For modeling real-world problems and constructing intelligent systems, integration of different methodologies and techniques has been the quest and focus of significant interdisciplinary research effort. The advantages of such a hybrid system are that the strengths of its partners are combined and complementary to each other’s weakness. In particular, object orientation provides a hierarchical data abstraction scheme and a mechanism for information hiding and inheritance. However, the classical object-oriented data model cannot deal with uncertainty and imprecision pervasive in real world problems. Meanwhile, probability theory and fuzzy logic provide measures and rules for representing and reasoning with uncertainty and imprecision. That has led to intensive research and development of fuzzy and probabilistic object-oriented databases, as collectively reported in De Caluwe (1997), Ma (2005), and Marín & Vila (2007).
The key issues in research on extending the classical object-oriented data models to deal with uncertainty and imprecision are:
Modeling partial subclass relationship.
Definition of partial class membership.
Representation of uncertain and/or imprecise attribute values.
Representation and execution of class methods.
Expression of partial applicability of class properties.
Mechanism for inheritance under uncertainty and imprecision.
In the classical object-oriented data model, a class hierarchy defines the subclass/super-class relation on classes. A class A is derived as a subclass of a class B, which is then called A’s super-class, either by narrowing the crisp value ranges of B’s attributes or by adding new properties to B’s ones. In the probabilistic and fuzzy case, due to the uncertain applicability of class properties or the imprecision of attribute value ranges, the inclusion between classes naturally becomes graded, which could be computed on the basis of the value ranges of their common attributes (George & Buckles & Petry, 1993, Rossazza & Dubois & Prade, 1997).
As discussed in Baldwin, Cao, Martin, and Rossiter (2000), a set of classes with a graded inclusion or inheritance relation actually forms a network rather than a hierarchy, because if a class A has some inclusion degree into a class B based on a fuzzy matching of their descriptions, then B usually also has some inclusion degree into A. Moreover, naturally, a concept is usually classified into sub-concepts that are totally subsumed by it, though the sub-concepts can overlap each other, as assumed in Dubitzky, Büchner, Hughes, and Bell (1999) for instance.
Uncertain and imprecise attribute values lead to partial membership of an object into a class, and there are different measures proposed. Yazici and George (1999), for instance, defined for each class a membership function on a set of objects. Bordogna, Pasi, and Lucarella (1999) used linguistic labels to express the strength of the link of an object to a class. Dubitzky et al. (1999) defined membership as similarity degrees between objects and classes. Blanco, Marín, Pons, and Vila (2001) mentioned different measures, including probabilistic one, to be used for membership degrees. Nevertheless, it is to be answered how measures of different meanings, such as possibility and probability, on various levels of a model are integrated coherently.
Key Terms in this Chapter
Fuzzy Class Hierarchy: An extended conventional class hierarchy where each link between a class and one of its subclasses is associated with an inclusion degree in [0, 1].
Probabilistic Interpretation of Selection Expression: A probability interval for an object satisfying a selection expression.
Uncertain Inheritance: An object may inherit a property of a class to a certain degree only.
Fuzzy-Probabilistic Triple Value: An imprecise and uncertain value expressed by lower and upper bound probability distributions on a set of fuzzy set values.
Fuzzy Property: A property that is applicable to a class with a certain possibility degree.
Fuzzy Class Membership: A class is considered as a fuzzy set on a set of objects, for which each object is a member of a class to a certain degree.
Uncertain Path Expression: A sequence of nested properties associated with a probability interval expressing the uncertainty of its applicability to a class or object.
Probabilistic Property: A property that is applicable to a class with a certain probability.
Probabilistic Class Hierarchy: An extended conventional class hierarchy where each link between a class and one of its subclasses is associated with a conditional probability for an object of the class belonging to that subclass.
Probabilistic Class Membership: There is a probability for an object belonging to a class.
Fuzzy-Probabilistic Selection Condition: A condition of a selection operation on a fuzzy-probabilistic object-oriented database, which is a selection expression associated with a probability interval.
Fuzzy-Probabilistic Selection Expression: An expression of a selection operation on a fuzzy-probabilistic object-oriented database, which includes constraints on uncertain path expressions and fuzzy-probabilistic triple values.