Brief Summary and Future Trends

Fuzzy logic (Chapter I) allows us to bring the operation of information systems closer to the working methods of humans. People frequently deal with fuzzy concepts (for example, terms such as “almost all,” “the majority,” “approximately 8,” “high,” or “low”), which include a certain vagueness or uncertainty and which traditional information systems do not understand and therefore cannot use.

Some Applications of Fuzzy Databases With FSQL

The applications of databases are immense. In almost all of them, the advantages of the fuzzy databases can be applied, exploiting their innovative features and possibilities without losing usefulness. Even the model presented here permits an easy use of those advantages in already existing traditional databases. The Type 1 fuzzy attributes are traditional attributes that admit fuzzy queries on them (by using labels, approximate values, fuzzy comparators, etc.). Imprecise information is a common phenomenon in any context, so it is not unusual to receive information in an incomplete or inexact way. In traditional databases, if information other than precise information exists, the value NULL is stored, preventing the storage of any known information, because the facts are not precise.

Mapping Fuzzy EER Model Concepts to Relations

This chapter shows the transformation of the FuzzyEER model to a logical design by using relational databases. The FuzzyEER-to-Relational mapping algorithm is based on the “classical” EER-to-Relational mapping algorithm, published in Elmasri and Navathe (2000) and summarized in the first section of this chapter, but other versions are very similar (De Miguel, Piattini, & Marcos, 1999; Silverschatz, Korth, & Sudarshan, 2002). The FuzzyEER-to-Relational mapping algorithm includes additional rules for mapping fuzzy concepts. The following sections translate the FuzzyEER concepts, that is, the definitions in Chapter IV, to the FIRST-2 schema, which was exposed in Chapter V. Thus, this chapter relates Chapter IV with Chapter V, obtaining a fuzzy relational database. In addition, we need a comprehensive fuzzy database language with statements for data definition, query, and update. This language is FSQL (Fuzzy SQL), and we describe it in Chapter VII.

FSQL

The SQL language was essentially developed by Chamberlin and Boyce (1974) and Chamberlin et al. (1976). In 1986, the American National Standard Institute (ANSI) and the International Standards Organization (ISO) published the standard SQL-86 or SQL1 (ANSI, 1986). In 1989, an extension of the SQL standard, called SQL-89, was published, and SQL2 or SQL-92 was published in 1992 (ANSI, 1992). SQL2 basically provided new types, constraints (such as checks or unique predicates), it supported subqueries in UPDATE and DELETE operations, and in the FROM clause, operator IN, ANY and ALL, CASE constructor, JOIN, UNION, INTERSECT and EXCEPT operators and the modification of base table through views. In the latest version of SQL standard, SQL 2003, major improvements have been made in a number of key areas. Firstly, it has additional object-relational features, which were first introduced in SQL-1999. Secondly, SQL 2003 standard revolutionizes SQL with comprehensive OLAP features and data-mining applications. Thirdly, SQL 2003 integrates popular XML standards into SQL (SQL/XML). Finally, numerous improvements have been made throughout the SQL 2003 standard to refine existing features.

FuzzyEER

In this chapter we present the FuzzyEER Model, which is an extension of the EER Model with fuzzy semantics and notations. The Entity-Relationship Model was introduced by Chen (1976). Since then, numerous modifications and extensions of its modeling capabilities have been suggested. We will mainly use the approach by Elmasri and Navathe (2000) because it is very popular, general, and has an international scope. With regard to the fuzzy attributes, the following aspects will be defined in the FuzzyEER Model: imprecise attributes, fuzzy attributes associated to one or more attributes or with an independent meaning, as well as degrees of fuzzy membership to the model itself. Furthermore, the following concepts will also be defined: fuzzy aggregation, fuzzy entity, weak fuzzy entity, fuzzy relationship, and defined specialization with fuzzy degrees. Fuzzy constraints are very important, and we review them in this chapter as well.

