Commutative N-polyregular functions

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.