We determine all functional closure properties of finite $\mathbb{N}$-weighted automata, even all multivariate ones, and in particular all multivariate polynomials. We also determine all univariate closure properties in the promise setting, and all multivariate closure properties under certain assumptions on the promise, in particular we determine all multivariate closure properties where the output vector lies on a monotone algebraic graph variety.

翻译：我们确定了有限$\mathbb{N}$加权自动机的所有函数闭包性质，包括所有多元闭包性质，特别地，确定了所有多元多项式闭包性质。我们还确定了在承诺设定下的所有单变量闭包性质，以及在关于承诺的特定假设下的所有多元闭包性质，特别地，确定了输出向量位于单调代数图簇上的所有多元闭包性质。