Rivero L.Encyclopedia of database technologies and applications.2006

Z.M.Ma

Northeastern University, China

INTRODUCTION

A major goal for database research has been the incorporation of additional semantics into the data model. Classical data models often suffer from their incapability of representing and manipulating imprecise and uncertain information that may occur in many real-world applications. Since the early 1980s, Zadeh’s fuzzy logic (Zadeh, 1965) has been used to extend various data models. The purpose of introducing fuzzy logic in database modeling was to enhance the classical models such that uncertain and imprecise information can be represented and manipulated. This resulted in numerous contributions, mainly with respect to the popular relational model or to some related form of it.

Also, rapid advances in computing power have brought opportunities for databases in emerging applications in CAD/CAM (Computer-Aided Design/Computer-Aided Manufacturing), multimedia, and geographic information systems (GIS). These applications characteristically require the modeling and manipulation of complex objects and semantic relationships. It proved that the objectoriented paradigm lends itself extremely well to the requirements. Because the classical relational database model and its extension of fuzziness do not satisfy the need of modeling complex objects with imprecision and uncertainty, many current researches have concentrated on fuzzy object-oriented database models to deal with complex objects and uncertain data together.

BACKGROUND

Database modeling can be carried out at two different levels: conceptual data modeling and database modeling. Therefore, we have conceptual data models (e.g., ER/ EER—Entity-Relationship/Extended Entity-Relationship and UML) and logical database models (relational databases and object-oriented databases). Logical database models are often created through mapping conceptual data models into logical database models. This conversion is called conceptual design of databases.

In order to deal with imprecise and uncertain information in database modeling, fuzzy set has been applied. Let U be a universe of discourse, then a fuzzy value on U

is characterized by a fuzzy set F in U. A membership function F: U → [0,1] is defined for the fuzzy set F, whereF (u), for each u U, denotes the degree of membership of u in the fuzzy set F. Thus the fuzzy set F is described as follows.

F = { (u1)/u1, (u2)/u2, ..., (un)/un}

When µF (u) is viewed to be a measure of the possibility that a variable X has the value u in this approach, where X takes values in U, a fuzzy value is described by a possibility distribution πX (Zadeh, 1978).

πX = {πX (u1)/u1, πX (u2)/u2, ..., πX (un)/un}

Here, πX (ui), ui U, denotes the possibility that ui is true. Let πX and F be the possibility distribution representation and the fuzzy set representation for a fuzzy value, respectively. It is apparent that πX = F is true (Raju & Majumdar, 1988).

Also, fuzzy data can be represented by similarity relations in domain elements (Buckles & Petry, 1982), in which the fuzziness comes from the similarity relations between two values in a universe of discourse, not from the status of an object itself. Similarity relations are thus used to describe the degree similarity of two values from the same universe of discourse. A similarity relation Sim on the universe of discourse U is a mapping: U × U → [0, 1] such that:

b.for x, y U, Sim (x,y) = Simi (y,x), and

(symmetry)

c.for x,y,z U,Sim(x,z)≥maxy (min(Sim(x,y),Sim(y,z))).

(transitivity)

MAJOR ISSUES AND SOLUTIONS

Fuzzy Relational Databases

Fuzzy information has been extensively investigated in the context of the relational databases. The followings are some major issues in current studies of fuzzy relational databases (Ma & Mili, 2002b):

Fuzzy Database Modeling

Fuzzy Relational Database Models

One can find several kinds of fuzzy relational database models. One of the fuzzy relational data models is based on similarity relations (Buckles & Petry, 1982), or proximity relation (Shenoi & Melton, 1989), or resemblance (Rundensteiner, Hawkes, & Bandler, 1989). The other one is based on possibility distribution (Prade & Testemale, 1984), which can further be classified into two categories: tuples associated with possibilities and attribute valuesrepresented possibility distributions (Raju & Majumdar, 1988). The form of an n-tuple in each of the previously mentioned fuzzy relational model can be expressed, respectively, as:

t = <p1, p2, …, pi, …, pn >, t = <a1, a2, …, ai, …, a n, d> and t = <πA1, πA2, …, πAi, …, πAn>,

where pi Di with Di being the domain of attribute Ai, aiDi, d (0, 1), πAi is the possibility distribution of attribute Ai on its domain Di, and πAi (x), x Di, denotes the possibility that x is the actual value of t [Ai].

