A geometric test-synthesis procedure for a threshold device

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A Boolean function may be defined as a mapping from the vertices of an n-dimensional hypercube whose vertices are n-tuples of “1” and “−1” to (0, 1). A threshold function is then a Boolean functin whose “1” or “TRUE” vertices are separable from the “0” or “FALSE” vertices by a hyperplane. Using this geometric representation, one may show that a threshold device realization may be approximated as follows: Form the vector sum of the TRUE vertices. The components of the resultant vector can then serve as the weights. The threshold is approximated by 2n−1 minus the number of TRUE vertices. If the function under consideration is a threshold function, the vector formed from the first approximation and the threshold may be repeatedly rotated until it converges on a position such that its components form an exact threshold device realization. A specific procedure for performing these rotations is given. If the Boolean function is not a threshold function, the procedure must oscillate. The procedure can thus be used both as a test and a synthesis procedure. Further, whether or not the function is a threshold function, the procedure can be stopped at any point to yield an approximation to a threshold device realization. The desirable features of this procedure are: (1), it is simpler than the usual procedure of solving a set of inequalities; and (2), it will yield approximate realizations.