Comparative Analysis of Deterministic and Nondeterministic Decision Trees for Decision Tables from Closed Classes

In this paper, we consider classes of decision tables with many-valued decisions closed under operations of the removal of columns, the changing of decisions, the permutation of columns, and the duplication of columns. We study relationships among three parameters of these tables: the complexity of a decision table (if we consider the depth of the decision trees, then the complexity of a decision table is the number of columns in it), the minimum complexity of a deterministic decision tree, and the minimum complexity of a nondeterministic decision tree. We consider the rough classification of functions characterizing relationships and enumerate all possible seven types of relationships.


Introduction
In this paper, we consider closed classes of decision tables with many-valued decisions and study relationships among three parameters of these tables: the complexity of a decision table (if we consider the depth of decision trees, then the complexity of a decision table is the number of columns in it), the minimum complexity of a deterministic decision tree, and the minimum complexity of a nondeterministic decision tree.
A decision table with many-valued decisions is a rectangular table in which columns are labeled with attributes, rows are pairwise different and each row is labeled with a nonempty finite set of decisions.Rows are interpreted as tuples of values of the attributes.For a given row, it is required to find a decision from the set of decisions attached to the row.To this end, we can use the following queries: we can choose an attribute and ask what is the value of this attribute in the considered row.We study two types of algorithms based on these queries: deterministic and nondeterministic decision trees.One can interpret nondeterministic decision trees for a decision table as a way to represent an arbitrary system of true decision rules for this table that cover all rows.We consider in some sense arbitrary complexity measures that characterize the time complexity of decision trees.Among them, we distinguish so-called limited complexity measures, for example, the depth of decision trees.
Decision tables with many-valued decisions often appear in data analysis, where they are known as multi-label decision tables [7,30,31].Moreover, decision tables with many-valued decisions are common in such areas as combinatorial optimization, computational geometry, and fault diagnosis, where they are used to represent and explore problems [2,21].
The depth of deterministic and nondeterministic decision trees for computation Boolean functions (variables of a function are considered as attributes) was studied quite intensively [4,12,16,29].Note that the minimum depth of a nondeterministic decision tree for a Boolean function is equal to its certificate complexity [9].
We study classes of decision tables with many-valued decisions closed under four operations: removal of columns, changing of decisions, permutation of columns, and duplication of columns.The most natural examples of such classes are closed classes of decision tables generated by information systems [22].An information system consists of a set of objects (universe) and a set of attributes (functions) defined on the universe and with values from a finite set.A problem over an information system is specified by a finite number of attributes that divide the universe into nonempty domains in which these attributes have fixed values.A nonempty finite set of decisions is attached to each domain.For a given object from the universe, it is required to find a decision from the set attached to the domain containing this object.
A decision table with many-valued decisions corresponds to this problem in a natural way: columns of this table are labeled with the considered attributes, rows correspond to domains and are labeled with sets of decisions attached to domains.The set of decision tables corresponding to problems over an information system forms a closed class generated by this system.Note that the family of all closed classes is essentially wider than the family of closed classes generated by information systems.In particular, the union of two closed classes generated by two information systems is a closed class.However, generally, there is no an information system that generates this class.
Various classes of objects that are closed under different operations are intensively studied.Among them, in particular, are classes of Boolean functions closed under the operation of superposition [25], minor-closed classes of graphs [27], classes of read-once Boolean functions closed under removal of variables and renaming of variables [13], languages closed under taking factors [3], etc. Deci-sion tables represent an interesting mathematical object deserving mathematical research, in particular, the study of closed classes of decision tables.
This paper continues the study of closed classes of decision tables that started by work [15] and frozen for various reasons for many years.In [15], we studied the dependence of the minimum depth of deterministic decision trees and the depth of deterministic decision trees constructed by a greedy algorithm on the number of attributes (columns) for conventional decision tables from classes closed under operations of removal of columns and changing of decisions.
In the present paper, we study so-called t-pairs (C, ψ), where C is a class of decision tables closed under the considered four operations and ψ is a complexity measure for this class.The t-pair is called limited if ψ is a limited complexity measure.For any decision table T ∈ C, we have three parameters: ψ i (T ) -the complexity of the decision table T .This parameter is equal to the complexity of a deterministic decision tree for the table T , which sequentially computes values of all attributes attached to columns of T .ψ d (T ) -the minimum complexity of a deterministic decision tree for the table T .ψ a (T ) -the minimum complexity of a nondeterministic decision tree for the table T .
We investigate the relationships between any two such parameters for decision tables from C. Let us consider, for example, the parameters ψ i (T ) and ψ d (T ).Let n ∈ N. We will study relations of the kind ψ i (T ) ≤ n ⇒ ψ d (T ) ≤ u, which are true for any table T ∈ C. The minimum value of u is the most interesting for us.This value (if exists) is equal to We will also study relations of the kind ψ i (T ) ≥ n ⇒ ψ d (T ) ≥ l.In this case, the maximum value of l is the most interesting for us.This value (if exists) is equal to

