## 2 Determination of the Support Set of Features Using Test Theory Algorithms  
This section discusses the apparatus of test theory, which is of practical importance for identifying homogeneous groups of objects. The basic concepts of test theory (TT) are a test table (decision table), a test (support set), a dead-end test, and a conditional test (decision tree) [3]. Furthermore, TT uses the concepts of a representative set (decision rule, template) and a non-deterministic decision tree (a system of decision rules). All applications of TT can be divided into two groups: when complete information about the problem is known (i.e., the entire decision table) and when we have incomplete information about the problem (i.e., only some rows of the entire decision table are known). Decision tables and decision trees as well as rules and tests are popular ways to represent data and knowledge. Decision trees, decision rule systems, and some test-based structures can be thought of as predictors or algorithms for solving problems given a finite set of attributes. Let \(k\) be a natural number (\(k>2\)). A \(k\)-valued decision table is a rectangular corner table whose elements belong to the set \(\{0,\ldots,k-1\}\). The columns of this table are marked with conditional attributes \(f_{0}\), \(\ldots,f_{n}\). The rows of the table are pairwise distinct, and each row is labeled with a non-negative integer, the value of the decision attribute \(d\). Denote by \(N(T)\) the number of rows in the decision table \(T\). A decision tree over \(T\) is a finite tree with a root, in which each terminal node is labeled with a decision (non-negative integer), and each non-terminal node (we will call these worker nodes) is labeled with an attribute from the set \(\{f_{1},\ldots,f_{n}\}\). Each worker node starts at most \(k\) edges. These edges are pairwise labeled with different numbers from the set \(\{0,\ldots,k-1\}\). Let \(G\) be a decision tree over \(T\). For a given row \(r\) of \(T\), the evaluation of this tree starts at the root of \(G\). If the node in question is a finite node, then the result of evaluation of \(G\) is the number attached to that node. Assume that the node currently under consideration is a worker node labeled with the \(f\) attribute. If the value of \(f\) in the row under consideration is equal to the number assigned to one of the edges outgoing from the node, then we pass along this edge. Otherwise, the computation of \(G\) terminates at the considered non-terminal node without result. We will assume that \(G\) is a decision tree for \(T\) if, for any row \(T\), the evaluation of \(G\) ends at the final node, which is labeled with the solution attached to the row in question. Denote by \(h(G)\) the depth of \(G\), which is the maximum length of the path from the root to the end node. Denote by \(h(T)\) the minimum depth of the decision tree for the table \(T\). We also take into account the average depth of decision trees. To do this, suppose that we are given a probability distribution \(P=(p_{1},\ldots,p_{N(T)})\) for \(T\), where \(p_{1},\ldots,p_{N(T)}\) are positive real numbers and \(\sum_{i=1}^{N(T)}p_{i}=1\). Then, the average depth of \(G\) with respect to \(P\) is equal to \(h_{avg}(G,P)=\sum_{i=1}^{N(T)}p_{i}l_{i}\), where \(l_{i}\) is the path length from the root to the end node as the computation for the \(i\)th row. Denote by \(h_{avg}(T,P)\) the minimum average depth of the decision tree for the table \(T\) with respect to \(P\). The value \(H(P)=-\sum_{i=1}^{N(T)}p_{i}\log_{2}p_{i}\) represents the entropy of the distribution probabilities \(P\)[3]. The decision rule over \(T\) is an expression of the form \(f_{i_{1}}=b_{1}\wedge\ldots\wedge f_{i_{m}}=b_{m}\to t\), where \(f_{i_{1}},\ldots,f_{i_{m}}\in\{f_{1},\ldots,f_{n}\},\ b_{1},\ldots,b_{m}\in\{0, \ldots,k-1\}\), and \(t\) represents the value of the solution attribute \(d\). The number \(m\) is the length of the rule.