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

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 function...

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Vydáno v:Automatic control and computer sciences Ročník 57; číslo 7; s. 706 - 717
Hlavní autoři: Bystrov, L. Y., Kuzmin, E. V.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Moscow Pleiades Publishing 01.12.2023
Springer Nature B.V
Témata:
Boolean
Boolean functions
Codes
Computer Science
Control Structures and Microprogramming
Equilibrium
Formulas (mathematics)
Intersections
Lower bounds
Monotone functions
Variables
Voters
Boolean functions with fictitious variables
Dedekind’s problem
disjunctive normal form
monotone boolean functions
majority voting functions self-dual boolean functions
equilibrium sets
free voting functions
ISSN:0146-4116, 1558-108X
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Shrnutí: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 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 . For the class 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 is presented for the first time.
Bibliografie:ObjectType-Article-1
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content type line 14
ISSN:0146-4116
1558-108X
DOI:10.3103/S0146411623070027