Eight Variables

The number of threshold functions of eight variables is counted by ILLIAC II, the computer of the University of Illinois. Sets of optimum weights of majority elements realizing these functions also are investigated. Actually, canonical positive self-dual threshold functions of nine variables are investigated instead of directly investigating threshold functions of eight variables because it is easier to deal with them. The number and optimum weights of threshold functions of eight variables are easily obtained from these functions of nine variables and their realization. First a linear program to minimize the total input weight is considered. Canonical positive self-dual functions of nine variables are generated by modifying Winder's method and tested by the simplex method to find whether or not they are threshold functions. The number of canonical positive self-dual threshold functions of exactly nine variables and the number of all threshold functions of exactly eight variables are 172 958 and 17 494 930 604 032, respectively. It had been an open question whether there exists any completely monotonic function of exactly eight variables which is not a threshold function. But the computational result verified that there was no such function. Unlike the case of threshold functions of seven or fewer variables, there are some threshold functions of exactly eight variables whose extreme optimum weights include fractional numbers. Other properties of these functions are also explored. Next a linear program to minimize the threshold is considered. The linear program is considerably different from the previous linear program to minimize the total input weight with respect to uniqueness of an optimum solution. The number of threshold functions which have multiple optimum solutions is counted.