Equisolvability of series vs. controller´s topology in synchronous language equations

Given a plant MA and a specification M_C, the largest solution of the FSM equation M_X · M_A ≤ M_C contains all possible discrete controllers M_X. Often we are interested in computing the complete solutions whose composition with the plant is exactly equivalent to the specification. Not every solution contained in the largest one satisfies such property, that holds instead for the complete solutions of the series topology. We study the relation between the solvability of an equation for the series topology and of the corresponding equation for the controller´s topology. We establish that, if M_A is a deterministic FSM, then the FSM equation M_X · M_A ≤ M_C is solvable for the series topology with an unknown head component iff it is solvable for the controller´s topology. Our proof is constructive, i.e., for a given solution M_B of the series topology it shows how to build a solution M_D of the controller´s topology and vice versa.