Generation of Self-Dual and Self-Complementary Dual Functions

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 Tj is effective if Emin ≪ Tj ≪ Emax. 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 Wn and Wn+m are constrained by If ¿ⁿ i=1 Wi = ¿^n+m i=1 Wi. ¿ⁿ i=1 |Wi| = ¿^n+m i=1 |Wi|, then optimal realization vectors seem to be obtained.