Finite-State Transducers in the Wheeler Setting

Finite-state transducers and Wheeler automata are two well-established frameworks in formal language theory. While transducers extend finite-state automata by associating output words to input words, Wheeler automata are automata whose underlying graph admits a co-lexicographic sorting of states, giving rise to the class of Wheeler languages, a proper subclass of star-free regular languages with efficient indexing properties. In this work, we introduce the notion of sequential Wheeler transducers, a class of deterministic one-way transducers combining the Wheeler condition on the underlying automaton with a monotonicity requirement on the output function. We establish several fundamental properties of this class: closure under composition, and closure of Wheeler languages under inverse image of Wheeler transductions. We then develop a minimization theory by refining Choffrut's syntactic equivalence $\sim_f$ into a relation $\sim_f^c$, and prove a Myhill-Nerode-style theorem characterizing exactly the functions realizable by a sequential Wheeler transducer. Finally, we give a machine-independent characterization of Wheeler functions in terms of the behavior of the function. These results lay the groundwork for a broader structural theory of Wheeler transducers, and we outline open problems concerning decidability, complexity, non-deterministic extensions, and logical characterizations.