As an alternative to requiring that the FST generate every output string, we can define an extended inversion function, that includes the inverted FST, but also generates output values for transformed values that are not generated by the FST.
in cases where the transformed value is not generated by the FST, we can assume that one or more of the transformed tags was chosen incorrectly; and make the minimal set of changes to those tags that results in a string that is generated by the FST.