DEVELOPING THE MINIMIZATION OF A POLYNOMIAL NORMAL FORM OF BOOLEAN FUNCTIONS BY THE METHOD OF FIGURATIVE TRANSFORMATIONS

This paper reports a study that has established the possibility of improving the effectiveness of the method of figurative transformations in order to minimize Boolean functions on the Reed-Muller basis. Such potential prospects in the analytical method have been identified as a sequence in the procedure of inserting the same conjuncterms of polynomial functions followed by the operation of super-gluing the variables.
The extension of the method of figurative transformations to the process of simplifying the functions of the polynomial basis involved the developed algebra in terms of the rules for simplifying functions in the Reed-Muller basis. It was established that the simplification of Boolean functions of the polynomial basis by a figurative transformation method is based on a flowchart with repetition, which is actually the truth table of the predefined function. This is a sufficient resource to minimize functions that makes it possible not to refer to such auxiliary objects as Karnaugh maps, Weich charts, cubes, etc.
A perfect normal form of the polynomial basis functions can be represented by binary sets or a matrix that would represent the terms of the functions and the addition operation by module two for them.
The experimental study has confirmed that the method of figurative transformations that employs the systems of 2-(n, b)-design, and 2-(n, x/b)-design in the first matrix improves the efficiency of minimizing Boolean functions. That also simplifies the procedure for finding a minimum function on the Reed-Muller basis. Compared to analogs, this makes it possible to enhance the performance of minimizing Boolean functions by 100‒200 %.
There is reason to assert the possibility of improving the efficiency of minimizing Boolean functions in the Reed-Muller basis by a method of figurative transformations. This is ensured by using more complex algorithms to simplify logical expressions involving a procedure of inserting the same function terms in the Reed-Muller basis, followed by the operation of super-gluing the variables.


Introduction
An arbitrary Boolean function f (x 1 , x 2 , …, x n ) can be represented in an Exclusive-OR Sum-Of-Product form (ESOP), created by double conjunction operations (AND) and the amount by mod 2 (EXOR) and a unity constant; the inversion of an arbitrary variable is produced by the opera-tion x x ⊕ = 1 . At the same time, depending on which variables of the ESOP conjuncterms f (all or some of them) have or do not have an inversion sign, which determines the so-called polarity of variables, the classes of AND/EXOR expressions of Boolean functions ESOP are distinguished. In a general case, they are termed Reed-Muller expres-sions (RM-polynomials) [1]. The taxonomy of RM-polynomials, the relationship between different classes, and the complexity of their implementation are considered in [1][2][3][4][5][6].
For comparison, we shall illustrate possible polynomials by following the examples: x x x x x 1  The development of microelectronic technology has ensured the creation of elements that form multiple disjunctions with exception (EXOR elements). This, in turn, ensured the synthesis of similar two-level AND/EXOR circuits that contain the same elements in the second cascade. The structure of these schemes is described by formulas similar to a disjunctive normal form (DNF), in which disjunction operators with an exception are used instead of disjunction operators. Such formulas are called ESOP -the exclusive sum of products.
The advantages of these formulas are justified by the fact that the number of logic elements in their respective schemes is usually less. For example, after minimizing the DNF of arbitrary Boolean functions, the four variables contain an average of 4.13 conjuncterms, and ESOP contains only 3.66 [4,7]. When considering Boolean functions typical of schemes that implement arithmetic operations, the win is even greater. In addition, AND/EXOR circuits are more easily diagnosed [7,8].
In the case of representing Boolean functions by Zhegalkin polynomials, the optimization problem does not arise since the solution is unambiguous. The optimization problem appears for non-fully defined Boolean functions. If the value of the function remains undefined in k sets, 2 k different function determination and, accordingly, 2 k different Zhegalkin polynomials forming a given function are possible. The choice among them of the simplest polynomial is a complex combinatorial problem. The task becomes even more difficult when implementing functions in the form of ESOP (which contain literals with different inversions) or when it is necessary to implement a system of Boolean functions.
As stated in [2], effective algorithms for minimizing ESOP of the Boolean functions f(x 1 , x 2 , …, x n ) do not exist. Such a conclusion, however, could be some start, which would eventually move to its own opposite.
The evolution of the visual-matrix form of the analytical method of simplification of logic functions is the result of continuous optimization, in particular [9]. In this regard, theoretical studies on minimizing ESOP of Boolean functions remain relevant, in particular, to improve the following factors: -the visual and matrix methods of minimizing logic functions in the ESOP class; -the cost of a technique to minimize ESOP of logic functions.

