Synthesis of Computable Regular Functions of Infinite Words

Regular functions from infinite words to can be equivalently specified by MSO-transducers, streaming $\omega$-string transducers as well deterministic two-way with look-ahead. In their one-way restriction, the latter define class of rational functions. Even though regular are robustly characterised several finite-state devices, even subclass may contain which not computable (by a Turing machine input). This paper proposes decision procedure for following synthesis problem: given function $f$ (equivalently one aforementioned transducer model), is and if it is, synthesize computing it. For functions, we show that computability equivalent continuity, therefore problem boils down deciding continuity. We establish generic characterisation continuity preserving languages under inverse image (such functions). exploit this decidability (and hence computability) done in $\mathsf{NLogSpace}$ (it was already known $\mathsf{PTime}$ Prieur). similar fashion, also effectively characterise uniform relate notion computability, offers stronger efficiency guarantees.