Relational queries over interpreted structures

We rework parts of the classical relational theory when the underlying doma in is a structure with some interpreted operations that can be used in quer ies. We identify parts of the classical theory that go through 'as before' when interpreted structure is present, parts that go through only for class es of nicely behaved structures, and parts that only arise in the interpret ed case,The first category includes a number of results on language equival ence and expressive power characterizations for the active-domain semantics for a variety of logics. Under this semantics, quantifiers range over elem ents of a relational database. The main kind of results we prove here are g eneric collapse results: for generic queries, adding operations beyond orde r, does not give us extra power. The second category includes results an the natural semantics, under which quantifier range over the entire interpreted structure. We prove, for a var iety of structures, natural-active collapse results, showing that using unr estricted quantification does not give us any extra power. Moreover, for a variety of structures, including the real field, we give a set of algorithm s for eliminating unbounded quantifications in favor of bounded ones. Furth ermore, we extend these collapse results to a new class of higher-order log ics that mix unbounded and bounded quantification. We give a set of normal forms for these logics, under special conditions on the interpreted structu res. As a by-product, we obtain an elementary proof of the fact that parity test is not definable in the relational calculus with polynomial inequalit y constraints. We also give examples of structures with nice model-theoreti c properties over which the natural-active collapse fails.