A string-to-string function is said to be rational (resp. regular) when it is computed by a functional nondeterministic one-way finite-state transducer (resp. a deterministic two-way transducer). Filiot et al. [arXiv:1301.5197] have shown that it is decidable whether a given regular function is rational (the converse is always true); it was later shown that the problem is EXPSPACE-complete [Unpublished result by Ismaël Jecker, obtained by combining arXiv:1701.02502 arXiv:2101.05895].

(1) Is there a "simple" effective characterization in terms of local behaviors of two-way transducers that compute rational functions?

(2) Is it possible to extend this to infinite words?

Gaëtan Douéneau-Tabot suggests that (1) might be doable with factorisation forests, using the kind of techniques he has applied to another transducer membership problem in [arXiv:2112.10212]. If this works, then it might be possible to tackle (2) using factorisation forests over infinite words [10.1016/j.tcs.2009.10.013].