The two functions U di
Cψ and L di Cψ describe how the behavior of the parameter ψ d (T ) depends on the behavior of the parameter ψ i (T ) for tables from C.
There are 18 similar functions for all ordered pairs of parameters ψ i (T ), ψ d (T ), and ψ a (T ).These 18 functions well describe the relationships among the considered parameters.It would be very interesting to point out 18-tuples of these functions for all t-pairs and all limited t-pairs.But this is a very difficult problem.
In this paper, instead of functions, we will study types of functions.With any partial function f : N → N, we will associate its type from the set {α, β, γ, δ, }.For example, if the function f has an infinite domain, and it is bounded from above, then its type is equal to α.If the function f has an infinite domain, is not bounded from above, and the inequality f (n) ≥ n holds for a finite number of n ∈ N, then its type is equal to β, etc.Thus, we will enumerate 18-tuples of types of functions.These tuples will be represented in tables called the types of t-pairs.We will prove that there are only seven realizable types of t-pairs and only five realizable types of limited t-pairs.
First, we will study 9-tuples of types of functions U bc Cψ , b, c ∈ {i, d, a}.These tuples will be represented in tables called upper types of t-pairs.We will enumerate all realizable upper types of t-pairs and limited t-pairs.After that, we will extend the results obtained for upper types of t-pairs to the case of types of t-pairs.We will also define the notion of a union of two t-pairs and study the upper type of the resulting t-pair depending on the upper types of the initial t-pairs.This paper is based on the work [17] in which similar results were obtained for classes of problems over information systems.We generalized proofs from [17] to the case of decision tables from closed classes and use some results from this paper to prove the existence of t-pairs and limited t-pairs with given upper types.
The paper consists of eight sections.In Sect.2, basic definitions are considered.In Sect.3, we provide the main results related to types of t-pairs and limited t-pairs.In Sects.4-6, we study upper types of t-pairs and limited t-pairs.Section 7 contains proofs of the main results and Sect.8 -short conclusions.
Definition 1.We now define the set of decision tables M k (F ).An arbitrary decision table T from this set is a rectangular table with n ∈ N \ {0} columns labeled with attributes f 1 , . . ., f n ∈ F , where any two columns labeled with the same attribute are equal.The rows of this table are pairwise different and are filled in with numbers from E k .Each row is interpreted as a tuple of values of attributes f 1 , . . ., f n .For each row in the table, a set from P(N) is attached, which is interpreted as a set of decisions for this row.
Example 1.Three decision tables T 1 , T 2 , and T 3 from the set M 2 (F 0 ), where We correspond to the table T the following problem: for a given row of T , we should recognize a decision from the set of decisions attached to this row.To this end, we can use queries about the values of attributes for this row.
We denote by At(T ) the set {f 1 , . . ., f n } of attributes attached to the columns of T .By Π(T ), we denote the intersection of the sets of decisions attached to the rows of T , and by ∆(T ), we denote the set of rows of the table T .Decisions from Π(T ) are called common decisions for T .The table T will be called degenerate if ∆(T ) = ∅ or Π(T ) = ∅.We denote by M c k (F ) the set of degenerate decision tables from M k (F ).
Example 2. Two degenerate decision tables D 1 and D 2 are shown in Fig. 2. Definition 2. A subtable of the table T is a table obtained from T by removal of some of its rows.Let Θ(T ) = {(f, δ) : f ∈ At(T ), δ ∈ E k } and Θ * (T ) be the set of all finite words in the alphabet Θ(T ) including the empty word λ.Let α ∈ Θ * (T ).We now define a subtable T α of the table Then T α consists of all rows of T that in the intersection with columns f i1 , . . ., f im have values δ 1 , . . ., δ m , respectively.
Example 3. Two subtables of the tables T 1 and T 2 (depicted in Fig. 1) are shown in Fig. 3.
Fig. 3. Subtables T1(f1, 1) and T2(f1, 0)(f2, 0)(f3, 0) of tables T1 and T2 shown in Fig. 1 We now define four operations on the set M k (F ) of decision tables: Example 4. Decision tables T 1 ,T 2 ,T 1 , and T 2 depicted in Fig. 4 are obtained from decision tables T 1 and T 2 shown in Fig. 1 by operations of changing the decisions, removal of columns, permutation of columns, and duplication of columns, respectively.Let T ∈ M k (F ).The closure of the table T is a set, which contains all tables that can be obtained from T by the operations of removal of columns, changing of decisions, permutation of columns, and duplication of columns and only such tables.We denote the closure of the table In particular, the empty set of tables is a closed class.
Example 5. We now consider a closed class C 0 of decision tables from the set where the decision table Q is depicted in Fig. 5.The closed class C 0 contains all tables depicted in Fig. 6 and all tables that can be obtained from them by operations of duplication of columns and permutation of columns.