Literature review and problem statement
Generalized rules for simplifying logic expressions in the format of the theory of polynomial sets are considered in paper [10]. The rules reported there are based on the proposed theorems for different initial conditions for the transformation of paired conjuncterms, the Hamming distance between which can be arbitrary. These rules can be useful for minimizing arbitrary logic functions with n variables in the theoretical format of polynomials. The advantages of the proposed simplification rules are illustrated by examples.
A method to search for the exact ESOP expression for a not fully specified arbitrary logic function up to six input variables is proposed in work [11]. To this end, the weights of all 5-variable functions are entered in the table, which is used in the proposed approach and speeds up the computation time. It is believed that this is the first paper concerning the exact solutions of minimization for not fully defined Boolean functions.
Work [12] reports the results of a study into minimizing AND-XOR expressions for functions with a large number of literals. The object of the study is the use of a simple greedy algorithm for finding a minimum function, based on a set of local transformations, to the expressions of the Boolean functions of the Reed-M ller basis with positive polarity. It is noted that experiments with large functions demonstrate good results. The number of literals was reduced by an average of 23 %. It is assumed that much better results could be achieved once a more complex non-greedy algorithm for finding minimal functions is applied.
Minimization of multilevel representation of Boolean functional systems based on Shannon extension with finding equal coefficients (exact within inversion), and using Zhegalkin polynomials for these purposes, is proposed in [13]. Zhegalkin polynomials are easy to compare and easy to obtain the inversion of a function, and that significantly reduces the calculation time. The application of the program that implements the proposed algorithms makes it possible to receive smaller areas of VLSI circuits compared to chains that are synthesized using minimized DNF and Shannon expansion schemes where coefficient inversion is not taken into consideration.
Paper [14] reports a method for determining the upper limit of complexity in the implementation of arbitrary Boolean functions, which can be implemented by Zhegalkin polynomials. A computational method for improving these boundaries is proposed.
The representation of Boolean functions by reverse circuits, on the elements of Toffoli, is considered in paper [15]. Interest in this issue is associated with the implementation of «cold» calculations. This means that when performing such calculations, there is no heat dissipation. In general, reversible schemes implement reversible functions. Therefore, the Toffoli-Fredkin method is used to represent a Boolean function by a reversible function. The paper describes an algorithm for finding the minimum representation of a Boolean function in the class of reversing chains built on Toffoli elements. The algorithm uses ESOP of Boolean functions, as well as the class of polarized Zhegalkin polynomials or the Reed-Muller forms. The results of the computational algorithm that minimizes Boolean functions in the class of reversible circuits are presented.
Minimizing logic has been attracting considerable attention lately, as it is important for many applications to have the most compact images possible. Paper [16] proposes a fast minimizing algorithm (FMA) for the Reed-Muller fixed polarity expressions (FPRM). The basic FMA idea is to find the minimum FPRM function with the fewest conjuncterms. This uses the proposed binary differential search evolution (BDE) algorithm. The authors described experimental results involving 24 MCNC control circuits, which show that FMA surpasses the genetically based effectiveness of finding the minimum Reed-Muller expressions. It is assumed that the use of the differential evolution algorithm to minimize FPRM was first considered in paper [16]. FMA can be expanded so that it is possible to obtain a minimum mixed polarity of the expression on the polynomial basis.
The above literary sources [10][11][12][13][14][15][16] mostly report algorithms and methods for minimizing Boolean functions in the Reed-Muller basis by using theoretical objects of adjacent theory. Specifically, Hamming distance tables, greedy (nongreedy) algorithms, Shannon extensions, genetic algorithm, differential evolution algorithm to minimize FPRM, etc. Not all methods provide an accurate solution to minimization. An obligatory technological point for these algorithms and methods are automated calculations. In the complex search for the optimal function, compensation may be an approximate synthesis -the tendency of logic synthesis, when some results of the logical specification change within the permissible optimality of the digital circuit to be designed.
A method of figurative transformations based on binary combinatorial systems with repeated 2-(n, b)-design, 2-(n, x/b)-design belongs to the classical analytical method by qualification. They do not allow for the approximate result of minimization and do not exclude the manual technique for minimizing Boolean functions, including in the Reed-Muller basis.
Thus, the algorithms and methods that employ theoretical objects of adjacent theory, software tools for them, which cover the general procedure for minimizing the logic functions of the polynomial basis [10][11][12][13][14][15][16] and a method of figurative transformations, follow different approaches (principles of minimization). Therefore, they consider different prospects regarding the possibility of algorithmic minimization of logic functions of the polynomial basis.
The prospect for a figurative transformation method, which is a descendant of the analytical method, regarding the proper minimization of logic functions in the Reed-Muller basis is to create the necessary algebra in terms of the rules for the equivalent transformation of polynomial functions. As well as identify the reserves of the analytical method, such as the sequence in the procedure of inserting the same conjuncterms of polynomial functions, followed by the operation of super-gluing the variables. Thus, the classical analytical method still has the prospect of increasing its hardware capabilities to minimize functions on the Reed-Muller basis. And this is the reason to believe that the software and technological base, represented by the algorithms and methods with theoretical objects of adjacent theories [10][11][12][13][14][15][16], is insufficient for theoretical research into the optimal minimization of Boolean functions on the Reed-Muller basis.
This determines the need to investigate equivalent figurative transformations in order to minimize logic functions on the Reed-Muller basis. In particular, the peculiarities of relatively complex algorithms of simplification of functions with the procedure of inserting the same conjuncterms of polynomial functions followed by the operation of super-gluing the variables, a stack of logical operations for the first binary matrix of a polynomial function [9], ways to simplify arbitrary functions in the Reed-Muller basis.
In practical terms, a figurative transformation method would provide an expansion of the capabilities of technology for designing digital components based on the basis

