The paper deals with the problem of "compound" and "cascade threshold" element "synthesis" of an arbitrary Boolean function from the "multithreshold weight threshold vector" (MTWTV). The above synthesis procedure is presented to reveal a unique feature of the multithreshold weight threshold vector, from which several realizations of threshold element nets can be obtained from one of the multithreshold weight threshold vectors.

An algorithm for the realization of k-threshold threshold realizable functions is presented. Instead of solving the set of linear inequalities, where the unknowns are the weights corresponding to the input variables, incremental weights are sought. The procedure reduces to that of resolving contradicting pairs of vertices by the incremental weights. The minimum number of thresholds are sought for each complementation and permutation of input variables. A definition of an optimal multithreshold weight threshold vector is derived from the reliability viewpoint. The desired solution is obtained through a search of all possible obtainable realization vectors of the function. For single-threshold realizable functions, permutation and complementation of input variables need not be considered if the input variables of the function are ordered and positivized. The procedure is systematic and has been programmed in FORTRAN IV. As a comparison with Haring and Ohori's tabulation on the 221 equivalence classes of four variable Boolean functions under the NPN1operation, it can be seen that 42 of the 221 equivalence classes need fewer numbers of thresholds for their realization. For the same number of thresholds, 58 equivalence classes have less absolute sum of weights. Finally, with the number of thresholds and absolute sum of weights being equal, 36 equivalence classes have lower threshold values.

This paper deals with input tolerance considerations of a multithreshold threshold element. The concept of a multilevel minus-plus-one model[9]of a threshold element is introduced. It is shown that the multilevel minus-plus-one model is equivalent to the zero-one model[9]of a multithreshold element. The equivalence is established through a set of defining equations similar to that of a single threshold threshold element. For a nonlinearly separable Boolean function, more than one multithreshold weight threshold vector may exist for its realization, even though the number of thresholds and the sum of the absolute magnitude of all the input weights are the same; i.e., Σni=1|wi|. From reliability considerations, the optimal synthesis of the multithreshold threshold elements is shown to depend on the magnitudes of max (|T1|, |Tk|) in addition to the minimum number of thresholds needed to realize the Boolean function and the min Σni=1|wi|. An example is given for illustrative purposes of choosing an optimal realization. For a given margin of operation with the error model assumed, it is shown that one need only work from the normalized set of inequalities. The solution with margins specified is obtained from the optimal solutions of the normalized system of inequalities.

Properties of self-dual and self-complementary dual functions are discussed. Necessary and sufficient conditions of self-dual and self-complementary dual functions are obtained in terms of the multithreshold weight threshold vector. In particular, self-dual and self-complementary dual functions are shown to be realizable only by an odd and even number of effective thresholds, respectively. A threshold Tjis effective if Eminjmax. It is shown that n+1 variable self-dual and self-complementary dual functions can always be generated from 1- and 2-effective-threshold weight threshold vectors of n-variable Boolean functions, respectively. If the number of effective thresholds exceeds 2, constraints on the thresholds must be met in order to generate n+1 variable self-dual and self-complementary dual functions. Such generations of self-dual and self-complementary dual functions are shown to correspond to the functional forms of self-dualization and self-complementary dualization of an n-variable Boolean function. Moreover, they are realized by the same threshold vector T. Furthermore, it is shown that if an n-variable Boolean function Fn(X) is self-dual or self-complementary dual with weight threshold vector [Wn; T], then an n+m variable self-dual or self-complementary dual Boolean function Fn+m(X), where m is any positive integer, can be realized by a weight threshold vector [Wn+m; T]. The above cited weight vectors Wnand Wn+mare constrained by If Σni=1Wi= Σn+mi=1Wi. Σni=1|Wi| = &3x003A3;n+mi=1|Wi|, then optimal realization vectors seem to be obtained.