It is clear that one can combine two kinds of fuzziness in possibility-based fuzzy relational databases, where attribute values may be possibility distributions and tuples are connected with membership degrees. Such fuzzy relational databases are called possibility-distri- bution-fuzzy relational models (Umano & Fukami, 1994). Another possible extension is to combine possibility distribution and similarity (proximity or resemblance) relation, and the extended possibility-based fuzzy relational databases are hereby proposed (G. Q. Chen, Vandenbulcke, & Kerre, 1992; G. Q. Chen, Kerre, & Vandenbulcke, 1994, 1996; Ma & Mili, 2002; Ma, Zhang, & Ma, 2000; Ma, Zhang, & Mili, 2002), where possibility distribution and resemblance relation arise in a relational databases simultaneously.

Fuzzy Data Integrity Constraints and

Formalizations

Fuzzy data dependencies, mainly including fuzzy functional dependency (FFD) and fuzzy multivalued dependency (FMVD), have extensively been studied in the context of fuzzy relational databases. There are some papers that focus only on FMVD (Bhattacharjee & Mazumdar, 1998; Jyothi & Babu, 1997; Tripathy & Sakena, 1990; ). And some papers focus only on FFD, where we can classify two kinds of papers: The first one focuses on the axiomatization of FFD (G. Q. Chen, Kerre, & Vandenbulcke, 1994, 1995; Cubero & Vila, 1994; Liao, Wang, & Liu, 1999; Liu, 1992, 1993a, 1993b; Saxena & Tyagi, 1995) and the second focuses on the lossless join and decomposition (Bhuniya &

Niyogi, 1993; Bosc & Pivert, 2003; Raju & Majumdar, 1988). The later is the basis on which to implement the F normalization of fuzzy relational databases (G. Q. Chen, Kerre, & Vandenbulcke, 1996). Also, there are some papers that focus both on FFD and FMVD and present

the axiomatization of FFD and FMVD (Liu, 1997; Ma, Zhang, Ma, & Mili,, 2002; Sözat & Yazici, 2001; Yazici & Sözat, 1998).

In addition, fuzzy data dependencies can be applied in data handling. In Bosc, Dubois, and Prade (1998), FFD was used for redundancy elimination. In Intan and Mukaidono (2000), FFD was used for approximate data querying. In Chang and Chen (1998), Liao, Wang, and Liu (1999), and Ma, Zhang, and Mili (2002), FFD was used for fuzzy data compression.

Query and Data Processing

Classical relational databases suffer from a lack of flexibility in query. The given selection condition and the contents of the relation are all crisp. A query is flexible if the following conditions can be satisfied (Bosc & Pivert, 1992):

•A qualitative distinction between the selected tuples is allowed.

•Imprecise conditions inside queries are introduced when the user cannot define his or her needs in a definite way, or when a prespecified number of responses are desired and therefore a margin is allowed to interpret the query.

Here, typically, the former case occurs when the queried relational databases contain incomplete information and the query conditions are crisp, and the later case occurs when the query conditions are imprecise, even if the queried relational databases do not contain incomplete information.

