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<p>The application of the object-oriented (O-O) paradigm in the database management field has gained much attention in recent years. Several experimental and commercial O-O database management systems have become available. However, the existing O-O DBMSs still lack a solid mathematical foundation for the manipulation of O-O databases, the optimization of queries, and the design and selection of storage structures for supporting O-O database manipulations. This paper presents an association algebra (A-algebra) to serve as a mathematical foundation for processing O-O databases, which is analogous to the relational algebra used for processing relational databases. In this algebra, objects and their associations in an O-O database are uniformly represented by association patterns which are manipulated by a number of operators to produce other association patterns. Different from the relational algebra, in which set operations operate on relations with union-compatible structures, the A-algebra operators can operate on association patterns of homogeneous and heterogeneous structures. Different from the traditional record-based relational processing, the A-algebra allows very complex patterns of object associations to be directly manipulated. The pattern-based query formulation and the A-algebra operators are described. Some mathematical properties of the algebraic operators are presented together with their application in query decomposition and optimization. The completeness of the A-algebra is also defined and proven. The A-algebra has been used as the basis for the design and implementation of an object-oriented query language, OQL, which is the query language used in a prototype Knowledge Base Management System OSAM*.KBMS.</p>
association algebra; object-oriented databases; A-algebra; mathematical properties; query decomposition; completeness; object-oriented query language; OQL; Knowledge Base Management System; OSAM*.KBMS; database theory; object-oriented databases; query languages; query processing

H. Lam, M. Guo and S. Su, "Association Algebra: A Mathematical Foundation for Object-Oriented Databases," in IEEE Transactions on Knowledge & Data Engineering, vol. 5, no. , pp. 775-798, 1993.
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