A Tabular Method for the Synthesis of Multithreshold Threshold Elements

Abstract

This paper presents a tabular method for synthesizing Boolean functions having four or less variables with multithreshold threshold elements. The method is similar to that used for conventional single-threshold threshold elements. All 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> functions of four variables are divided into 221 equivalence classes by variable complementations and/or permutations and/or function complementation. Each equivalence class is characterized by a subset of its corresponding Rademacher-Walsh coefficients, the size of the subset being determined by the number of thresholds required to realize that equivalence class. An arbitrary Boolean function of four or less variables is synthesized by systematically calculating subsets of its Rademacher-Walsh coefficients until, through simple equivalence operations, the equivalence class of the function is found in a table of the 221 equivalence classes. The table indicates a multithreshold realization of the given function. The table shows that any 4-variable function can be realized with at most five thresholds, or by a network of conventional, or single-threshold, threshold elements with at most three gates in which each gate has the identical weight vector for the four input variables.