The Application of Majority Voting Functions to Estimate the Number of Monotone Self-Dual Boolean Functions

Abstract

One of the problems of modern discrete mathematics is Dedekind’s problem on the number of monotone Boolean functions. For other precomplete classes, general formulas for the number of functions of the classes had been found, but it has not been found so far for the class of monotone Boolean functions. Within the framework of this problem, there are problems of a lower level. One of them is the absence of a general formula for the number of Boolean functions of intersection $MS$ of two classes—the class of monotone functions and the class of self-dual functions. In the paper, new lower bounds are proposed for estimating the cardinality of the intersection for both an even and an odd number of variables. It is shown that the majority voting function of an odd number of variables is monotone and self-dual. The majority voting function of an even number of variables is determined. Free voting functions, which are functions with fictitious variables similar in properties to majority voting functions, are introduced. Then the union of a set of majority voting functions and a set of free voting functions is considered, and the cardinality of this union is calculated. The resulting value of the cardinality is proposed as a lower bound for $\left| {MS} \right|$. For the class $MS$ of monotone self-dual functions of an even number of variables, the lower bound is improved over the bounds proposed earlier, and for functions of an odd number of variables, the lower bound for $\left| {MS} \right|$ is presented for the first time.