Complexity of higher-order queries

While relational algebra and calculus are a well-established foundation for classical database query languages, it is less clear what the analog is for higher-order functions, such as query transformations. Here we study a natural way to add higher-order functionality to query languages, by adding database query operators to the λ-calculus as constants. This framework, which we refer to as λ-embedded query languages, was introduced in [BPV10]. That work had a restricted focus: the containment and equivalence problems for query-to-query functions, in the case where only positive relational operators are allowed as constants. In this work we take an in-depth look at the most basic issue for such languages: the evaluation problem. We give a full picture of the complexity of evaluation for λ-embedded query languages, looking at a number of variations: with negation and without; with only relational algebra operators, and also with a recursion mechanism in the form of a query iteration operator; in a strongly-typed calculus as well as a weakly-typed one. We give tight bounds on both the combined complexity and the query complexity of evaluation in all these settings, in the process explaining connections with Datalog and prior work on λ-calculus evaluation.

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