This paper studies which functions computed by $\mathbb{Z}$-weighted automata can be realized by $\mathbb{N}$-weighted automata, under two extra assumptions: commutativity (the order of letters in the input does not matter) and polynomial growth (the output of the function is bounded by a polynomial in the size of the input). We leverage this effective characterization to decide whether a function computed by a commutative $\mathbb{N}$-weighted automaton of polynomial growth is star-free, a notion borrowed from the theory of regular languages that has been the subject of many investigations in the context of string-to-string functions during the last decade. Furthermore, we open the road to a generalization of our results to non-commutative functions, by formalizing a canonical computational model for $\mathbb{N}$-weighted automata of polynomial growth based on the notion of residual transducer.

翻译：本文研究在两种额外假设下，由ℤ加权自动机计算的哪些函数可由ℕ加权自动机实现：交换性（输入中字母的顺序无关紧要）和多项式增长（函数的输出受输入大小的多项式限制）。我们利用这一有效刻画，判定由多项式增长的交换ℕ加权自动机计算的函数是否为星号无关的——这一概念借用于正则语言理论，在过去十年中在字符串到字符串函数的研究背景下受到广泛关注。此外，我们通过基于剩余转换器概念，为多项式增长的ℕ加权自动机形式化一种规范计算模型，为将我们的结果推广到非交换函数开辟了道路。