Takahashi presented a fuzzy query language for relational databases (Takahashi, 1991) and fuzzy database query languages and their relational completeness theorem (Takahashi, 1993). Bosc and Lietard (1996) presented the concepts of fuzzy integrals and database flexible querying. Bosc and Pivert (1995) presented a relational database language called SQLf for fuzzy querying. Also, S. M. Chen and Jong (1997) and Y. C. Chen and Chen (2000) presented fuzzy query translation techniques for relational database systems and techniques of fuzzy query processing for fuzzy database systems, respectively. In addition, based on matching strengths of answers in fuzzy relational databases, Chiang, Lin, and Shis (1998) presented a method for fuzzy query processing. Yang et al. (2001) focused on nested fuzzy SQL queries in a fuzzy relational database.

Fuzzy logic techniques have been used in multimedia database querying (Dubois, Prade, & Sedes, 2001). In Kacprzyk, Zadrozny, and Ziokkowski, (1987), a “humanconsistent” database querying system based on fuzzy logic with linguistic quantifiers was presented. Using clustering techniques, Kamel, Hatfield, and Ismail (1990) presented a fuzzy query processing method.

In addition to query processing in fuzzy relational databases, there are also studies focusing on the operations of relational algebra in fuzzy relational databases (Ma & Mili, 2002; Umano & Fukami, 1994). In Zhang and Wang (2000), a type of fuzzy equi-join was defined using fuzzy equality indicators.

Fuzzy Nested Relational Databases

In Yazici et al. (1999), an extended nested relational data model (also known as an NF2 data model) was introduced for representing and manipulating complex and uncertain data in databases, and the extended algebra and the extended SQL-like query language were hereby defined. Also, physical data representation of the model and the core operations that the model provides were also introduced. Ma and Mili (2002) based possibility distribution rather than the similarity relations in Yazici et al. (1999), an extended possibility-based fuzzy nested relational database model was introduced and its algebra is hereby developed.

It should be noted that the NF2 data model is able to handle complex-valued attributes and may be better suited to some complex applications such as office automation systems, information retrieval systems, and expert database systems. But it is difficult for NF2 data model to represent complex relationships among objects and attributes. Some advanced abstracts in data modeling (e.g., class hierarchy, inheritance, superclass/subclass, and encapsulation), which are needed by many real applications, are not supported by the NF2 data model. Therefore, in order to model uncertain data and complexvalued attributes as well as complex relationships among objects, current efforts have focused on conceptual data models and object-oriented databases (OODB) with imprecise and uncertain information.

Fuzzy Object-Oriented Databases

The incorporation of imprecise and uncertain information in object-oriented databases has increasingly received attention where fuzziness is witnessed at the levels of object instances and class hierarchies. Based on similarity relationship, George, Srikanth, Petru, and Buckles, 1996 used the range of attribute values to represent the set of allowed values for an attribute of a given class. Depending on the inclusion of the actual

Fuzzy Database Modeling

attribute values of the given object into the range of the attributes for the class, the membership degrees of an object to a class can be calculated. The weak and strong class hierarchies were defined based on monotone increase or decrease of the membership of a subclass in its superclass. Design and implementation issues in such a fuzzy object-oriented data model were presented in Yazici, George, and Aksoy (1998).

A UFO (i.e., uncertainty and fuzziness in an objectoriented databases model) was proposed by Van Gyseghem and Caluwe (1998) to model fuzziness and uncertainty by means of fuzzy set theory and generalized fuzzy set, respectively. That the behaviour and structure of the object are incompletely defined results in a gradual nature for the instantiation of an object. The partial inheritance, conditional inheritance, and multiple inheritances are permitted in fuzzy hierarchies.

In Lee, Xue, Hsu, and Yang, (1999), an approach to OO (object-oriented) modeling based on fuzzy logic is proposed to formulate imprecise requirements along four dimensions: fuzzy class, fuzzy rules, fuzzy class relationships, and fuzzy associations between classes.

