An Approach to Single-Threshold-Element Synthesis

An approach based on correlation techniques is developed for the analysis and synthesis of threshold elements. A necessary and sufficient condition for linear separability is derived. Single-threshold-element realizability of arbitrary Boolean functions of n variables is shown to be dependent on a (n+1)-tuple of real numbers (characteristic vector), derivable solely from the given Boolean function. The task of finding appropriate weights for a given threshold function is converted to the minimization of a functional. Approximation to this functional relates the desired weights with the characteristic vector. As the degree of approximation increases, exact solution for the weights becomes possible at the expense of increased computational complexity. The absolute value of characteristic vector components is used to tabulate threshold functions compactly. A table is given for functions of up to six variables which permits one to determine directly, whenever it is possible, the weights and threshold that realize a given function without Boolean manipulation of that function into a standard form.