Realization of Logical Circuits with Majority Logical Function as Symmetrical Function

The paper deals with the “production” and design of symmetrical functions, particularly aimed at the design of circuits with majority elements, which lead to interesting solutions of logical structures. The solutions are presented in several examples, which show the applicability of the procedures to the design of FPGA morphology on chips.


Introduction
Binary logical circuits designed with respect to Boolean symmetrical, particularly majority, output functions are certainly worth attention.The article, therefore, makes an evaluation both by controlling binary one-digit adders and by using functions interpreted by arithmetic polynomials.It also demonstrates how effectively the Shannon decomposition of the output functions can be used in designing a circuit with majority elements.

Let a Boolean function f xx x m m
:{ , } { , } : , , , 0 ® K and y be given.If we denote the set { } x i i m =1 of arguments x 1 by the symbol X, we can briefly write f (X) instead of f (x 1 , x 2 , …, x m ).In addition, instead of f (x 1 , x 2 , …, x i-1 , s i , x i+1 , …, x m ), in which s i Î{ , } 0 1 , let us simply write f (x= i s i ).Let where n £ m esp., , ,  , , )  s s s

K K
The partial derivation ¶ ¶ f X x i ( ) [1] of function f X ( ) by the argument x will be termed the Boolean function

K
defining the conditions under which f X ( ) changes its value while the value of the argument x is changed.
For example, for y x x x x x = Ú 2 3 1 2 3 , when ( ) the function y changes its value while the value of the argument x 1 is changed under one condition for w y x

Boolean formulae and arithmetic polynomials
Let us have where k = 1, 2, … , m and i = 0,1, …, 2 m -1, the function f (X) is represented by a normal disjunctive formula (ndf f (X)).If there holds where k = 1, 2, … , m and j = 0,1, …, 2 m -1, the conjuncts presented are termed orthogonal, i.e., all conjuncts of a complete normal disjunctive formula of the symmetrical Boolean function (see Paragraph 4) are mutually orthogonal.If all conjuncts ndf f (X) are mutually orthogonal, we can also write Note that the function f (X) can also be conveniently expressed by the Boolean (Zegalkin) polynomial [4].
Since, as can easily be confirmed, the following equality holds: x y x y xy Ú = + - where , this can be done either by applying the equality x y x y xy Ú = + -, or orthogonalizing all conjuncts ndf f (X) and applying the absorption x xy . Note that if the Boolean function f X ( ) is expressed by the arithmetic polynomial A X ( ), then Ú be given.Express the given formula by means of the arithmetic polynomial A x x x ( , , )

Symmetrical Boolean function
Let the bijection X « X: , , , , , , K K « be a set of all permutations of arguments from X; the function f X ( ) is called sym- be a set of integers P j (called operational or characteristic numbers) such that 0 The symmetrical function with characteristic numbers P j will be denoted { } For example: { } S x y The symmetrical function { } S P m is elementary; for the length { } cndf S P m of the completely normal disjunctive for- 1 2 , , , ( ) There also holds [2] { as well as For example be expressed by the composition For example: , resp.
Symmetrical functions are discussed in greater detail, e.g., in [2,3,4].The majority function { } Maj M m refers to the symmetrical function For the three-variable majority function # ## can be used.There obviously holds x x x x

Numerical representation of symmetrical functions
We might construct a minimal normal disjunctive formula to a given symmetrical function [4]  Further, consider a one-digit binary half-adder or an adder (Fig. 1) with which a i , b i are binary augmenters, S i is the sum in i position, and C i -and C i + denote the transfer from the position i -1 to the position i + 1, respectively.The half--adder can be modeled by a system of output functions 2 ; by analogy for the adder we obtain { } It is therefore sufficient to provide the half-adder with an inverse disjunctor (Fig. 1a) and the adder with a decoder (Fig. 1b) and we obtain the products and Example 2.: Design a structural model with adders or half--adders modeled with a system of output symmetrical functions S -Fig.2. Indeed,   If the Boolean function f X ( ) is symmetrical, it can be suitably expressed by an arithmetic polynomial in the form for example for The parametric notation Hence b b ) (

Circuits with majority elements
Let us limit ourselves to majority elements modeled with the function { } Maj 2 3 .
Since, as can be easily confirmed, there holds When stating the formula which expresses the derivation of the function we will preferably use a map, to each field of which we will write the value in the form of a fraction: the resulting value of the remainder function formula as well as the weight of its derivation is evident (Fig. 5).There also holds

Conclusion
It appears that it is feasible to produce symmetrical Boolean functions in a sufficiently simple way by a suitable control of one-digit binary adders or by numerical representation of values of the respective arithmetic polynomials, and to design logical circuits with majority elements by applying the Shannon decomposition of the given output function through effective selection of the arguments by which the decomposition is carried.
Boolean function f (x 1 , x 2 , …,x m ) can be expressed, without loss of generality, by the Shan- the elementary symmetrical Boolean functions the representation of which in the form of normal disjunctive formulae (ndf) does not contain negated variables with the symbol { } S n m (n = 0, 1, …, m).
or decompose the given Czech Technical University in Prague Acta Polytechnica Vol. 45 No. , according to the constructed formulae, a structural model of the given function in one of the structurally complete systems of statistical elements [5].