Based on the extension of a graphs-based model object model, a fuzzy object-oriented data model was defined in Bordogna, Pasi, and Lucarella (1999). The notion of strength expressed by linguistic qualifiers was proposed, which can be associated with the instance relationship as well as an object with a class. Fuzzy classes and fuzzy class hierarchies were thus modeled in the OODB. The definition of graph-based operations to select and browse such a fuzzy objectoriented database that manages both crisp and fuzzy information was proposed in Bordogna and Pasi (2001).

Based on possibility theory, vagueness and uncertainty were represented in class hierarchies in Dubois, Prade, and Rossazza (1991), where the fuzzy ranges of the subclass attributes defined restrictions on that of the superclass attributes, and then the degree of inclusion of a subclass in the superclass was dependent on the inclusion between the fuzzy ranges of their attributes. Also based on possibility distribution theory, Ma, Zhang, and Ma (2004) extended under the fuzzy information environment some major notions in ob- ject-oriented databases such as objects, classes, ob- jects–classes relationships, subclass/superclass, and multiple inheritances. A generic model for fuzzy ob- ject-oriented databases and some operations were hereby developed.

In addition to major focus on fuzzy object-oriented database model, there are studies that mainly concentrate on issues related to fuzzy object-oriented databases. Fuzzy types were added into fuzzy object-ori- ented databases to manage vague structures in Marín,

Fuzzy Database Modeling

Vila, and Pons (2000) and Marín, Pons, and Vila (2001). And in Marín, Medina, Pons, Sánchez, and Vila (2003), complex object comparison in a fuzzy context was developed. In Cross (2001, 2003), fuzzy relationships in object models were investigated.

Also, some efforts have been paid to the establishment of a consistent framework for a fuzzy objectoriented model based on the standard for the object data management group (ODMG) object data model (Cross , Caluwe, & Van Gyseghem, 1997). And some special fuzzy object-oriented databases (e.g., fuzzy deductive object-oriented databases; Yazici & Koyuncu, 1997; Koyuncu & Yazici, 2003), fuzzy and probabilistic object bases (Cao & Rossiter, 2003) have been developed. In addition, fuzzy object-oriented database have been applied in some areas such as GIS (Cross & Firat, 2000) and multimedia (Majumdar, Bhattacharya, & Saha, 2002). Concerning most recent research and application issues about fuzzy object-oriented databases, refer to Ma (2005).

Fuzzy Conceptual Data Models and

Fuzzy Database Design

Fuzzy Conceptual Data Models

Based on fuzzy set theory, Zvieli and Chen (1986) introduced three levels of fuzziness in the ER model. At the first level, entity sets, relationships, and attribute sets may be fuzzy, namely, they have a degree of membership in the model. The second level is related to the fuzzy occurrences of entities and relationships. The third level concerns the fuzzy values of attributes of special entities and relationships. By using fuzzy set theory, the fuzzy extensions of several major EER concepts were introduced in G. Q. Chen and Kerre (1998), including superclass/subclass, generalization/specialization, category, and the subclass with multiple superclasses.

In addition to the ER/EER model, the IFO data model (Abiteboul & Hull, 1987) is a mathematically defined conceptual data model that incorporates the fundamental principles of semantic database modeling within a graph-based representational framework. The extensions of the IFO to deal with fuzzy information were proposed in Vila et al., 1996). In (Vila, Cubero, Medina, & Pons, 1996), several types of imprecision and uncertainty, such as the values without semantic representation, the values with semantic representation and disjunctive meaning, the values with semantic representation and conjunctive meaning, and the representation of uncertain information, were incorporated into the attribute domain of the object-based data model. In addition

Conceptual Design of Fuzzy Databases

Traditional databases are generally designed from conceptual data models. By mapping, conceptual data models are converted into database models. It is shown that less research has been done on modeling fuzzy information in the conceptual data model. It is particularly true in developing design methodologies for implementing fuzzy databases.

