  Data exchange heavily relies on the notion of incomplete database instances.
Several semantics for such instances have been proposed and include open (OWA),
closed (CWA), and open-closed (OCWA) world. For all these semantics important
questions are: whether one incomplete instance semantically implies another;
when two are semantically equivalent; and whether a smaller or smallest
semantically equivalent instance exists. For OWA and CWA these questions are
fully answered. For several variants of OCWA, however, they remain open. In
this work we adress these questions for Closed Powerset semantics and the OCWA
semantics of Libkin and Sirangelo, 2011. We define a new OCWA semantics, called
OCWA*, in terms of homomorphic covers that subsumes both semantics, and
characterize semantic implication and equivalence in terms of such covers. This
characterization yields a guess-and-check algorithm to decide equivalence, and
shows that the problem is NP-complete. For the minimization problem we show
that for several common notions of minimality there is in general no unique
minimal equivalent instance for Closed Powerset semantics, and consequently not
for the more expressive OCWA* either. However, for Closed Powerset semantics we
show that one can find, for any incomplete database, a unique finite set of its
subinstances which are subinstances (up to renaming of nulls) of all instances
semantically equivalent to the original incomplete one. We study properties of
this set, and extend the analysis to OCWA*.
