First-order definability of rational transductions: An algebraic approach

The algebraic theory of rational languages has provided powerful decidability results. Among them, one of the most fundamental is the definability of a rational language in the class of aperiodic languages, i.e., languages recognized by finite automata whose transition relation defines an aperiodic congruence. An important corollary of this result is the first-order definability of monadic second-order formulas over finite words. Our goal is to extend these results to rational transductions, i.e. word functions realized by finite transducers. We take an algebraic approach and consider definability problems of rational transductions in a given variety of congruences (or monoids). The strength of the algebraic theory of rational languages relies on the existence of a congruence canonically attached to every language, the syntactic congruence. In a similar spirit, Reutenauer and Schützenberger have defined a canonical device for rational transductions, that we extend to establish our main contribution: an effective characterization of V-transductions, i.e. rational transductions realizable by transducers whose transition relation defines a congruence in a (decidable) variety V. In particular, it provides an algorithm to decide the definability of a rational transduction by an aperiodic finite transducer. Using those results, we show that the FO-definability of a rational transduction is decidable, where FO-definable means definable in a first-order restriction of logical transducers á la Courcelle.