  We study two formalisms that allow to compare transducers over words under
origin semantics: rational and regular resynchronizers, and show that the
former are captured by the latter. We then consider some instances of the
following synthesis problem: given transducers T1, T2, construct a rational
(resp. regular) resynchronizer R, if it exists, such that T1 is contained in
R(T2) under the origin semantics. We show that synthesis of rational
resynchronizers is decidable for functional, and even finite-valued, one-way
transducers, and undecidable for relational one-way transducers. In the two-way
setting, synthesis of regular resynchronizers is shown to be decidable for
unambiguous two-way transducers. For larger classes of two-way transducers, the
decidability status is open.
