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：我们考虑多正则函数——一类具有多项式输出规模的特殊字符串到字符串函数。我们证明：一个多正则函数的输出规模为 $\mathcal O(n^k)$ 当且仅当它能被一个维度为 $k$ 的 MSO 解释所定义，即通过一元二阶逻辑 MSO 在某个 $k$ 元输入位置元组中解释每个输出位置的字符串到字符串变换。我们还证明这一刻画不能推广到石传感器（描述多正则函数的另一模型）：对于每个 $k \in \{1,2,\ldots\}$，存在一个输出规模为二次的多正则函数，其计算至少需要 $k$ 个石子。