Consider a Generalized Sequential Machine (GSM; or nearly equivalently -- an FSM transducer). These machines are closed under compositions. A composition of two GSMs $f(x)$ and $g(x)$ can be written as a function $h(x) = f(g(x))$. I am interested in the inverse problem:
Given a transducer $h(x)$, decompose it into a composition of two transducers $f(g(x))$. (One of the $f,g$ might be given.)
Has this question been studied?
This question that occurred to me many years ago and is similar to a recent question by Dave Lang. After some back-and-forth we agree that it seems basic, yet not to under-studied. Although, Dave Lang did find a potentially related paper: Weber, A. (1996). Decomposing a k-valued transducer into k unambiguous ones. Informatique théorique et applications, 30(5): 379-413.
I suspect that this is linked to the Krohn-Rhodes decomposition and that through the natural correspondence between TMs and transducers, we might find hidden structure that could be useful in termination analysis and automated theorem proving.