Deterministic and Nondeterministic Decision Trees
A finite directed tree with the root is a finite directed tree in which exactly one node has no entering edges.This node is called the root.Nodes of the tree, which have no outgoing edges are called terminal nodes.Nodes that are neither the root nor the terminal are called worker nodes.A complete path in a finite directed tree with the root is any sequence of nodes and edges starting from the root node and ending with a terminal node ξ = v 0 , d 0 , . . ., v m , d m , v m+1 , where d i is the edge outgoing from the node v i and entering the node v i+1 , i = 0, . . ., m. Definition 9. A decision tree over the set of decision tables M k (F ) is a labeled finite directed tree with the root with at least two nodes (the root and a terminal node) possessing the following properties: • The root and the edges outgoing from the root are not labeled.
• Each worker node is labeled with an attribute from the set F .
• Each edge outgoing from a worker node is labeled with a number from E k .
• Each terminal node is labeled with a number from N.
We denote by T k (F ) the set of decision trees over the set of decision tables M k (F ).Definition 10.A decision tree from T k (F ) is called deterministic if it satisfies the following conditions: • Exactly one edge leaves the root.
• Edges outgoing from each worker node are labeled with pairwise different numbers.
Let Γ be a decision tree from T k (F ).Denote by At(Γ ) the set of attributes attached to worker nodes of Denote by Θ * (Γ ) the set of all finite words in the alphabet Θ(Γ ) including the empty word λ.We correspond to an arbitrary complete path ξ = v 0 , d 0 , . . ., v m , d m , v m+1 in Γ , a word π(ξ).If m = 0, then π(ξ) = λ.Let m > 0 and, for i = 1, . . ., m, the node v i be labeled with an attribute f ji and the edge d i be labeled with the number δ i .Then π(ξ) = (f j1 , δ 1 ) • • • (f jm , δ m ).We denote by τ (ξ) the number attached to the terminal node of the path ξ.We denote by P ath(Γ ) the set of complete paths in the tree Γ .Definition 11.Let T ∈ M k (F ).A nondeterministic decision tree for the table T is a decision tree Γ over M k (F ) satisfying the following conditions: -For any row r ∈ ∆(T ) and any complete path ξ ∈ P ath(Γ ), if r ∈ ∆(T π(ξ)), then τ (ξ) belongs to the set of decisions attached to the row r.
Example 6. Nondeterministic decision trees Γ 1 and Γ 2 for decision tables T 1 and T 2 shown in Fig. 1 are depicted in Fig. 7. Definition 12.A deterministic decision tree for the table T is a deterministic decision tree over M k (F ), which is a nondeterministic decision tree for the table T .
Example 7. Deterministic decision trees Γ 1 and Γ 2 for decision tables T 1 and T 2 shown in Fig. 1 are depicted in Fig. 8.

