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Abstract : |
We study the problem of maintaining recursively-defined views, such as the transitive closure of a relation, in traditional relational languages that do not have recursion mechanisms. In particular, we show that the transitive closure cannot be maintained in relational calculus under deletion of edges. We use new proof techniques to show this result. These proof techniques generalize to other languages, for example, to the language for nested relations that also contains a number of aggregate functions. Such a language is considered in this paper as a theoretical reconstruction of SQL. Our proof techniques also generalize to other recursive queries. Consequently, we show that a number of recursive queries cannot be maintained in an SQL-like language. We show that this continues to be true in the presence of certain auxiliary relations. We also relate the complexity of updating transitive closure to that of updating the samegeneration query and show that the latter is strictly harder than the former. Then we extend this result to that of updating queries based on context-free sets. 1 Problem Statement and Summary It is well known that relational calculus (equivalently, first-order logic) cannot express recursive queries such as transitive closure [1]. However, in a real database system, it is reasonable to store both the relation and, |
