Application of Conditional Probability in Constructing Fuzzy Functional Dependency

Intan, Rolly (2015) Application of Conditional Probability in Constructing Fuzzy Functional Dependency. Petra Press. ISBN 979-602-71225-2-9

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Abstract

In real-world application, information is mostly imprecise or ambiguous. Therefore, the motivation of extending classical (crisp) relational database [Codd, 1970] to fuzzy relational database by Buckles and Petry [1982] stems from the need to be able to process and represent vague, imprecise and partially known (incomplete) information. The concept of fuzzy relational database proposed by Buckles and Petry [1982] are necessary to be extended to a more generalized concept of fuzzy relational database, since the data value in domain attributes of the fuzzy relational model is still considered as a subset of atomic data. In this case, each data value stored in the more generalized concept of fuzzy relational database is considered as a fuzzy subset. An important feature of a relational database is to express constraints in sense of relation of data, known as integrity constraints (ICs). For instance, if a relational database contains information about student ID-number, course, unit, term and grade, some constrains such as: “A given ID-number, course, and term give a unique grade”, “number of courses are no more than 6 courses for a given ID-number and term” and “total units are no more than 16 for a given ID-number and term” might be hold. Many types of integrity constraints have been provided since 1970s along with the Codd’s relational database, such as multi-valued dependency proposed by Fagin [1977], join dependency [Nicolas, 1978] [Rissanen, 1978], etc. Among them, functional depen¬dencies (FDs) [Berstein, Swenson, & Tsichritzis, 1975] are one of the most important and widely used in database design. As we extend the classical relational database to fuzzy relational database, it would be necessary to consider integrity constraints that may involve fuzzy value. In fact, fuzzy integrity constraints, such as: “The higher an education someone has, the higher salary he should get”, “almost equally qualified employees should get more or less equal salary” will arise naturally and usefully in real-world application. Therefore, the objective of extending FDs to fuzzy functional dependencies (FFDs) is in necessary to apply FDs in fuzzy relational database [Intan, Mukaidono, 2000a, 2003, 2004]. Various definitions and the notion of a fuzzy functional dependency have been devised since 1988. Among them, Raju and Majumdar [1988] defined FFD based on the membership function of the fuzzy relation; Tripathy, [1990] proposed definition of the FFD in terms of fuzzy Hamming weight; Kiss, [1991] constructed FFD using weighted tuples; Chen [1995], Cubero [1994] and W. Liu [1992,1993] introduced definition of the FFD based on the equality of two possibility distributions, and they used a certain type of implication and expression of cut off; Liao [1997] gave design of the FFD by introducing semantic proximity. In this book, some properties of conditional probability and its relation with fuzzy sets are studied and discussed as an alternative concept to measure similarity of fuzzy labels. Even it could be understood that interpretation of numerical value between fuzzy sets and probability measures are philosophically distinct, basic operations, such as, intersection and union of two fuzzy values can be interpreted as maximum intersection and minimum union of two events. Considering this reason, it is necessary to define three approximate conditional probabilities of two fuzzy events based on minimum, independent and maximum probability intersection between two (fuzzy) events. Moreover, conditional probability of two fuzzy events can be interpreted as probabilistic matching of two fuzzy sets [Baldwin, Martin, Pilsworth, 1995], [Baldwin, Martin, 1996] and as basis of getting similarity of two fuzzy sets and constructing equivalence classes inside their domain attribute. By using this property and Cartesian product operation of fuzzy sets, a concept of fuzzy functional dependency (FFD) is proposed and defined to express integrity constraints that may involve fuzzy value, called fuzzy integrity constraints. It can be proved that the concept of FFD satisfies classical/ crisp relational database by example. Also, inference rules which are similar to Armstrong’s Axioms [Armstrong, 1974] for the FFDs are both sound and complete. Next, a concept of partial FFD is introduced to express the fact as usually found in data that a given attribute domain X do not determine Y completely, but in the partial area of X, it might determine Y. For instance, in the relation between two domains student’s name and student’s ID, student’s ID determines student’s name. It means a given student’s ID certainly gives a unique student’s name. On the other hand, a given student’s name may give more than one student’s ID because it is possible to have more than one student who has the same name. However, in a partial area of student’s name where some students have unique names, student’s name can be considered to determine student’s ID. In addition, approximate data reduction and projection of relations are investigated in order to get relation among the partitions of data values. Here, data values might be considered as crisp as well as fuzzy data. Finally, this book discusses the application of FFDs in constructing fuzzy query relation for query data and approximate natural join of two or more fuzzy query relations in the framework of extended query system [Intan, Mukaidono, 2001, 2002]. The structure of the book is following. In Chapter 2, some basic definitions and notations, such as conditional probability, classical relational database, functional dependency, fuzzy sets, transformation fuzzy set and probability, and fuzzy relational database are recalled. Chapter 3 firstly introduces conditional probability of two fuzzy sets based on the possibility theory [Baldwin, Martin, Pilsworth, 1995]. The next, it provides three approximate interpretations in constructing conditional probability of two fuzzy events (sets) based on minimum, independent and maxi¬mum probability intersection between two (fuzzy) events [Intan, Mukaidono, 2004]. Chapter 4 is devoted to the construction of FFDs based on the concept of conditional probability relations. It is proved that inference rules (Reflexivity, Augmentation and Transitivity) which are similar to Armstrong’s Axioms for FFDs are both sound and complete. A special attention will be given to partial FFD in order to find relation between two partial areas of two attribute domains [Intan, Mukaidono, 2004]. In Chapter 5, the application of FFDs in approximating data reduction and query data are presented [Intan, Mukaidono, 2001, 2002]. This chapter also discussed two other operations called projection and join operations in the relation to approximate data reduction and extended query system respectively [Intan, Mukaidono, 2004]. This book will be closed by summary including suggestion for future work in Chapter 6.

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