The aim and objectives of the study
The purpose of this work is to extend the method of figurative transformations to minimize Boolean functions in the class of Exclusive-OR Sum-Of-Product forms (ESOP) and perfect Exclusive-OR Sum-Of-Product forms (PESOP). That could simplify the performance of minimizing functions in the Reed-Muller basis by refining Reed-Muller's algebra in terms of algebraic rules for the equivalent transformation of ESOP functions.
To accomplish the aim, the following tasks have been set: -to establish the hermeneutics of logic operations for the class of equivalent binary matrices of functions of the Reed-Muller basis; -to establish patterns in the use of algorithms for the equivalent transformation of Boolean functions in the Reed-Muller basis, consisting of the procedure of inserting two identical ESOP conjuncterms with the following operation of super-gluing the variables; -to devise a method for the orthogonalization of logic functions using figurative transformations in order to establish singular functions; -to refine Reed-Muller's algebra in terms of the necessary algebraic rules for the equivalent transformation of ESOP functions; -to analyze the results from simplifying functions in the ⊕ The completeness of this basis is proven by the following ratios, which demonstrate that it is reduced to a well-known full basis ¬ ∨ { } , : Similar to disjunction and conjunction operations, the sum by module two has the properties of commutativity and associativity, and is also generalized in the case of a large number of variables.
Multiple amount for module two of:

ReedMuller Algebra
An algebra over a set of logic functions with two binary operations & and ⊕ is the Reed-Muller (Zhegalkin) algebra. The following interrelations are satisfied in the Reed-Muller algebra: x y x y xy x y x y Identity (2) has an illustration of the representation: For the function of addition modulo two, there are displaceable and connecting laws, as well as the distribution law with respect to the conjunction.
In addition, the following formulas hold: x y x y x y xy x y x y y Logic identities for two variables are given in Table 1. The Reed-Muller algebra provides the creation of classical rules of equivalent transformation to simplify logic expressions in the ESOP class by analytical method. The difference between a Reed-Muller polynomial and a Zhegalkin polynomial is that the Zhegalkin polynomial represents 2-level logic, and the Reed-Muller polynomial represents 3-level logic. However, a Reed-Muller polynomial generally contains fewer literals.
In some cases, the transformation of the Reed-Muller polynomial to the mixed basis produces 2-level logic. Such as: The 3-level logic of the Reed-Muller polynomial is transformed into 2-level mixed basis logic.

