  A possible world of an incomplete database table is obtained by imputing
values from the attributes (infinite) domain to the place of \texttt{NULL} s. A
table satisfies a possible key or possible functional dependency constraint if
there exists a possible world of the table that satisfies the given key or
functional dependency constraint. A certain key or functional dependency is
satisfied by a table if all of its possible worlds satisfy the constraint.
Recently, an intermediate concept was introduced. A strongly possible key or
functional dependency is satisfied by a table if there exists a strongly
possible world that satisfies the key or functional dependency. A strongly
possible world is obtained by imputing values from the active domain of the
attributes, that is from the values appearing in the table. In the present
paper, we study approximation measures of strongly possible keys and FDs.
Measure $g_3$ is the ratio of the minimum number of tuples to be removed in
order that the remaining table satisfies the constraint. We introduce a new
measure $g_5$, the ratio of the minimum number of tuples to be added to the
table so the result satisfies the constraint. $g_5$ is meaningful because the
addition of tuples may extend the active domains. We prove that if $g_5$ can be
defined for a table and a constraint, then the $g_3$ value is always an upper
bound of the $g_5$ value. However, the two measures are independent of each
other in the sense that for any rational number $0\le\frac{p}{q}<1$ there are
tables of an arbitrarily large number of rows and a constant number of columns
that satisfy $g_3-g_5=\frac{p}{q}$. A possible world is obtained usually by
adding many new values not occurring in the table before. The measure $g_5$
measures the smallest possible distortion of the active domains. We study
complexity of determining these approximate measures.
