Deriving a module of a multi agent system via Finite State Machine equation solving

In multiagent systems, there is a problem of constructing an agent that can work in different contexts satisfying different specifications. One of ways is to solve a system of corresponding automata equations. Since in general, the complexity of solving such equations is exponential w.r.t. to the number of states of the context and specification, the question arises whether a system of equations can be reduced to a single equation. In this paper, we consider two special cases when a system of equations under the parallel composition over Finite State Machines can be reduced to a single equation. For each case, it is shown how a corresponding single equation can be derived.