# Continuity of Functional Transducers: A Profinite Study of Rational Functions

## TL;DR

This paper investigates the continuity of word-to-word functions implemented by transducers within certain language classes, establishing decidability results and comparing algebraic approaches using profinite analysis.

## Contribution

It introduces a decidability framework for continuity of functional transducers over standard language classes using profinite methods, and compares algebraic approaches based on automaton structure.

## Key findings

- Decidability of continuity for functional transducers in standard language classes.
- Comparison between automaton-based and algebraic approaches to transducer analysis.
- Conditions under which continuity properties propagate to larger classes.

## Abstract

A word-to-word function is continuous for a class of languages~$\mathcal{V}$ if its inverse maps $\mathcal{V}$_languages to~$\mathcal{V}$. This notion provides a basis for an algebraic study of transducers, and was integral to the characterization of the sequential transducers computable in some circuit complexity classes. Here, we report on the decidability of continuity for functional transducers and some standard classes of regular languages. To this end, we develop a robust theory rooted in the standard profinite analysis of regular languages. Since previous algebraic studies of transducers have focused on the sole structure of the underlying input automaton, we also compare the two algebraic approaches. We focus on two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses?