  It is known that the composition of schema mappings, each specified by
source-to-target tgds (st-tgds), can be specified by a second-order tgd (SO
tgd). We consider the question of what happens when target constraints are
allowed. Specifically, we consider the question of specifying the composition
of standard schema mappings (those specified by st-tgds, target egds, and a
weakly acyclic set of target tgds). We show that SO tgds, even with the
assistance of arbitrary source constraints and target constraints, cannot
specify in general the composition of two standard schema mappings. Therefore,
we introduce source-to-target second-order dependencies (st-SO dependencies),
which are similar to SO tgds, but allow equations in the conclusion. We show
that st-SO dependencies (along with target egds and target tgds) are sufficient
to express the composition of every finite sequence of standard schema
mappings, and further, every st-SO dependency specifies such a composition. In
addition to this expressive power, we show that st-SO dependencies enjoy other
desirable properties. In particular, they have a polynomial-time chase that
generates a universal solution. This universal solution can be used to find the
certain answers to unions of conjunctive queries in polynomial time. It is easy
to show that the composition of an arbitrary number of standard schema mappings
is equivalent to the composition of only two standard schema mappings. We show
that surprisingly, the analogous result holds also for schema mappings
specified by just st-tgds (no target constraints). This is proven by showing
that every SO tgd is equivalent to an unnested SO tgd (one where there is no
nesting of function symbols). Similarly, we prove unnesting results for st-SO
dependencies, with the same types of consequences.
