## 2 Determination of the Support Set of Features Using Test Theory Algorithms
This rule is called realizable for the row \(r=(\delta_{1},\ldots,\delta_{n})\), if \(\delta_{i_{1}}=b_{1},\ldots,\delta_{i_{m}}=b_{m}\). A rule is said to be true for \(T\) if any row \(r\) of the matrix \(T\) for which this rule is realizable is labeled with a decision attribute value of \(t\). Denote by \(L(T,r)\) the minimum length of a rule over \(T\) that is true for \(T\) and realizable for \(r\). We will say that the rule under consideration is a rule for \(T\) and \(r\) if this rule is true for \(T\) and realizable for \(r\). A system of decision rules \(S\) is called a complete system of decision rules for \(T\) if every rule from \(S\) is true for \(T\) and for every row of \(T\) there is a rule from \(S\) that is implemented for this row. Denote by \(L(S)\) the maximum length of a rule from \(S\), and by \(L(T)\), we denote the minimum value of \(L(S)\) among all complete systems of decision rules \(S\) for \(T\). The parameter \(L(S)\) will be called the depth of \(S\). The criterion for \(T\) is such a subset of columns (conditional signs) that at the intersection with these columns, any two rows with different solutions are different. The power of a test will sometimes be referred to as the length of that test. A reduction for \(T\) is a test for \(T\) for which every proper subset is not a test. It is clear that each test has a reduction as a subset. Denote by \(R(T)\) the minimum cardinality of a reduct for \(T\)[3]. Appropriate algorithms are needed to build and optimize tests that decide rules and trees. The most interesting are the following optimization problems:  
* The task of minimizing the power of a test: Given a decision table \(T\), it is required to construct a test that has the minimum power;
* The problem of minimizing the length of the decision rule: For a given decision table \(T\) and a row \(r\), it is required to construct a decision rule over \(T\) that is true for \(T\), realizable for \(r\), and has a minimum length;
* The problem of optimizing the system of decision rules: For a given decision table \(T\), it is required to build a complete system of decision rules \(S\) with the minimum value of the parameter \(L(S)\);
* The problem of minimizing the depth of a decision tree: For a given decision table \(T\), it is required to build a decision tree that has a minimum depth. Test theory algorithms can be used both independently to classify objects based on a set of dead-end tests and as part of other approaches, namely LAD and FCA to identify sets of essential features sufficient to divide objects into classes. The initial set of features can be excessively large, which makes it difficult to search for concepts in such a multidimensional space. In addition, the presence of a large number of features complicates the interpretation of the results. There are different ways of selecting the most informative features. Some of them are based on statistical criteria that evaluate a feature by how effectively it separates examples of different classes. These are such criteria as Envelope Eccentricity, Separation Measure, System Entropy, Pearson Correlation, and Signal-to-Noise Correlation [27]. This approach can significantly reduce the set of features but does not take into account their combinatorial influence. The combinatorial influence of features is taken into account in test theory when forming dead-end tests as well as in logical data analysis when forming a support set of features. To use this method, features must be binarized. Binarization of features occurs through their discretization using cut-points. Determination of the number and location of cut-points can be performed both through frequency sampling and by creating a complete set of potential cut-points and their further selection. Let there be two classes of observations \(K^{+}\) and \(K^{-}:\ K^{+}\cup K^{-}=K,\ K^{+}\cap K^{-}=\omega\). Observations are described by real features.