Interpretation of the truth table of a logic function by the binary combinatorial systems with repetition
For some set A, a new set M(A) can be considereda set of all its subsets -a Boolean. М k (A) denotes the set of all subsets A that have k elements.
Assume А = {a, b, c}, then: , , , , , , , ; The number of all k-element subsets of the set of n elements is: The following equality holds: Since C n k is the number of k-element subsets of the set of n elements, the sum in the left-hand side of expression (7) is the number of all subsets.
The set А = {a, b, c}, in addition to recalculating its elements, can also determine the number of positions on which the element a is located. For example, a can mean the first position, b can mean the second position of the set А = {a, b, c}, etc. Subsets of the set А = {a, b, c}, in this case, are the subsets containing the element α at k positions, k = 0, …, n, where n is the number of positions of the set A. In a general case, the element a can occupy several positions on the set A, so the element a is repeated on the set A.
Let α = 1, then the positions where the element a is missing are denoted by zero.
For the set А = {a, b, c}, which defines position numbers, we accept α = 1. Then the subsets of set A take the following form: The number of all k-element subsets of the set А = {a, b, c}, which determines the position numbers, is calculated from formula (7).
Configuration (8) is a complete combinatorial system with a repeated element α, which is denoted: where n is the bit size of the system block; b is the number of blocks of the complete system, determined from formula b n = 2 , the number 2 before the brackets denotes the binary structure of configuration (8). For example, 2-(4, 16)-design is the complete binary combinatorial system with a repe tition consisting of 4-bit blocks, the number of blocks is 16.
It is easy to see that configuration (8), which makes up the complete combinatorial system with repetition, can be interpreted as a truth table of the logic function f(a, b, c), with a full set of minterms or maxterms (Table 2).
Another interpretation variant is demonstrated by a truth table that contains a combinatorial system with repeated 2-(2, 4)-design in the table configuration variant when there is one column with the same variable values (Table 3). Table 3 The truth table of the logic  In this case, the combinatorial system with repeated 2-(2, 4)-design (combinatorial representation) would interpret the login operation, in a given case -super-gluing the variables.
The procedure of reducing the complete perfect disjunctive normal form (PESOP) of a logic function yields unity. For example, reducing a 2-variable full PESOP takes the following form: Since the complete PESOP uniquely determines the complete combinatorial system with repeated 2-(n, b)-design and vice versa, this gives reason to remove all blocks of the full combinatorial system with a repetition from the truth table of the assigned function.
The visual representation of the logic operation of super-gluing the variables involving 2-(2, 4)-design takes the following form: The algebraic notation of the logic operation of supergluing the variables is as follows: Similarly, other operations of equivalent transformation of logic expressions are interpreted.
In a general case, the truth table of the logic function, in addition to the configuration of the complete combinatorial system with repeated 2-(n, b)-design, can also contain the configuration of an incomplete combinatorial system with repeated 2-(n, x/b)-design. In this case, x is the number of blocks of an incomplete combinatorial system with repetition. The properties of an incomplete combinatorial system with repeated 2-(n, x/b)-design also make it possible to set rules that ensure the effective minimization of Boolean functions.
Configuration (8) can also be interpreted as a truth table of the logic function ESOP f (a, b, c), with a full set of conjuncterms of the function in the Reed-Muller basis and the operation of adding modulo two for them.

Results of minimizing Boolean functions in the ReedMuller basis by a figurative transformation method
Equivalent figurative transformations when minimizing functions in the Reed-Muller basis yield the following result: -determining the hermeneutics of logic operations for the class of equivalent binary matrices of functions in the Reed-Muller basis; -a protocol with relatively complex algorithms for simplifying logic expressions, which consists of the procedure for inserting two identical conjuncterms of the functions of a polynomial basis with the following operation of super-gluing the variables. This protocol increases the efficiency of the procedure, which makes it possible, in particular, to simplify logic functions in the Reed-Muller basis with a relatively large number of input variables manually; -ensuring the method of orthogonalization of specified logic functions for the establishment of singular functions; -creating algebra in terms of the rules for equivalent transformation of Boolean functions in the Reed-Muller basis.