In Chaudhry, Moyne, & Rundensteiner (1999), the fuzzy relational databases were designed by using the fuzzy ER model proposed in Zvieli and Chen (1986). In Yazici, Buckles, and Petry (1999), based on similarity relations (Buckles & Petry, 1982), the IFO (Abiteboul and Hull, 1987) model was extended to the ExIFO (Extended IFO) model to represent uncertainties at the levels of the attribute, the object, and class. Also, a mapping process to transform the ExIFO model into the fuzzy extended NF2 relations, including uncertain properties that are represented in both models, was also described. In Ma, Zhang, Ma, and Chen (2001), a full fuzzy EER model and the graphical representations were presented and the formal design methodology for fuzzy object-oriented databases from a fuzzy entity-relation- ship model was provided.

FUTURE TRENDS

Compared with fuzzy relational databases, the research on fuzzy conceptual data models and fuzzy object-ori- ented databases is receiving increasing attention (Ma, 2005). In addition, the incorporation of probabilistic uncertain information into fuzzy data models should be interesting. Finally, building FDBMS and developing prototype systems of fuzzy databases are important topics in the area of fuzzy databases.

CONCLUSION

This article reviews fuzzy database modeling technologies, including fuzzy conceptual data models and database models. Concerning fuzzy database models, fuzzy relational databases and fuzzy object-oriented databases are discussed, respectively. In fuzzy relational databases, the issues of data presentation and model structure, data dependencies and formalization, and data query and processing have extensively studied.

A major goal for database research has been the incorporation of additional semantics into the data model. Fuzzy database modeling can support imprecise and uncertain data storage, processing, and retrieval activities related to data management in intelligent information systems.

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KEY TERMS

Conceptual Design of Fuzzy Databases: It refers to the conversion of fuzzy conceptual data models to fuzzy database models. The focus is on developing the rules of mapping fuzzy conceptual data models to fuzzy database models.

Fuzzy Conceptual Data Models: Being the extension of traditional conceptual data models, such as ER/ EER and UML fuzzy conceptual data models, which can model imperfect data and semantic relationships at a high level of data abstraction.

Fuzzy Database Modeling

Fuzzy Database Modeling: In addition to the issues of fuzzy database models, fuzzy conceptual data models, and conceptual design of fuzzy databases, fuzzy database modeling has a focus on fuzzy database systems and discusses (fuzzy) query, (fuzzy) data handling, database theory (e.g., fuzzy data dependencies and formalization in fuzzy relational databases), database implementation, and and so forth.

Fuzzy Database Models: The database models which have the ability to store and handle fuzzy data, mainly including fuzzy relational database model, fuzzy nested relational database model, and fuzzy object-oriented database model.

Fuzzy Set: The set in which elements are associated with membership degrees in (0, 1) to indicate how they

belong to the set. Fuzzy-set theory was originated by L.

A. Zadeh in 1965 and can be used for imprecise informa- F tion processing.

Possibility Distribution: For a fuzzy set, when the membership degree associated with an element is viewed to be a measure of the possibility that a variable has the value of the element, the fuzzy set becomes a possibility distribution.

Similarity Relation: Similarity relation is also used to describe fuzzy data, whereby the fuzziness comes from the similarity relations between two values in a universe of discourse, not from the status of an object itself. So similarity relations are used to represent the degree similarity of two values from the same universe of discourse.

Generic model management (gMMt) is a novel view on classical and modern metadata management problems. The present article surveys the goals, components, pros and cons of gMMt, and major problems cited in the literature. It argues that some methodology developed in abstract mathematics can be extremely helpful for the field and is capable of providing it with a convenient notation, semantic foundations and truly generic specification patterns. The two other articles, titled Mathematics of Generic Specifications for Model Management, I (further referred to as Math-I , see p. 351), and Mathematics of Generic Specifications for Model Management, II (further referred to as Math-II, see p. 359), give some evidence to these claims by demonstrating how the machinery works in a series of examples.

