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.     

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.     

Handling indefinite and maybe information in logical fuzzy relational databases

In this article, fuzzy set theory uses an extension of the classical logical relational database model. A logical fuzzy relational database model was developed with the aim of manipulating imprecise information and adding deduction capabilities to the database system. The essence of this work is the detailed discussion on fuzzy definite, fuzzy indefinite, and fuzzy maybe information and the development of an information theoretical approach of query evaluation on the logical fuzzy relational database. We define redundancies among fuzzy tuples and the operator of their removal. A complete set of fuzzy relational operations in relational algebra and the calculus of linguistically quantified propositions are included also. © 2004 Wiley Periodicals, Inc.     

Normalization and lossless join decomposition of similarity‐based fuzzy relational databases

Fuzzy relational database models generalize the classical relational database model by allowing uncertain and imprecise information to be represented and manipulated. In this article, we introduce fuzzy extensions of the normal forms for the similarity‐based fuzzy relational database model. Within this framework of fuzzy data representation, similarity, conformance of tuples, the concept of fuzzy functional dependencies, and partial fuzzy functional dependencies are utilized to define the fuzzy key notion, transitive closures, and the fuzzy normal forms. Algorithms for dependency preserving and lossless join decompositions of fuzzy relations are also given. We include examples to show how normalization, dependency preserving, and lossless join decomposition based on the fuzzy functional dependencies of fuzzy relation are done and applied to some real‐life applications. © 2004 Wiley Periodicals, Inc. Int J Int Syst 19: 885–917, 2004.     

Handling fuzzy information in 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 paper, we focus on such fuzzy relational databases. We classify two kinds of fuzzy data redundancies and define their removal. On this basis, we define fuzzy relational operations in relational algebra, which, being similar to the conventional relational databases, are complete and sound. In particular, we investigate fuzzy querying strategies and give the form of fuzzy querying with SQL. © 2002 Wiley Periodicals, Inc.     

Rewriting rules to permeate complex similarity and fuzzy queries within a relational database system

In recent years, the availability of complex data repositories (e.g., multimedia, genomic, semistructured databases) has paved the way to new potentials as to data querying. In this scenario, similarity and fuzzy techniques have proven to be successful principles for effective data retrieval. However, most proposals are domain specific and lack of a general and integrated approach to deal with generalized complex queries, i.e., queries where multiple conditions are expressed, possibly on complex as well as on traditional data. To overcome such limitations, much work has been devoted to the development of middleware systems to support query processing on multiple repositories. On a similar line, We present a formal framework to permeate complex similarity and fuzzy queries within a relational database system. As an example, we focus on multimedia data, which is represented in an integrated view with common database data. We have designed an application layer that relies on an algebraic query language, extended with MM-tailored operators, and that maps complex similarity and fuzzy queries to standard SQL statements that can be processed by a relational database system, exploiting standard facilities of modern extensible RDBMS. To show the applicability of our proposal, we implemented a prototype that provides the user with rich query capabilities, ranging from traditional database queries to complex queries gathering a mixture of Boolean, similarity, and fuzzy predicates on the data.     

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     

Fuzzy rough set techniques for uncertainty processing in a relational database

This paper concerns the modeling of imprecision, vagueness, and uncertainty in databases through an extension of the relational model of data: the fuzzy rough relational database, an approach which uses both fuzzy set and rough set theories for knowledge representation of imprecise data in a relational database model. The fuzzy rough relational database is formally defined, along with a fuzzy rough relational algebra for querying. Comparisons of theoretical properties of operators in this model with those in the standard relational model are discussed. An example application is used to illustrate other aspects of this model, including a fuzzy entity–relationship type diagram for database design, a fuzzy rough data definition language, and an SQL‐like query language supportive of the fuzzy rough relational database model. This example also illustrates the ease of use of the fuzzy rough relational database, which often produces results that are better than those of conventional databases since it more accurately models the uncertainty of real‐world enterprises than do conventional databases through the use of indiscernibility and fuzzy membership values. ©2000 John Wiley & Sons, Inc.     

