  The synthesis problem asks to automatically generate, if it exists, an
algorithm from a specification of correct input-output pairs. In this paper, we
consider the synthesis of computable functions of infinite words, for a
classical Turing computability notion over infinite inputs. We consider
specifications which are rational relations of infinite words, i.e.,
specifications defined non-deterministic parity transducers. We prove that the
synthesis problem of computable functions from rational specifications is
undecidable. We provide an incomplete but sound reduction to some parity game,
such that if Eve wins the game, then the rational specification is realizable
by a computable function. We prove that this function is even computable by a
deterministic two-way transducer. We provide a sufficient condition under which
the latter game reduction is complete. This entails the decidability of the
synthesis problem of computable functions, which we prove to be
ExpTime-complete, for a large subclass of rational specifications, namely
deterministic rational specifications. This subclass contains the class of
automatic relations over infinite words, a yardstick in reactive synthesis.
