This correspondence gives a method which generates prime implicants in a simple one-pass procedure. Starting from the lowest (or highest) points in the function lattice, it generates successive subcubes in a unique sequence. Each subcube is formed only once.

This note attempts to show that, in a vertex weight method [1], every contradiction equation bears a one-to-one correspondence with the summability pair C1S, C2S, where C1S= {X11, X12, ..., X1k}⊆ C1C2S= {X21, X22,..., X2k} ⊆ C2and vector sums of the vertices plz check [Eqa] The vertices, Xki's, K = 1, or 2, are not necessarily distinct, and C1, C2are two disjoint sets of vertices in Enspace. As a consequence, the contradiction equation is a necessary and sufficient condition that the homogeneous system, solved for a threshold function of order r, has no solution. This tells that the threshold function is of order greater than r.

In this paper we shall study the problem of nonlinear separation. As usual, a Boolean function F of n binary variables, xl,..., xn, xiΣ{1,0}, i=1,..., n, will be represented by a set of vertices C in the n-dimensional Euclidean space Enwhere each vertex has n binary-valued components, {1,0}.

In this note the successive-higher-ordering method for testing and realization of threshold functions is applied to the realization of a threshold function F, such that it will contain a given function F1, which may or may not be a threshold function, and such that it will be contained in F1+Fø, where Føis the function representing the don't care vertices. Before the application of the successive-higher-ordering method, the given functions F1and F0= (F1+Fø) are first changed into unate functions by the successive positivizing of the functions with respect to the variables, one at a time. Some theorems relating to this formation of unate functions are presented. A systematic procedure for testing and realization is developed. An example is given for illustration.

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.