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 selfdual 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 respectively by an odd and even number of effective thresholds only. A threshold, Tj, is effective if EMIN ≪ Tj ≪ EMAX. It is shown that an n+1 variable self-dual and self-complementary dual function can always be generated from a 1- and 2-effective threshold weight threshold vector of an n-variable Boolean function, respectively. If the number of effective thresholds exceeds 2, then 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-cornplementary 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 $\rarr$;. Furthemore, it is shown that if an n-variable Boolean function Fn(X$\rarr$) is self-dual or self-complementary dual with weight threshold vector [Wn$\rarr$; T$\rarr$], then an n+m variable self-dual or self-complementary dual Boolean function Fn+m(X$\rarr$), where m is any positive integer, can be realized by a weight threshold vector [Wn+m$\rarr$; T$\rarr$]. The above-cited weight vectors Wn and Wn+m, are constrained by Sigma i=1n wi = Sigma i=1n+m wi. If Sigma |wi|= Sigma i=1n+m |wi|, then optimal realization vectors seem to be obtained. Similarly n-m variable self-dual or self-complementary dual functions can like-wise be obtained; however, realization vectors are in general not optimal in nature. Since self-complementary dual functions can always be obtained for 2-threshold realizable functions, the present discussion suggests the classification of 2-threshold realizable functions by self-complementary dual functions.