Variable-Threshold Threshold Elements

Variable-threshold threshold elements (V-T threshold elements) are threshold elements in which the weights are fixed while the threshold may be varied. When the threshold is varied, a set of functions is generated. A concept termed complete uniformity, analogous to complete monotonicity, is shown to be a necessary condition for a set of functions to be realizable by a single V-T threshold element. Theorems which aid in checking complete uniformity are given. Necessary and sufficient conditions for realization of a set of Boolean functions by a single V-T threshold element are given for both exact and inclusion realization. These provide synthesis techniques and methods of applying them efficiently are described. As in the constant-logic problem, the final solution involves the solution of sets of linear inequalities. A method of solving these inequalities using backtrack programming is given; this method is applicable to the constant-logic problem as well.