Testing and Realization of Threshold Functions by Successive Higher Ordering of Incremental Weights

In this paper, a modification or generalization of Sheng´s secondary ordering method for testing and realization of threshold functions is presented. Instead of assigning integral values to the incremental weights according to secondary ordering, a search for successively higher ordering is made, and incremental weights of higher orders are successively substituted back into the inequalities until finally no more higher ordering can be found. If the given function is a threshold function, it will turn out that the sum of the coefficients of all the terms on the left side of each of the inequalities will be greater than the sum of the coefficients of all the terms on the right side. Then a minimal integral assignment can be made by assigning unity to every incremental weight of any order appearing in the final set of inequalities. If the given function is not a threshold function, a contradiction will be revealed. Some theorems are proved to justify the method. A complete procedure for testing and realization is given. An example is worked out in detail to illustrate this method.