The many faces of query monotonicity

Abstract

Monotonicity, based on the partial order defined by the ‘is a subset of’ relation, is a well understood property of queries. For nested relations, other partial orders leading to different notions of monotonicity are possible. Monotonicity can be used for simple negative comparison of the expressive power of two languages by showing that one is monotone and the other is not. Using this approach we study three questions related to the expressive power of practically useful subsets of well known programming languages for nested relations. First, we show that logic programming languages over nested relations can be regarded as Datalog with user-defined algebraic expressions. This leads to a modular integration of recursion with the monotone subset of the algebra. Second, we prove that the equivalence of the powerset algebra and the complex object Datalog breaks down for their monotone subsets. Third, for the class of positive existential queries over nested relations, which generalize the relational tableau set queries, we show that the use of intermediate types does not enhance their expressive power, in contrast to the known result for general existential queries. We also show that this class does not contain the powerset operator, hence it is a candidate for a tractable tableau query system for nested relations. Finally, the (monotone) Bancilhon-Khoshafian calculus for complex objects is shown to be incomparable to the monotone subsets of most known languages.