Representation of Fuzzy Knowledge in Relational Databases

The Relational Model was developed by E.F. Codd of IBM and published in 1970. It is currently the most used and has been a milestone in the history of databases, revolutionizing the market. In fact, relational databases have been the most widespread of all databases. On a theoretical level, many Fuzzy Relational Database models (Chapter II), which are based on the relational model, extend this so that vague and uncertain information can be stored and/or treated with or without fuzzy logic (see Chapter I). The FuzzyEER Model (see Chapter IV) is an extension of the EER Model for creating conceptual schemas with fuzzy semantics and notations. This extension is a good eclectic synthesis between different models (see Chapter III) and provides new and useful definitions: fuzzy attributes, fuzzy entities, fuzzy relationships, fuzzy specializations, and so forth.

State of the Art in Fuzzy Database Modeling

On occasion, the term imprecision embraces several meanings that we should differentiate. For example, as you saw in Chapter II, the information you have may be incomplete or fuzzy (diffuse or vague), you may not know whether it is certain (uncertainty), perhaps you are totally ignorant of the information (unknown), you may know that the information cannot be applied to a specific entity (undefined), or you may not even know whether the data can be applied to the entity in question (total ignorance or a value of null) (Umano & Fukami, 1994). Each of these terms depends on the context in which it is applied. The management of uncertainty in database systems is a very important problem (Motro, 1995), as the information is often vague. Motro states that fuzzy information is content-dependent, and he classifies it as follows: • Uncertainty: It is impossible to determine whether the information is true or false. For example, “John may be 38 years old.” • Imprecision: The information available is not specific enough. For example, “John may be between 37 and 43 years old,” “John is 34 or 43 years old” (disjunction), “John is not 37 years old” (negative), or even a simple unknown. • Vagueness: The model includes elements (predicates or quantifiers) that are inherently vague, for example, “John is in his early years” or “John is at the end of his youth.” However, after these concepts have been defined, this case would match the previous one (imprecision). • Inconsistency: It contains two or more pieces of information that cannot be true at the same time. For example, “John is 37 and 43 years old, or he is 35 years old”; this is a special case of disjunction. • Ambiguity: Some elements of the model lack complete semantics (or a complete meaning). For example, “It is unclear whether the salaries are annual or monthly.”

Fuzzy Database Approaches

Both the problem of representation and the treatment of imprecise information have been widely discussed. Many references can be found in the corresponding bibliography. Nevertheless, all the known models aimed at solving this problem have their own advantages, disadvantages, and constraints. The term imprecision encompasses various meanings, which might be interesting to highlight. It alludes to the facts that the information available can be incomplete, that we don’t know whether the information is true (uncertainty), that we are totally unaware of the information (unknown), or that such information is not applicable to a given entity (undefined). Sometimes these meanings are not disjunctive and can be combined in certain types of information. This chapter deals with the main published models aimed at solving the problem of representation and treatment of imprecise information in relational databases. This problem is not trivial, because it requires relations structure modification, and thus, the operations on these relations also need to be modified. To allow the storage of imprecise information and the making of an inaccurate query of such information, a wide variety of case studies that do not occur in the classic model, without imprecision, is required.

Introduction to Fuzzy Logic

This book mixes concepts of different areas of knowledge or technologies, such as databases, system architecture design, SQL language, programming concepts and logic, mathematics, and so forth. These concepts are introduced where they correspond, although we have not intended this book to be an introduction to databases or information systems. An important part of this book utilizes the fuzzy logic. For that reason we will begin by introducing some basic concepts of the theory of fuzzy sets as well as the notation used in this book. In this summary we will focus on the semantic aspects and those of representation associated with this important theoretical tool. In written sources we can find a large number of papers dealing with this theory, which was first introduced by L.A. Zadeh1 in 1965 (Zadeh, 1965). A compilation of some of the most interesting articles published by Zadeh on the theme can be found in Yager et al. (1987). Dubois and Prade (1980, 1988) and Zimmerman (1991) bring together the most important aspects behind the theory of fuzzy sets and the theory of possibility.