Logical and Algebraic Characterizations of Rational 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.
Keywords
Cite
@article{arxiv.1705.03726,
title = {Logical and Algebraic Characterizations of Rational Transductions},
author = {Emmanuel Filiot and Olivier Gauwin and Nathan Lhote},
journal= {arXiv preprint arXiv:1705.03726},
year = {2023}
}