Complexity Measures
Denote by F * the set of all finite words over the alphabet F including the empty word λ.Definition 14.The complexity measure ψ will be called limited if it possesses the following properties: (c) For any α ∈ F * , the inequality ψ(α) ≥ |α| holds, where |α| is the length of α.
We extend an arbitrary complexity measure ψ onto the set T k (F ) in the following way.Let Γ ∈ T k (F ).Then ψ(Γ ) = max{ψ(ϕ(ξ)) : The value ψ(Γ ) will be called the complexity of the decision tree Γ .We now consider an example of a complexity measure.Let w : F → N \ {0}.We define the function ψ w : F * → N in the following way: The function ψ w is a limited complexity measure over M k (F ) and it is called a weighted depth.If w ≡ 1, then the function ψ w is called the depth and is denoted by h.
Let ψ be a complexity measure over M k (F ) and T be a decision table from M k (F ) in which rows are labeled with attributes f 1 , . . ., f n .The value be called the complexity of the decision table T .We denote by ψ d (T ) the minimum complexity of a deterministic decision tree for the table T .We denote by ψ a (T ) the minimum complexity of a nondeterministic decision tree for the table T .

Information Systems
Let A be a nonempty set and F be a nonempty set of functions from A to E k .Definition 15.Functions from F are called attributes and the pair U = (A, F ) is called an information system.Definition 16.A problem over U is any (n + 1)-tuple z = (ν, f 1 , . . ., f n ), where n ∈ N \ {0}, ν : E n k → P(N), and f 1 , . . ., f n ∈ F .The problem z can be interpreted as a problem of searching for at least one number from the set z(a) = ν(f 1 (a), . . ., f n (a)) for a given a ∈ A. We denote by P robl(U ) the set of problems over the information system U .
We correspond to the problem z a decision table T (z) ∈ M k (F ).This table has n columns labeled with attributes f 1 , . . ., f n .A tuple δ = (δ 1 , . . ., δ n ) ∈ E n k is a row of the table T (z) if and only if the system of equations has a solution from the set A. This row is labeled with the set of decisions ν( δ).Let T ab(U ) = {T (z) : z ∈ P robl(U )}.One can show the set T ab(U ) is a closed class of decision tables.
Closed classes of decision tables based on information systems are the most natural examples of closed classes.However, the notion of a closed class is essentially wider.In particular, the union T ab(U 1 ) ∪ T ab(U 2 ), where U 1 and U 2 are information systems, is a closed class, but generally, we cannot find an information system U such that T ab(U ) = T ab(U 1 ) ∪ T ab(U 2 ).

Types of T-Pairs
First, we define the notion of t-pair.Definition 17.A pair (C, ψ) where C is a closed class of decision tables from M k (F ) and ψ is a complexity measure over M k (F ) will be called a test-pair (or, t-pair, in short).If ψ is a limited complexity measure then t-pair (C, ψ) will be called a limited t-pair.
Let (C, ψ) be a t-pair.We have three parameters ψ i (T ), ψ d (T ) and ψ a (T ) for any decision table T ∈ C. We now define functions that describe relationships among these parameters.Let b, c ∈ {i, d, a}.Definition 18.We define partial functions U bc Cψ : N → N and L bc Cψ : N → N by If the value U bc Cψ (n) is definite, then it is the unimprovable upper bound on the values ψ b (T ) for tables T ∈ C satisfying ψ c (T ) ≤ n.If the value L bc Cψ (n) is definite then it is the unimprovable lower bound on the values ψ b (T ) for tables T ∈ C satisfying ψ c (T ) ≥ n.
Let g be a partial function from N to N. We denote by Dom(g) the domain of g.Denote Dom + (g) = {n : n ∈ Dom(g), g(n) ≥ n} and Dom − (g) = {n : n ∈ Dom(g), g(n) ≤ n}.
-If Dom(g) is an infinite set and g is a bounded from above function, then typ(g) = α.-If Dom(g) is an infinite set, Dom + (g) is a finite set, and g is an unbounded from above function, then typ(g) = β.-If both sets Dom + (g) and Dom − (g) are infinite, then typ(g) = γ.
-If Dom(g) is an infinite set and Dom − (g) is a finite set, then typ(g) = δ.
-If Dom(g) is a finite set, then typ(g) = .
Example 8.One can show that typ(1) = α, typ( log Definition 20.We now define the table typ(C, ψ), which will be called the type of t-pair (C, ψ).This is a table with three rows and three columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices i, d, a.The pair typ(L bc Cψ ) typ(U bc Cψ ) is in the intersection of the row with index b ∈ {i, d, a} and the column with index c ∈ {i, d, a}.

