  We propose to study transformations on graphs, and more generally structures,
by looking at how the cut-rank (as introduced by Oum) of subsets is affected
when going from the input structure to the output structure. We consider
transformations in which the underlying sets are the same for both the input
and output, and so the cut-ranks of subsets can be easily compared. The purpose
of this paper is to give a characterisation of logically defined transductions
that is expressed in purely structural terms, without referring to logic:
transformations which decrease the cut-rank, in the asymptotic sense, are
exactly those that can be defined in monadic second-order logic. This
characterisation assumes that the transduction has inputs of bounded treewidth;
we also show that the characterisation fails in the absence of any assumptions.
