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.

翻译：本文研究在两种额外假设下——交换性（输入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约束）——由$\mathbb{Z}$-加权自动机计算的哪些函数可以由$\mathbb{N}$-加权自动机实现。我们利用这一有效刻画来判定一个由多项式增长的交换$\mathbb{N}$-加权自动机计算的函数是否是无星号的，这一概念借用于正则语言理论，在过去十年间在字符串到字符串函数的研究中已成为众多探讨的主题。