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，这是一类具有多项式输出规模的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 $\mathcal O(n^k)$ 当且仅当它可以通过维度为 $k$ 的 MSO 解释来定义，即一种字符串到字符串的变换，其中每个输出位置都是通过一元二阶逻辑（MSO）在某 $k$ 元输入位置元组中解释得到的。我们还证明，这一刻画并不适用于卵石换能器（另一种描述多正则函数的模型）：对于每个 $k \in \{1,2,\ldots\}$，存在一个具有二次输出规模的多正则函数，其计算至少需要 $k$ 个卵石。