We consider polyregular functions, which are certain string-to-string functions that have polynomial output size. We prove that a polyregular function has output size $\mathcal O(n^k)$ if and only if it can be defined by an MSO interpretation of dimension $k$, i.e. a string-to-string transformation where every output position is interpreted, using monadic second-order logic MSO, in some $k$-tuple of input positions. We also show that this characterization does not extend to pebble transducers, another model for describing polyregular functions: we show that for every $k \in \{1,2,\ldots\}$ there is a polyregular function of quadratic output size which needs at least $k$ pebbles to be computed.

翻译：我们考虑多规则函数，这些函数是具有多项式输出大小的某些字符串到字符串的函数。我们证明，对于多规则函数，如果它可以由一个维数为$k$的MSO解释来定义，则其输出大小为$\mathcal O(n^k)$，即，一个字符串到字符串的转换，其中每个输出位置都是在一些$k$个输入位置上使用单调二阶逻辑MSO解释的。我们还表明，这种表征不适用于卵石传感器，另一种描述多规则函数的模型：我们展示对于每个$k∈{1,2,…}$，存在一个二次输出大小的多规则函数，它至少需要$k$个卵石才能计算出来。