). The fuzzy possibility measure takes on regular fuzzy variable (RFV) values, so it generalizes the scalar possibility measure in the literature. One of the interesting consequences of the FPS is that it leads to a new definition of T2 fuzzy set on the Euclidean space ?^m,\Re^m, which we call T2 fuzzy vector, as a map to the space instead of on the space. More precisely, we define a T2 fuzzy vector as a measurable map from an FPS to the space ?^m\Re^m of real vectors. In the current development, we are suggesting that T2 fuzzy vector is a more appropriate definition for a T2 fuzzy set on ?^m.\Re^m. In the literature, a T2 fuzzy set is usually defined via its T2 membership function, whereas in this paper, we obtain the T2 possibility distribution function as the transformation of a fuzzy possibility measure from a universe to the space ?^m\Re^m via T2 fuzzy vector. Second, we develop the product fuzzy possibility theory. In this part, we give a general extension theorem about product fuzzy possibility measure from a class of measurable atom-rectangles to a product ample field, and discuss the relationship between a T2 fuzzy vector and T2 fuzzy variables. We also prove two useful theorems about the existence of an FPS and a T2 fuzzy vector based on the information from a finite number of RFV-valued maps. The two results provide the possible interpretations for the concepts of the FPS and the T2 fuzzy vector, and thus reinforce the credibility of the approach developed in this paper. Finally, we deal with the arithmetic of T2 fuzzy variables in fuzzy possibility theory. We divide our discussion into two cases according to whether T2 fuzzy variables are defined on single FPS or on different FPSs, and obtain two theorems about T2 fuzzy arithmetic.     

Fuzzy database query languages and their relational completenesstheorem

Two fuzzy database query languages are proposed. They are used to query fuzzy databases that are enhanced from relational databases in such a way that fuzzy sets are allowed in both attribute values and truth values. A fuzzy calculus query language is constructed based on the relational calculus, and a fuzzy algebra query language is also constructed based on the relational algebra. In addition, a fuzzy relational completeness theorem such that the languages have equivalent expressive power is proved     

Generalizing database relational algebra for the treatment of incomplete or uncertain information and vague queries

This paper deals with relational databases which are extended in the sense that fuzzily known values are allowed for attributes. Precise as well as partial (imprecise, uncertain) knowledge concerning the value of the attributes are represented by means of [0,1]-valued possibility distributions in Zadeh's sense. Thus, we have to manipulate ordinary relations on Cartesian products of sets of fuzzy subsets rather than fuzzy relations. Besides, vague queries whose contents are also represented by possibility distributions can be taken into account. The basic operations of relational algebra, union, intersection, Cartesian product, projection, and selection are extended in order to deal with partial information and vague queries. Approximate equalities and inequalities modeled by fuzzy relations can also be taken into account in the selection operation. Then, the main features of a query language based on the extended relational algebra are presented. An illustrative example is provided. This approach, which enables a very general treatment of relational databases with fuzzy attribute values, makes an extensive use of dual possibility and necessity measures.     

Fuzzy rough set based attribute reduction for information systems with fuzzy decisions

Fuzzy rough set is a generalization of crisp rough set to deal with data sets with real value attributes. A primary use of fuzzy rough set theory is to perform attribute reduction for decision systems with numerical conditional attribute values and crisp (symbolic) decision attributes. In this paper we define inconsistent fuzzy decision system and their reductions, and develop discernibility matrix-based algorithms to find reducts. Finally, two heuristic algorithms are developed and comparison study is provided with the existing algorithms of attribute reduction with fuzzy rough sets. The proposed method in this paper can deal with decision systems with numerical conditional attribute values and fuzzy decision attributes rather than crisp ones. Experimental results imply that our algorithm of attribute reduction with general fuzzy rough sets is feasible and valid.     

A ranking method for multiple attribute decision-making problems based on the possibility degrees of trapezoidal intuitionistic fuzzy numbers

To solve multiple attribute decision-making problems with attribute values or decision values characterized by trapezoidal intuitionistic fuzzy numbers (TIFNs), we define a trapezoidal intuitionistic fuzzy induced ordered weighted arithmetic averaging (TIFIOWA) operator, which is an extension of the induced ordered weighted arithmetic averaging operator. We derive and prove some related properties and conclusions of the TIFIOWA operator. To compare the TIFNs, we define possibility degrees of the TIFNs. Based on the possibility degrees of the TIFNs and the TIFIOWA operator, we construct a new method to determine the order of alternatives in multiple attribute decision making and to choose the best alternative. Finally, a numerical example shows that the developed method is feasible and effective.     

Generalization of strategies for fuzzy query translation in classical relational databases

Users of information systems would like to express flexible queries over the data possibly retrieving imperfect items when the perfect ones, which exactly match the selection conditions, are not available. Most commercial DBMSs are still based on the SQL for querying. Therefore, providing some flexibility to SQL can help users to improve their interaction with the systems without requiring them to learn a completely novel language. Based on the fuzzy set theory and the α-cut operation of fuzzy number, this paper presents the generic fuzzy queries against classical relational databases and develops the translation of the fuzzy queries. The generic fuzzy queries mean that the query condition consists of complex fuzzy terms as the operands and complex fuzzy relations as the operators in a fuzzy query. With different thresholds that the user chooses for the fuzzy query, the user’s fuzzy queries can be translated into precise queries for classical relational databases.     

On the primitivity of the division of fuzzy relations

 Allowing for flexible queries enables database users to express preferences inside elementary conditions and priorities between conditions. The division is one of the algebraic operators defined in order to query regular databases. This operation aims at the selection of A-elements which are connected with (at least) a given subset of B-elements, e.g., the stores which ordered all the items supplied by a given manufacturer. It is mainly used in the framework of the relational model of data, although it makes sense in object-oriented databases as well. In the relational context, the division is a non-primitive operation which may be expressed in terms of other operations, namely projection, Cartesian product and set difference. When fuzzy predicates appear, this operator needs to be extended to fuzzy relations and this requires the replacement of the usual implication by a fuzzy one. This paper proposes two types of meaning of the extended division and it investigates the issue of the primitivity of the extended operation (i.e., if the division of fuzzy relations is expressible in terms of other operations). The final objective is to decide whether this operator is necessary or not for the purpose of flexible querying and to help the design of a query language supporting flexible queries, among which those conveying a division of fuzzy relations.     

SQLf: a relational database language for fuzzy querying

An important issue in extending database management systems functionalities is to allow the expression of imprecise queries to enable these systems to satisfy the user needs more closely. This paper deals with imprecise querying of regular relational databases. The basic idea is to extend an existing query language, namely SQL. In this context, two important points must be considered: one concerns the integration in the extended language of many propositions that have been made elsewhere, in particular those concerning fuzzy aggregation operators; and the second point is to know whether the equivalences which are valid in SQL still hold in the extended language. Both these topics are investigated in this paper     

Implementation of fuzzy classification in relational databases using conventional SQL querying

In this paper, a framework for implementing fuzzy classifications in information systems using conventional SQL querying is presented. The fuzzy classification and use of conventional SQL queries provide easy-to-use functionality for data extraction similar to the conventional non-fuzzy classification and SQL querying. The developed framework can be used as data mining tool in large information systems and easily integrated with conventional relational databases. The benefits of using the presented approach include more flexible data analysis and improvement of information presentation at the report generation phase. To confirm the theory, a prototype was developed based on the stored procedures and database extensions of Microsoft SQL Server 2000.