Aperiodic two-way transducers and FO-transductions

TL;DR

This paper establishes a precise correspondence between aperiodic two-way transducers and FO-transductions, extending classical language theory results to transducers and providing a logical characterization of their expressive power.

Contribution

It introduces a notion of aperiodicity for two-way transducers and proves their equivalence to FO-transductions, bridging automata theory and logic for transductions.

Findings

Aperiodic two-way transducers correspond exactly to FO-transductions.

Classical equivalences for FO-definability extend to transducers.

Aperiodicity characterizes the expressive power of these transducers.

Abstract

Deterministic two-way transducers on finite words have been shown by Engelfriet and Hoogeboom to have the same expressive power as MSO-transductions. We introduce a notion of aperiodicity for these transducers and we show that aperiodic transducers correspond exactly to FO-transductions. This lifts to transducers the classical equivalence for languages between FO-definability, recognition by aperiodic monoids and acceptance by counter-free automata.