The role of prime compatibles in the minimization of finite state machines

A. Grasselli and F. Luccio (1965) proved that a minimum state cover of an incompletely specified finite-state machine could be found by only considering prime compatibles. It was conjectured that in practice one could restrict even further the set of compatibles being considered to the set of maximal compatibles. The conditions under which a solution formed of maximal compatibles was guaranteed to be exact were determined, but the question of the practical relevance of prime compatibles remained open. It is shown here that state minimization problems which require the full generality afforded by prime compatibles for the solution to be optimal are actually found in practice. The proof relies on the concept of analogous machines-essentially machines that pose the same minimization problem. The main result is that for any incompletely specified machine there is an analogous machine that has to be minimized in the optimization of two interacting, completely specified machines