A Realization Procedure for Multithreshold Threshold Elements

A synthesis procedure is introduced for obtaining multithreshold threshold-element realizations of arbitrary Boolean functions (including functions with don´t cares). Through transforms of multithreshold threshold-element realizations, the procedure is also applicable to threshold-element-network realizations of functions. The procedure is useful for hand calculations for functions with a small number of variables, and has been programmed on a computer for functions having up to 11 variables¿the number of variables is presently limited by storage allocation. The function to be realized is first completely decomposed about its variables to form a tree. The functions resulting at the tips of the tree branches from the complete decomposition have simple ``value functions´´ (which specify a set of multithreshold realizations) that are uniquely determined by the weight chosen for the final variable in the decomposition. After choosing this weight, the given function is systematically reconstructed from the tree by choosing the weight of one variable at a time through the combined use of a theorem that specifies the allowable weights of the new variable and an analysis of the minimum possible number of thresholds required as a result of a given allowable weight. Experimental results of hand and computer calculations show the procedure to be fast in producing realizations with a small number of thresholds and a small sum of weight magnitudes.