WHY: FROM ELEMENT-AT-A-TIME TO MODEL-AT-A-TIME PROGRAMMING

Many data management routines include metadata applications that manipulate descriptions of data, usually called schemas, rather than the data itself. Typical examples are database design, schema integration and evolution, reverse engineering, data integration and translation, or data warehousing. Lately, the Web’s dramatically rapid invasion into the field has added to this list several new tasks: ontology engineering and integration, Web site design, and XML wrappers generation. It has also multiplied the importance and diversity of versions of classical tasks by a big coefficient of e-commerce applications. In the OO jargon, data schemas (more generally, metadata artifacts) are often called models, so applications listed above can be classified as model management (MMt).

Along with models, MMt includes specifying and operating relations between models, which are usually called model mappings in the literature. Some examples are mappings between ERor UML-diagrams on consecutive stages of design or between different releases of a database schema; mappings and their inverses between

ER-diagrams and SQL schemas implementing them; mappings between XML schemas to manage message translation; and various sorts of mappings between various UML diagrams either of one sort (homogeneous) or between different sorts (heterogeneous model transformation). The construct of view, well known in the relational data model, also presents nothing but a special syntax for specifying a mapping between relational schemas. In a sense, MMt is all about mappings.

A commonly accepted standard approach to implementing MMt tasks is to present models and mappings as collections of objects and to program manipulations with them via programming manipulations with objects they consist of. Bernstein (2003) calls this object-at-a-time programming. A better term might be element-at-a-time programming, to emphasize working on the level of elements from which models and mappings are built. Though it does the job, element-wise programming is very laborious and error-prone. In a sense, it is similar to record-at- a-time programming in data processing. As is well known, eliminating the latter in modern DBMSs raised data processing technology on a qualitatively new level in programmers’ productivity and semantic transparency.

Similarly, we can expect that a model/mapping-at-a- time programming environment, where the application programmer can think of MMt routines in terms of operations over models and mappings as integral entities, could essentially facilitate development and maintenance of metadata applications. To be really useful, such an environment should be generic, that is, be applicable to a wide range of MMt tasks involving a wide range of models of different types, i.e., of different metamodels. In this way we come to the idea of generic MMt (gMMt) environment manifested by Bernstein, Halevy, and Pottinger (2000).

WHAT: ABSTRACT, CONCRETE AND HEURISTIC PARTS OF GENERIC MMt

To achieve its goals, gMMt must resolve the following three major groups of problems, which are schematically presented in Figure 1.

Generic Model Management

Figure 1. Three parts of generic MMt

Generic MMt

G

First of all, we need a generic way of specifying collections of models and mappings. Then we need to find a set of basic operations with models and mappings so that any practically important MMt procedure could be presented as a composition of basic operations. In other words, like we need data definition and manipulation language in data management, in MMt we need model and mapping definition and manipulation language. Table 1 compares MMt concepts to be developed with their analogs in the relational view to data management. We call this part of MMt abstract since here the internal structure of models and mappings is encapsulated, and they are treated as holistic abstract entities.

Abstract MMt can be divided into two parts, homogeneous and heterogeneous, dependant on whether the models we deal with are of the same or different types (metamodels). The most important issue in heterogeneous MMt is model translation, and the most difficult

into an SQL schema and the latter into an XML DTD would be particular instances of the same specification pattern).

B.ConcretegMMt—Genericnessacrossmetamodels:

To implement abstract MMt patterns and operations, we need to have some concrete representation of models and mappings. Moreover, this representation should be universal with respect to data models (metamodels) so that such diverse models as relational schemas, XML DTDs, or various dialects of ERand UML-diagrams would all be instances of the same universal format. The problem is evidently far from being easy.

There is a reasonable fear that even if we find such a universal representation U, encoding models/mappings of some particular metamodel in U-terms can be bulky and unwieldy. Then an MMt system’s genericness (as many other sorts of genericness in software products) will be an asset for tool builders rather than for tool users. A counterproposal might be to implement model definition