  We demonstrate that for the $k$-variable theory $T$ of a finite structure
(satisfying certain amalgamation conditions), if finite models of $T$ can be
recovered from diagrams of finite {\em subsets} of model of $T$ in a certain
"efficient" way, then $T$ is rosy -- in fact, a certain natural
$\aleph_0$-categorical completion $T^{\lim}$ of $T$ is super-rosy of finite
$U^\thorn$-rank. In an appendix, we also show that any $k$-variable theory $T$
of a finite structure for which the Strong $L^k$-Canonization Problem is
efficient soluble has the necessary amalgamation properties up to taking an
appropriate reduct.
