Relational Queries Over Interpreted Structures

We rework parts of the classical relational theory when the underlying domain is a structure with some interpreted operations that can be used in queries. We identify parts of the classical theory that go through "as before" when interpreted structure is present, parts that go through only for classes of nicely-behaved structures, and parts that only arise in the interpreted case. The first category includes a number of results on language equivalence and expressive power characterizations for the active-domain semantics for a variety of logics. Under this semantics, quantifiers range over elements of a relational database. The main kind of results we prove here are generic collapse results: for generic queries, adding operations beyond order, does not give us extra power. The second category includes results on the natural semantics, under which quantifiers range over the entire interpreted structure. We prove, for a variety of structures, natural-active collapse results, showing that using unrestricted quantification does not give us any extra power. Moreover, for a variety of structures, including the real field, we give a set of algorithms for eliminating unbounded quantifications in favor of bounded ones.