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.     

Functional dependencies with null values, fuzzy values, and crispvalues

Based on the concepts of the semantic proximity, we present a definition of the fuzzy functional dependency, We show that the inference rules for fuzzy functional dependencies, which are the same as Armstrong's axioms for the crisp case, are correct and complete. We also show that dependent constraints with dull values constitute a lattice. Functional dependencies in classical relational databases and null functional dependencies can be viewed as a special case of fuzzy functional dependencies. By applying the unified functional dependencies to the relational database design, we can represent the data with fuzzy values, null values and crisp values under relational database management systems, By using fuzzy functional dependencies, we can compress the range of a fuzzy value and make this fuzzy value “clearer”     

Formal translation from fuzzy EER model to fuzzy XML model

XML has been the de facto standard of data representation and exchange over the Web. In addition, fuzzy data are inherent in the real-world applications. Although fuzzy information has been extensively investigated in the context of relational database model, the classical relational database model and its fuzzy extension to date do not satisfy the need of modeling and processing complex objects with imprecision and uncertainty on the Web. Based on fuzzy sets, this paper concentrates on fuzzy information modeling in the EER (enhanced entity-relationship or extended entity-relationship) model and the fuzzy XML model. In particular, the formal approach to mapping the fuzzy EER model to the fuzzy DTD (document type definition) model is developed.     

Reliability assessment for fuzzy multi-state systems

Fuzzy multi-state system (FMSS) is defined as a multi-state system (MSS) consisting of multi-state elements (MSE) whose performance rates and transition intensities are presented as fuzzy values. Due to the lack, inaccuracy or fluctuation of data, it is oftentimes impossible to evaluate the performance rates and transition intensities of MSE with precise values. This is true especially in continuously degrading elements that are usually simplified to MSE for computation convenience. To overcome these challenges in evaluating the behaviour of MSS, fuzzy theory is employed to facilitate MSS reliability assessment. Given the fuzzy transition intensities and performance rates, the state probabilities of MSE and MSS are also fuzzy values. A fuzzy continuous-time Markov model with finite discrete states is proposed to assess the fuzzy state probability of MSE at any time instant. The universal generating function with fuzzy state probability function and performance rate is applied to evaluate fuzzy state probability of MSS in accordance with the system structure. A modified FMSS availability assessment approach is introduced to compute the system availability under the fuzzy user demand. In order to obtain the membership functions of the indices of interest, parametric programming technique is employed according to Zadeh's extension principle. The effectiveness of the proposed method is illustrated and verified via reliability assessment of a multi-state power generation system.     

Analyzing fuzzy risk based on similarity measures between interval-valued fuzzy numbers

In this paper, we present a new method for handling fuzzy risk analysis problems based on the proposed new similarity measure between interval-valued fuzzy numbers. First, we present a new similarity measure between interval-valued fuzzy numbers. It considers the degrees of closeness between interval-valued fuzzy numbers on the X-axis and the degrees of differences between the shapes of the interval-valued fuzzy numbers on the X-axis and the Y-axis, respectively. We also prove three properties of the proposed similarity measure. Then, we make an experiment to compare the experimental results of the proposed method with the existing similarity measures between interval-valued fuzzy numbers. The proposed method can overcome the drawbacks of the existing methods. Finally, based on the proposed similarity measure between interval-valued fuzzy numbers, we present a new fuzzy risk analysis algorithm for dealing with fuzzy risk analysis problems. Because the proposed method allows the evaluating values of sub-components to be represented by interval-valued fuzzy numbers, it is more flexible than Chen and Chen’s method (2003).     