1. Hermeneutics of logic operations for the class of equivalent binary matrices of functions in the Reed Muller basis
Represent the logic function f x x x n 1 2 , ,...,

( ) in a perfect
Exclusive-OR Sum-Of-Product form (PESOP): where the symbol ⊕ 1 means that the sum by modulo two is taken only on sets of variables < > α α α To represent PESOP (9) by a binary equivalent or a matrix, variables with inversion x n must be replaced with 0 n , and variables without inversion x n -with 1 n , where n is a numeric index that determines the bit size of the variable character «1» or «0» in the conjuncterms of the Reed-Muller basis function. Then PESOP (9) can be represented by the following binary sets (tuples).
or the following matrix: We shall call matrix (11) an instance of the binary matrix class of the Reed-Muller basis functions.
The hermeneutics of logic operations for matrix (11) is that matrix (11) expresses the conjuncterms of the Reed-Muller basis function and the operation of adding modulo two for them. Our hermeneutics of logic operations in the Reed-Muller basis should be used when deriving the result of logic operations in the class of binary matrices of the Reed-Muller basis functions.

2. Patterns of using the logic operation of super gluing the variables for the polynomial normal form of Boolean functions
The implementation of logic operations on binary and algebraic function structures is to be highlighted in color if you know what we mean. That would provide for a better didactic of the method under consideration.
For the method of figurative transformations, the algorithm of function simplification involving the procedure of inserting two identical conjuncterms of polynomial functions with the following operation of super-gluing the variables may have the following variants, for example: 1. 2. x 3.
Comparatively complex algorithms for simplifying logic expressions involving the procedure of inserting two identical conjuncterms of polynomial basis functions with the following operation of super-gluing the variables (12), (14), (16) expand the variants of their application. This ensures an increase in the efficiency of the procedure for minimizing Boolean functions in ESOP by using a method of figurative transformations.

3. Singular functions
To solve optimization problems of logic synthesis, it is necessary to have PESOP functions and RM-polynomials with a minimum number of conjuncterms of the assigned function f(x 1 , x 2 , …, x n ). At the same time, if it is possible to choose an RM polynomial (except for PPRM-and NPRM-polynomials), then in the case of the same number of conjuncterms, preference is given to the RM-polynomial with the minimum total number of literals. And when the number of the latter is the same, the minimum RM polynomial is the one with fewer inverted literals. Thus, the cost of implementing the RM-polynomial of the assigned function f(x 1 , x 2 , …, x n ) can be estimated by the numerical ratio k k k l i n θ / / , where k θ , k l , k in is the number of conjuncterms, literals, and inverters, respectively [1]. A similar assessment of the cost of implementing a minimum function can be applied to the PESOP function and, to some extent, to logic functions on a mixed basis. Transitions between the Bull and Reed-Muller bases are carried out using a singular (special) function, the conjuncterms of which are pairwise orthogonal.
To convert the PESOP function into a normal polynomial form, it is necessary to orthogonalize the assigned function.
Example 1: It is required to orthogonalize function F x x x x 1 2 3 4 , , , The conjuncterms of function (19) are pairwise orthogonal. In algebraic notation, function (19) takes the following form (20): x x The result of the orthogonalization of assigned function (18) by figurative transformations (19) and algebraic technique (21) is the same.

