$\mathbb{N}$-polyregular functions arise from well-quasi-orderings

TL;DR

This paper extends the characterization of polyregular functions from the integer-valued case to the natural number-valued case by using well-quasi-orderings, providing a new canonical framework for understanding these functions.

Contribution

It introduces a novel approach to characterize $ $-polyregular functions via well-quasi-orderings, generalizing previous finite-index methods based on equivalence relations.

Findings

Characterization of $ $-polyregular functions using well-quasi-orderings.

Establishment of a canonical object for $ $-polyregular functions.

New insights into the structure of functions from $ $ to $ $.

Abstract

A fundamental construction in formal language theory is the Myhill-Nerode congruence on words, whose finitedness characterizes regular language. This construction was generalized to functions from $\Sigma^*$ to $\mathbb{Z}$ by Colcombet, Dou\'eneau-Tabot, and Lopez to characterize the class of so-called $\mathbb{Z}$-polyregular functions. In this paper, we relax the notion of equivalence relation to quasi-ordering in order to study the class of $\mathbb{N}$-polyregular functions, that plays the role of $\mathbb{Z}$-polyregular functions among functions from $\Sigma^*$ to $\mathbb{N}$. The analogue of having a finite index is then being a well-quasi-ordering. This provides a canonical object to describe $\mathbb{N}$-polyregular functions, together with a powerful new characterization of this class.