  We consider a system consisting of a sequential composition of Mealy
machines, called head and tail. We study two problems related to these systems.
In the first problem, models of both head and tail components are available,
and the aim is to obtain a replacement for the tail with the minimum number of
states. We introduce a minimization method for this context which yields an
exponential improvement over the state of the art. In the second problem, only
the head is known, and a desired model for the whole system is given. The
objective is to construct a tail that causes the system to behave according to
the given model. We show that, while it is possible to decide in polynomial
time whether such a tail exists, there are instances where its size is
exponential in the sizes of the head and the desired system. This shows that
the complexity of the synthesis procedure is at least exponential, matching the
upper bound in complexity provided by the existing methods for solving unknown
component equations.
