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.

翻译：我们提出通过研究子集在输入结构到输出结构过程中切割秩（由Oum引入）的变化，来探讨图（更一般地，结构）上的变换。我们考虑输入与输出底层集合相同的变换，因此子集的切割秩易于比较。本文旨在给出一种纯结构术语表述的逻辑定义转录刻画，无需涉及逻辑：渐近意义上削减切割秩的变换，恰好是那些可在一元二阶逻辑中定义的变换。该刻画假设转录具有有界树宽的输入；同时我们证明，在无任何假设的情况下该刻画不成立。