区间值关系数据库上模糊关联规则的预测方法

讨论了区间值关系数据库上模糊关联规则的挖掘算法与预测方法。采用一种比RFCM算法省时的FCMdd算法将记录在属性的取值划分成若干个模糊集,并提出区间值关系数据库上模糊关联规则的挖掘算法。仿真实例说明挖掘算法能够通过挖掘有意义的模糊关联规则来发现区间值关系数据库中蕴涵的关联性。区间值关系数据库上模糊关联规则的预测方法改进了标准可加性模型,并通过遗传算法调整模糊关联规则中三角模糊数的参数来提高预测的精度。     

A new approach for fuzzy risk analysis based on similarity measures of generalized fuzzy numbers

In this paper, we present a new method for fuzzy risk analysis based on similarity measures between generalized fuzzy numbers. First, we present a new similarity measure between generalized fuzzy numbers. It combines the concepts of geometric distance, the perimeter and the height of generalized fuzzy numbers for calculating the degree of similarity between generalized fuzzy numbers. We also prove some properties of the proposed similarity measure. We make an experiment to use 15 sets of generalized fuzzy numbers to compare the experimental results of the proposed method with the existing similarity measures. The proposed method can overcome the drawbacks of the existing similarity measures. Based on the proposed similarity measure between generalized fuzzy numbers, we present a new fuzzy risk analysis algorithm for dealing with fuzzy risk analysis problems, where the values of the evaluating items are represented by generalized fuzzy numbers. The proposed method provides a useful way to deal with fuzzy risk analysis problems.     

Fuzzy query translation for relational database systems

The paper presents a new method for fuzzy query translation based on the alpha-cuts operations of fuzzy numbers. This proposed method allows the retrieval conditions of SQL queries to be described by fuzzy terms represented by fuzzy numbers. It emphasizes friendliness and flexibility for inexperienced users. The authors have implemented a fuzzy query translator to translate user's fuzzy queries into precise queries for relational database systems. Because the proposed method allows the user to construct his fuzzy queries intuitively and to choose different retrieval threshold values for fuzzy query translation, the existing relational database systems will be more friendly and more flexible to the users.     

Relational similarity-based model of data part 1: foundations and query systems

We present a general rank-aware model of data which supports handling of similarity in relational databases. The model is based on the assumption that in many cases it is desirable to replace equalities on values in data tables by similarity relations expressing degrees to which the values are similar. In this context, we study various phenomena which emerge in the model, including similarity-based queries and similarity-based data dependencies. Central notion in our model is that of a ranked data table over domains with similarities which is our counterpart to the notion of relation on relation scheme from the classical relational model. Compared to other approaches which cover related problems, we do not propose a similarity-based or ranking module on top of the classical relational model. Instead, we generalize the very core of the model by replacing the classical, two-valued logic upon which the classical model is built by a more general logic involving a scale of truth degrees that, in addition to the classical truth degrees 0 and 1, contains intermediate truth degrees. While the classical truth degrees 0 and 1 represent nonequality and equality of values, and subsequently mismatch and match of queries, the intermediate truth degrees in the new model represent similarity of values and partial match of queries. Moreover, the truth functions of many-valued logical connectives in the new model serve to aggregate degrees of similarity. The presented approach is conceptually clean, logically sound, and retains most properties of the classical model while enabling us to employ new types of queries and data dependencies. Most importantly, similarity is not handled in an ad hoc way or by putting a “similarity module” atop the classical model in our approach. Rather, it is consistently viewed as a notion that generalizes and replaces equality in the very core of the relational model. We present fundamentals of the formal model and two equivalent query systems which are analogues of the classical relational algebra and domain relational calculus with range declarations. In the sequel to this paper, we deal with similarity-based dependencies.     

A new method for computing fuzzy functional dependencies in relational database systems

In this paper, we present a new method for computing fuzzy functional dependencies between attributes in fuzzy relational database systems. The method is based on the use of fuzzy implications. A literature analysis has shown that there is no algorithm that would enable the identification of attribute relationships in fuzzy relational schemas. This fact was the motive for development a new methodology in the analysis of fuzzy functional dependencies over a given set of attributes. Solving this, not so new problem, is not only research challenge having theoretical importance, but it also has practical significance. Possible applications of the proposed methodology include GIS, data mining, information retrieval, reducing data redundancy in fuzzy relations through implementation of logical database model, estimation of missing values etc.