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International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol 19, No (2011) 275−305 © World Scientific Publishing Company DOI: 10.1142/S0218488511007003 Int J Unc Fuzz Knowl Based Syst 2011.19:275-305 Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 02/11/15 For personal use only UNCERTAIN AND FUZZY OBJECT BASES: A DATA MODEL AND ALGEBRAIC OPERATIONS TRU H CAO Ho Chi Minh City University of Technology and John von Neumann Institute VNU-HCM, 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam tru@cse.hcmut.edu.vn HOA NGUYEN Faculty of Information Technology, Ho Chi Minh City Open University, 97 Vo Van Tan Street, District 3, Ho Chi Minh City, Vietnam hoa-hanh@hcm.vnn.vn Received 20 January 2010 Revised January 2011 Fuzzy set theory and probability theory are complementary for soft computing, in particular objectoriented systems with imprecise and uncertain object properties However, current fuzzy object-oriented data models are mainly based on fuzzy set theory or possibility theory, and lack of a rigorous algebra for querying and managing uncertain and fuzzy object bases In this paper, we develop an object base model that incorporates both fuzzy set values and probability degrees to handle imprecision and uncertainty A probabilistic interpretation of relations on fuzzy sets is introduced as a formal basis to coherently unify the two types of measures into a common framework The model accommodates both class attributes, representing declarative object properties, and class methods, representing procedural object properties Two levels of property uncertainty are taken into account, one of which is value uncertainty of a definite property and the other is applicability uncertainty of the property itself The syntax and semantics of the selection and other main data operations on the proposed object base model are formally defined as a fullfledged algebra Keywords: Fuzzy set theory; probability theory; object modeling; object base algebra Introduction It is undeniable that the real-world is pervaded by uncertain and imprecise information that we have to face, and make decisions on, in daily life Among the foundations for computer systems to deal with uncertainty and imprecision, probability theory and fuzzy set theory are major ones and complementary to each other Moreover, challenging realworld problems are also due to their large scales in practice For handling it, the objectoriented methodology has been proved as a key one for data modeling and system design and implementation In particular, there have been intensive research and development of fuzzy and probabilistic object-oriented databases, as collectively reported in Refs 1−4 Surveying research on extending the classical object-oriented data model to deal with uncertainty and imprecision, we identify the following key issues: (1) Modeling partial sub-class relationship; (2) Definition of partial class membership; (3) Representation of uncertain and/or imprecise attribute values; (4) Representation and execution of class 275 Int J Unc Fuzz Knowl Based Syst 2011.19:275-305 Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 02/11/15 For personal use only 276 T H Cao & H Nguyen methods; (5) Expression of partial applicability of class properties; and (6) Mechanism for inheritance under uncertainty and imprecision We discuss them in details in the following For the first issue, in the classical object-oriented model, a class hierarchy defines the subclass relation on classes, whereby a class is totally included in any of its super-classes However, in the probabilistic and fuzzy cases, 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.5,6 As discussed in Ref 7, 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, in practice, it is more natural to classify a concept into sub-concepts that are totally subsumed by it, than to think of overlapping between a concept and its sub-concepts, though the subconcepts can overlap each other, as assumed in Ref for instance For the second issue, when attribute values of an object are uncertain and imprecise, its matching degree with a class becomes graded, and there have been different measures proposed In Ref 9, for instance, a membership function on a set of objects was defined for each class In Ref 10 linguistic labels were used to express the strength of the link of an object to a class In Ref membership was defined as similarity degrees between objects and classes In Ref 11 different measures were mentioned, including a probabilistic one, for membership degrees Nevertheless, for the soundness of using measures of different meanings, such as possibilistic and probabilistic ones, it is to be answered how those measures are integrated coherently on a common ground For the third issue, many works on fuzzy object-oriented data models did not rely on probability theory, but used fuzzy sets or possibility distributions to represent imprecise attribute values The works10,11 also modeled uncertainty degrees for an attribute having a particular value However, much less concern was given for uncertainty over a set of values of an attribute and a foundation to combine probability degrees and fuzzy sets in the same model For the fourth issue, while class methods are common in classical object-oriented systems for modeling object behaviors and parameterized properties, they were often neglected in uncertain and fuzzy extended models In Refs and 11 methods were not considered In Ref 10 methods were mentioned but no formal representation and explicit manipulation were provided in the model In Refs and 9, which were for declarative and deductive in contrast to imperative and procedural models, methods were formally defined as Horn clauses and executed as in a theorem proving process For the fifth issue,12 introduced the notion of fuzzy property as an intermediate between the two extreme notions of required property and optional property, each of which was associated with a possibility degree of applicability of the property to the class Meanwhile,8 assumed that each property of a concept to have a probability degree for it occurring in exemplars of that concept Those are two typical works that model Int J Unc Fuzz Knowl Based Syst 2011.19:275-305 Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 02/11/15 For personal use only Uncertain and Fuzzy Object Bases 277 partial applicability of a property to a class of objects by possibility and probability degrees, respectively We note the distinction between the notion of uncertain property values and that of uncertain property applicability In the former case, a class or an object surely has a particular property but it is not sure which one among a given set of values the property takes Meanwhile, in the latter, it is even not sure if the class or the object has that property For example, “John owns a car whose brand is probably BMW” and “John probably owns a car” express different levels of uncertainty In Refs 7, 10, and 11, the two levels were mixed For the sixth issue, due to uncertain class membership and uncertain property applicability, inheritance of a class property by an object naturally becomes uncertain Uncertain inheritance was not considered in Refs 8, 9, and 10 In Ref 11, class membership degrees were used as thresholds to determine which properties in a class would be inherited with respect to an uncertainty degree In Ref 7, both membership of an object into a class and applicability of a property to the class were represented by probability intervals and combined into a support pair for the object to inherit the property For recent works,13 reviewed existing proposals and presented recommendations for the application of fuzzy set theory in a flexible generalized object model Further,14 focused on representing data as constraints on object attributes and answering queries as constraint satisfaction Meanwhile, for realization of fuzzy object-oriented data models,15 was concerned with implementation of their model on an existing platform In Ref 16, Fril++ was developed as a fuzzy object-oriented logic programming language The literature review of fuzzy relational and object-oriented databases17 missed those modeling uncertainty with probability theory A common disadvantage of current fuzzy object-oriented models is that they lack of a rigorous algebra for querying and managing object bases In contrast, Ref 18 introduced a probabilistic model to handle object bases with uncertainty, called POB (Probabilistic Object Base), and developed a full-fledged algebra for it However, the major shortcomings of the POB model are: (1) it does not allow imprecise attribute values; (2) it does not support class methods; and (3) it does not consider uncertain applicability of class properties To overcome the first two shortcomings,19,20 in turn extended POB to an uncertain and fuzzy object base model (UFOB) with class attributes and methods whose values could be fuzzy sets This paper extends UFOB further with class properties whose applicability to the class objects could also be uncertain, requiring and resulting in a new algebra of operations on uncertain and fuzzy object bases where previous definitions are to be extended accordingly Next, for the paper being self-contained, Section recalls the probabilistic interpretation of relations on fuzzy sets and the algebra on fuzzy-probabilistic triples introduced in Refs 19 and 20, as a basis to integrate fuzzy set values into the probabilitybased framework of POB Section describes properties of objects in UFOB, which can be imprecise attribute values, computational methods, and uncertainly applicable to classes Section presents the notion of instances and inheritance mechanism under 278 T H Cao & H Nguyen uncertainty and imprecision in object bases Sections and define the selection operation and other algebraic operations on the proposed object base model The definitions in Secs to are extensions of the corresponding ones,19,20 for modeling and computing with uncertain applicability of class properties Finally, Sec summarizes and concludes the paper Fuzzy Sets and Probability Int J Unc Fuzz Knowl Based Syst 2011.19:275-305 Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 02/11/15 For personal use only 2.1 Probabilistic interpretation of fuzzy set relations In this work, for extending the probabilistic model of POBs with fuzzy set values, we apply the voting model interpretation of fuzzy sets.21,22 That is, given a fuzzy set A on a domain U, this model defines a mass assignment mA(.) (i.e., probability distribution) on the power set of U, where the mass (i.e., probability value) assigned to a subset of U is the proportion of voters who have that subset as a crisp definition for the fuzzy concept A Example 1: Let us take the Dice example in Ref 22 Given the dice values from the set {1, 2, 3, 4, 5, 6}, suppose that a score high is defined by the discrete fuzzy set {3:0.2, 4:0.5, 5:0.9, 6:1}, i.e., the membership of value is 0.2, and so on The voting pattern for a group of 10 persons for this score could be as in Table Table Voting pattern for high dice values voters P1 P2 x x x scores P3 P4 P5 P6 P7 P8 P9 x x x x x x x x x x x x x x x x x x x x x x P10 x That is, all voters, P1 to P10, vote for value as a high score, while only two of them, P1 and P2, vote for as a high score, and so on In other words, the crisp definition of P10 for the high score is {6}, while that of P1 and P2 is {3, 4, 5, 6}, for instance An assumption made in this voting model is that any person who accepts a value as a high score also accepts all values that have higher membership in the fuzzy set high This model defines the following mass assignment on the power set of {1, 2, 3, 4, 5, 6}: {6}: 0.1 {5, 6}: 0.4 {4, 5, 6}: 0.3 {3, 4, 5, 6}: 0.2 where the mass assigned to a subset of {1, 2, 3, 4, 5, 6} (e.g mhigh({5, 6}) = 0.4) is the proportion of voters who have that subset as a crisp definition for the fuzzy concept high Uncertain and Fuzzy Object Bases 279 Int J Unc Fuzz Knowl Based Syst 2011.19:275-305 Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 02/11/15 For personal use only score This mass assignment corresponds to a family of probability distributions on {1, 2, 3, 4, 5, 6} On the basis of this voting model we introduce a probabilistic interpretation of the following binary relations on fuzzy sets We write Pr(e1 | e2) to denote the conditional probability of e1 given e2 Definition Let A be a fuzzy set on a domain U, B be a fuzzy set on a domain V, and θ be a binary relation from {=, ≤,