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.