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Issue No.01 - January (1968 vol.17)

pp: 57-66

Chao-Wei Mow , Purdue University, Lafayette, Ind.; Litton Industries, Inc., Guidance and Control Systems Division, Woodland Hills, Calif.

King-Sun Fu , School of Elec. Engrg., Purdue University, Lafayette, Ind.; Visiting Professor at the University of California, Berkeley, Calif.

ABSTRACT

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 T<inf>j</inf> is effective if E<inf>min</inf> ≪ T<inf>j</inf> ≪ E<inf>max</inf>. 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 F<inf>n</inf>(X) is self-dual or self-complementary dual with weight threshold vector [W<inf>n</inf>; T], then an n+m variable self-dual or self-complementary dual Boolean function F<inf>n+m</inf>(X), where m is any positive integer, can be realized by a weight threshold vector [W<inf>n+m</inf>; T]. The above cited weight vectors W<inf>n</inf> and W<inf>n+m</inf> are constrained by If ¿<sup>n</sup> <inf>i=1</inf> W<inf>i</inf> = ¿<sup>n+m</sup> <inf>i=1</inf> W<inf>i</inf>. ¿<sup>n</sup> <inf>i=1</inf> |W<inf>i</inf>| = ¿<sup>n+m</sup> <inf>i=1</inf> |W<inf>i</inf>|, then optimal realization vectors seem to be obtained.

CITATION

Chao-Wei Mow, King-Sun Fu, "Generation of Self-Dual and Self-Complementary Dual Functions",

*IEEE Transactions on Computers*, vol.17, no. 1, pp. 57-66, January 1968, doi:10.1109/TC.1968.5008870