On the Complexity Ofthe Boolean Minimalrealization Problem

One of the open problems in the max-plus-algebraic system theory for discrete event systems is the minimal realization problem. We consider a simpliied version of the general minimal realization problem: the boolean minimal realization problem, i.e., we consider models in which the entries of the system matrices are either equal to the max-plus-algebraic zero element or to the max-plus-algebraic identity element. We show that the corresponding decision problem (i.e., deciding whether or not a boolean realization of a given order exists) is decid-able, and that the boolean minimal realization problem can be solved in a number of elementary operations that is bounded from above by an exponential of the square of (any upper bound of) the minimal system order.