4. Equivalent transformations of logic functions on the ReedMuller basis
Since x x ⊕ =1, the Reed-Muller algebra allows the operation to glue variables: Proof: The rule of gluing the variables for ESOP (22) has an illustration of the combinatorial representation (23).
The logic operation of semi-gluing the variables for ESOP (24) is as follows: Proving rule (24) is based on formula (4). And, since the left-hand side of (24) is singular, in order to implement the simplification procedure by means of the operation of semi-gluing the variables, it is necessary to switch from ESOP to PESOP of the logic function.
The operation of semi-gluing the variables (24) has an illustration of the representation (25).
The hermeneutics of the logic operation of semi-gluing the variables (25) implies a transition from the ESOP function to the PESOP function.
Other variants of semi-gluing the variables: x x x x x The operation of semi-gluing the variables for ESOP can take the following form, for example [18]: Proof: The operation of semi-gluing the variables (28) has an illustration of the representation (30) [18].
Rule (28) in paper [18] is proved by using the Hamming distance.
The rule of semi-gluing the variables for PESOP logical functions.
For 4-variable conjuncterms of PESOP logic functions, the rule of super-gluing the variables can take the following form, for example: The super-gluing rule for ESOP (32) is based on the use of a complete combinatorial system with repeated 2-(2, 4)-design [19].
The super-gluing variable rule for ESOP, which uses a complete combinatorial system with repeated 2-(3, 8)-design [19] may take the following form, for example:

The rule of incomplete super-gluing the variables for ESOP logic functions.
Combinatorial properties of an incomplete combinatorial system with repeated 2-(n, x/b)-design [19] ensure a rule of incomplete super-gluing the variables in the Reed-Muller basis.
For a 2-variable ESOP function's conjuncterms, the rule of incomplete super-gluing the variables may take the following form, for example: In the second matrix of (35), the operation of semi-gluing the variables for ESOP of logic functions (24) was applied, the result of which is represented in the third matrix (35).
Generalized gluing the variables in the Reed-Muller basis can be carried out using the following transformations: Proof: The second and third matrices (37) involve an operation of absorbing the variables for PESOP (40) to (45).
The rule of generalized gluing the variables (38) has an illustration of representation (39): Both generalized variable gluing rules (36) and (38) have a 3-level logic. Applying rule (38) reduces the original expression by one literal.
The logic operation of variable absorption for ESOP is as follows: The validity of logic expression (40) is confirmed by a truth table (Table 4). Table 4 The truth table of the logic operation of absorbing the variables x x x xx Other variants of the operation of absorbing the variables for ESOP: x x x x x The created classical rules for the equivalent transformation of logic functions in the Reed-Muller basis ensure their effective simplification by a figurative transformation method.

5. Examples of minimizing Boolean functions in the ReedMuller basis by a figurative transformation method
Proving the advantage of logic expressions in a polynomial format comes down to a smaller number of logic elements in their respective schemes, compared to the expressions of the disjunctive (conjunctive) form. It is also promising to use a mixed basis.

Solution.
Since the conjuncterms of the initial function are pairwise orthogonal (a singular function), in order to simplify the assigned function, we select the Reed-Muller algebra. The minimization of f(x 1 , x 2 , x 3 , x 4 ) (Fig. 1) in a polynomial format is carried out using the following figurative transformations: ) .
Function f(x 1 ,x 2 ,x 3 ,x 4 ) MPNF (Fig. 1): The cost of (46) implementation is k k k l i n θ / / / / . = 3 6 3 To minimize the function in Fig. 1, an algorithm to simplify the function involving the procedure of inserting two identical ESOP conjuncterms with the following operation of super-gluing the variables (chapter 7.2) was applied.
The simplified function f(x 1 , x 2 , x 3 , x 4 ) (Fig. 1) in the mixed basis: The cost of (47) implementation is k k k l i n θ / / / / . = 2 6 2 Both functions (46) and (47) represent a 3-level logic. On a mixed basis, the minimum function (47) has better implementation indicators. Table 5 gives the results from minimizing the function f(x 1 , x 2 , x 3 , x 4 ) ( Fig. 1) in ESOP using a Karnaugh map [20] and the method of figurative transformations.
When contemplating Table 5, we see that the result of simplifying the function f (x 1 , x 2 , x 3 , x 4 ) (Fig. 1) by a figurative transformation method is a minimal function containing six literals. This is four literals less compared to [20].
Example 3: It is required, by using the method of figurative transformations, to simplify the Boolean function f (x 1 , x 2 , x 3 , x 4 ) in ESOP, which is set in the canonical form [18]: x x MPNF of function f(x 1 ,x 2 ,x 3 ) (48): When simplifying function (48), identity (3) is taken into consideration.
The results of minimizing (49) function (48) by the method of figurative transformations and by the method of splitting ESOP conjuncterms [18] coincide. The indicator of the implementation of function k k k l i n θ / / / / , = 3 9 5 where k θ is the number of simple implicants, k l is the number of input variables, k in is the number of inverted variables. However, the computational complexity of the procedure for minimizing the Boolean function with figurative transformations is less.
Function (48) can be simplified using the Zhegalkin polynomial.
Zhegalkin polynomials for the constituents of function (48) are given in Table 7.   The Zhegalkin polynomial corresponding to function (48) takes the following form: The simplification of the Zhegalkin polynomial is carried out with the help of figurative transformations. In the simplification of Zhegalkin polynomial, Zhegalkin algebra (Reed-Muller) is used. The simplification of the Zhegalkin polynomial turns the latter into a Reed-Muller polynomial (50): Minimal functions (49) and (50) match. Example 4: It is required, by using a figurative transformation method, to simplify the Boolean function f(x 1 , x 2 , x 3 , x 4 ) in ESOP, which is set in the canonical form [18]: MPNF of function f(x 1 , x 2 , x 3 , x 4 ) (51): 1  2  3  4  1 2 3 4  1 2  3 4 .
(52) Table 8 gives the results from minimizing the function f(x 1 , x 2 , x 3 , x 4 ) (51) in ESOP by splitting the ESOP conjuncterms [18] and by a figurative transformation method. Table 8 The result of minimizing the function f (x 1 , x 2 , x 3 , x 4 ) (51) in ESOP

