Algorithm for constructing logical operations to identify patterns in data

. Neural networks have proven themselves in solving problems when the input and output data are known, but the cause and effect relationship between them is not obvious. A well-trained neural network will find the right answer to a given request, but will not give any idea about the rules that form this data. The paper proposes an algorithm for constructing logical operations, in terms of multi-valued logic, to identify hidden patterns in poorly formalized areas of knowledge. As the basic elements are considered many functions of the multi-valued logic of generalized addition and multiplication. The combination of these functions makes it possible to detect relationships in the data under study, as well as the ability to correct the results of neural networks. The proposed approach was considered for classification problems, in the case of multidimensional discrete features, where each feature can take k-different values and is equivalent in importance to class identification.


Introduction
In practice, there are various approaches to the construction of machine learn-ing algorithms [1][2][3].Many of them successfully cope with the tasks, but at the same time, they do not give an idea about the laws of the processed data.Thus, it can be assumed that the neural network in weighted coefficients provides the rules for object recognition, but these rules are not explicit, and it can be difficult to determine the cause of the error.In this paper, we construct an algorithm for finding logical functions that provide an opportunity for more explicit interpreta-tion necessary for decision-making..

Formulation of the problem
The object will be represented by  -dimensional vector,  -the number of characteristic features of the object in question, the  -th coordinate of this vector is equal to the value of the  -th characteristic,  = 1, . ., .Information about any characteristic of the object may be missing.The dimensionality of the considered property of the   ∈ [2, . . ., ], N -object depends on the encoding method of the i-th characteristic [4].

It is necessary to construct a function such that 𝑌 = 𝑓(𝑋).
A function  = () is called a decisive function.The dependence under consideration can be approximated using a neural network built on the basis of elements that implement external summation and a continuous scalar function.
Of great interest are direct sums that allow you to simultaneously form the architecture of a computer network and configure its parameters, without resorting to solving complex optimization problems to achieve the correctness of its functioning [5].
The algorithm is built in the form of a tree.We assume that Σ( 1 , . . .,  −1 ,   ) = Σ( 1 , . . .,  −1 ) +   = (Σ( 1 , . . .,  −1 ),   ).Great Σ( 1 , . . .,  −1 ) suspense, but it must take on one of the meanings  = {0,1, . . .,  − 1}.This implies the need to fulfill one of k conditions: ( 1 , . . .,  −1 ) = 0, The set of occupied cells in the table corresponds to those necessary conditions of existence for the implementation of a given identity.And empty cells correspond to nonessential conditions, which means that each free cell generates three possible options: 0, 1 2 , 1.This makes it possible to establish the exact number of functions in the class of solutions (power).So, for this example, the following classes of solutions are possible: 1) Table 1 -the number of functions in the class of solutions is 9 (the property of commutativity was revealed in the process of finding a solution and is not predetermined); 2) Table 2, the number of functions in the class of solutions is 9.If the commutativity property is assumed to be given in advance, the number of functions in the class of solutions will be 3 (Table 3).
Statement.Commutativity reduces the power of many feasible solutions.This is due to the fact that the number of free cells in the truth table is reduced.Theorem.There is an algorithm that determines the possibility of expressing a given function in the form of a formula through the operations of generalized addition.
The proof of the theorem is based on the above algorithm for constructing the operation of generalized addition.An algorithm is applied to the function specified in the table, and then the results are intersected.If the resulting intersection set is not empty, then the decisive function is representable as a formula through the operation of generalized addition.If it is empty, then a given function cannot be represented with a single function, but you can select the minimum number of functions that meet the specified requirements.

Algorithm for constructing a decision function based on the generalized multiplication operation
When solving problems of constructive learning, there is a need to find functions that most effectively implement the specified training samples.
Let the input of the system be fed a vector of values and each input has the weight , the output of the system has the resulting offset y.You need to build (set in a table) a set of functions that satisfy the condition: , where is a function that can be represented through the operations of generalized addition and generalized multiplication.

Table 1 .
The

Table 2 .
The

Table 3 .
The truth table with commutativity is taken K V 2000 Zhurnal vychislitel'noj ma-tematiki i matematicheskoj fiziki