# Logical and Algebraic Characterizations of Rational Transductions

## TL;DR

This paper extends algebraic and logical characterizations from regular languages to rational transductions, providing decidability results for classifying transductions within certain logical and algebraic frameworks.

## Contribution

It introduces methods to decide if a rational transduction belongs to a specific algebraic or logical class, generalizing known results from languages to transductions.

## Key findings

- Decidability of class membership for rational transductions
- Transfer of logic-algebra equivalences from languages to transductions
- PSPACE-completeness of first-order definability for transductions

## Abstract

Rational word languages can be defined by several equivalent means: finite state automata, rational expressions, finite congruences, or monadic second-order (MSO) logic. The robust subclass of aperiodic languages is defined by: counter-free automata, star-free expressions, aperiodic (finite) congruences, or first-order (FO) logic. In particular, their algebraic characterization by aperiodic congruences allows to decide whether a regular language is aperiodic. We lift this decidability result to rational transductions, i.e., word-to-word functions defined by finite state transducers. In this context, logical and algebraic characterizations have also been proposed. Our main result is that one can decide if a rational transduction (given as a transducer) is in a given decidable congruence class. We also establish a transfer result from logic-algebra equivalences over languages to equivalences over transductions. As a consequence, it is decidable if a rational transduction is first-order definable, and we show that this problem is PSPACE-complete.