Method of splitting conjuncterms
Figurative transformation method When contemplating Table 8, we see that the result of minimizing the function f(x 1 , x 2 , x 3 , x 4 ) (51) by a figurative transformation method is a minimal function (52), containing 12 literals. This is 2 literals less compared to [18] and 3 literals less compared to [22].
The verification of the resulting MPNF (52) is given in Table 9. Table 9 shows that MPNF (52) Example 5: It is required, by using a figurative transformation method, to simplify the Boolean function f(x 1 , x 2 , x 3 , x 4 ) in ESOP, which is set in the canonical form (53) [23]:
MPNF of function f(x 1 ,x 2 ,x 3 ,x 4 ) (53): The cost of (55) implementation is that matches [23], however, the function represented by the eleventh matrix in (54): which represents a 3-level logic as well, but at the following cost of implementation: which is better than the implementation cost (56) of minimum function (55).
The verification of the resulting MPNF of assigned function (53) in a mixed basis (57) is given in Table 10. Table 10 Verification of MPNF in a mixed basis (57) -

Discussion of results of minimizing Boolean functions in the ReedMuller basis by a figurative transformation method
The mathematical apparatus of Boolean function minimization by a figurative transformation method is considered in works [24][25][26], and others. The technique of the method of figurative transformations is given in Table 11. Table 11 Figurative transformation method technique  Table 12. Table 12 Components added to the minimization of Boolean functions by a figurative transformation method The algebra that we created in terms of rules for simplifying functions with an illustration of equivalent figurative transformations of logic procedures makes it possible to spread the method of figurative transformations to minimize Boolean functions on the Reed-Muller basis.
A special feature of the method of figurative transformations is that the method is based on the binary combinatorial systems with repeated 2-(n, b)-design, 2-(n, x/b)-design. For example, the truth table (Table 2) of the logic function f(a, b, c) is a combinatorial system with repetition (chapter 6). This is a sufficient resource to minimize functions and makes it possible to do without auxiliary objects, such as Karnaugh maps, Weich diagrams, acyclic graph, non-directed graph, coverage tables, cubes, etc. The clarity of 2-dimensional binary matrices allows one to manually simplify Boolean functions (using a mathematical editor, such as MathType 7.4.0) within up to 64 input variables [9] for PESOP (DCNF) representation of the function.
The use of the method of figurative transformations to minimize functions in the Reed-Muller basis, to some extent, brings the problem of ESOP simplification to the level of a well-researched problem in the class of disjunctive-conjunctival normal forms (DCNF) of Boolean functions.
The algebra that we created for the equivalent transformation of functions in the Reed-Muller basis is represented by the following logic operations (Table 13): When simplifying Boolean functions on the Reed-Muller basis, it is advisable to use a mixed basis.
Example 6: It is required, by using a figurative transformation method, to simplify the Boolean function f(x 1 , x 2 , x 3 , x 4 )) in ESOP, which is set in the canonical form (58) [23]: and is represented by a truth table (Table 14). Table 14 Truth table of the logic function f (x 1 , x 2 , x 3 , x 4 ) (63)

Solution:
Function (58) is singular. Simplification of function (58) is to be performed for PESOP and ESOP.
Minimization in PESOP. Define the stack of logic operations of the first matrix of the function f(x 1 , x 2 , x 3 , x 4 ) as follows. We combine the sets of variables that contain unity in the far-right position of the set into a separate matrix. Such a matrix is presented in (59) first. In another separate matrix, we combine the sets of variables of function (58) which contain zeros in the extreme right position of the set. Such a matrix is given in (60) first. Simplification of function (58) in each matrix is performed separately. The cost of (61) implementation is: The minimal function f (x 1 , x 2 , x 3 ) (61) in a mixed basis: The cost of (63) implementation is: MPNF of function f(x 1 , x 2 , x 3 , x 4 ) (58): When simplifying function (65) in ESOP, identity (2) is taken into consideration.
The result of minimization (65) coincides with the result of minimization in [23].
The cost of (65) implementation is: The result of minimizing (68) the function (67) coincides with the result of minimization in [18,27] but the procedure for minimizing when using figurative transformations is easier.
The selection of the stack of logic operations [9] for ESOP functions is demonstrated by the following example.
Example 8: It is required to choose the optimal stack of logic operations to simplify the Boolean function f(x 1 , x 2 , x 3 , x 4 ) in ESOP set in the canonical form (69) [18]:
The procedure for splitting conjuncterms [18] and the corresponding stack of logic operations in the first matrix of function (69) produce the following result: Next, the underlined pair of terms in expression (70) is treated with the actual procedure of splitting conjuncterms [18]: The algorithm of ESOP function simplification, which consists of the procedure for inserting two identical conjuncterms with the following operation of super-gluing the variables (chapter 7. 2), and the corresponding stack of logic operations in the first matrix of function (69), produce the following result: The minimum ESOP functions (71) and (72) are the same but the simplification procedure based on the second stack is much simpler.
Limiting the use of the method of figurative transformations are cases when the switch function is represented on a mixed basis. In this case, the function must be represented by one logic basis.
The weak side of the method considered here is the small practical application of the method of figurative transformations to minimize Boolean functions in a polynomial format, followed by the design and manufacture of the corresponding computational components. The negative internal factors of the method are associated with additional time costs for establishing protocols for simplifying logic functions in the Reed-Muller basis, followed by the creation of a library of protocols that have an illustration of the corresponding figurative transformations.
The prospect of further research could be the search for new rules for the transformation of symmetrical logic functions and their minimization.

Conclusions
1. The perfect normal form of polynomial basis functions can be represented by equivalent binary sets (10) or an equivalent binary matrix (11), which, in this case, would give the conjuncterms of polynomial functions and the operation of adding modulo two for them. Such hermeneutics should be effectively used in simplifying logic functions and when deriving the result of logic operations in the binary matrix class of functions in the Reed-Muller basis.
2. Relatively complex algorithms for simplifying logic expressions involving the procedure of inserting two identical conjuncterms of polynomial functions with the following operation of super-gluing the variables (12), (14), (16)  Based on the results of our comparison, it was established that the effectiveness of the method of figurative transformations to minimize Boolean functions in the Reed-Muller basis gives grounds for its application in the procedures for minimizing logic functions since the method of figurative transformations is capable of the following: -to ensure the operational selection of the logic operations stack in the first binary matrix, which ultimately gives an optimal scenario for minimizing logic functions in the Reed-Muller basis; -to improve the efficiency of the procedure for minimizing logic functions in the Reed-Muller basis by implementing relatively complex algorithms for simplifying logic expressions, which consist of the procedure for inserting two identical conjuncterms of ESOP functions followed by the operation of super-gluing the variables.