Main Results
The main problem investigated in this paper is finding all types of t-pairs and limited t-pairs.The solution to this problem describes all possible (in terms of functions U bc Cψ , L bc Cψ types, b, c ∈ {i, d, a}) relationships among the complexity of decision tables, the minimum complexity of nondeterministic decision trees for them, and the minimum complexity of deterministic decision trees for these tables.We now define seven tables: Theorem 2. For any limited t-pair (C, ψ), the relation typ(C, ψ) ∈ {T 2 , T 3 , T 5 , T 6 , T 7 } holds.For any i ∈ {2, 3, 5, 6, 7}, there exists a limited t-pair (C, h) such that typ(C, h) = T i .

Possible Upper Types of T-Pairs
We begin our study by considering the upper type of t-pair, which is a simpler object than the type of t-pair.Definition 21.Let (C, ψ) be a t-pair.We now define table typ u (C, ψ), which will be called the upper type of t-pair (C, ψ).This is a table with three rows and three columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices i, d, a.The value typ(U bc Cψ ) is in the intersection of the row with index b ∈ {i, d, a} and the column with index c ∈ {i, d, a}.The table typ u (C, ψ) will be called the upper type of t-pair (C, ψ).
We divide the proofs of the propositions into a sequence of lemmas.
Lemma 1.Let T be a decision table from a set of decision tables M k (F ) and ψ be a complexity measure over M k (F ).Then the inequalities ψ a (T ) ≤ ψ d (T ) ≤ ψ i (T ) hold.
Proof.Let columns of the table T be labeled with attributes f 1 , . . ., f n .It is not difficult to construct a deterministic decision tree Γ 0 for the table T , which sequentially computes values of attributes f 1 , . . ., f n .Evidently, ψ(Γ 0 ) = ψ i (T ).Therefore ψ d (T ) ≤ ψ i (T ).If a decision tree Γ is a deterministic decision tree for T , then Γ is a nondeterministic decision tree for T .Therefore ψ a (T ) ≤ ψ d (T ).
Let (C, ψ) be a t-pair, n ∈ N and b, c ∈ {i, d, a}.The notation It is not difficult to prove the following statement.Let (C, ψ) be a t-pair and b, c, e, f ∈ {i, d, a}.The notation U bc Cψ U ef Cψ means that, for any n ∈ N, the following statements hold: (a) If the value Let be a linear order on the set {α, β, γ, δ, } such that α β γ δ .
Lemma  Proof.Using Lemma 4, we conclude that the function ψ i is unbounded from above on C. Let m ∈ N. Then there exists a decision table T ∈ C for which the inequality ψ i (T ) ≥ m holds.Let us consider a degenerate decision table T ∈ C obtained from T by replacing the sets of decisions attached to rows by the set {0}.It is clear that ψ i (T ) ≥ m.Let Γ be a decision tree, which consists of the root, the terminal node labeled with 0, and the edge connecting these two nodes.One can show that Γ is a deterministic decision tree for the table T .Therefore Let us prove by induction on n that, for any decision table T from C, if ψ i (T ) ≤ n, then ψ a (T ) ≤ m 0 , where m 0 = max{m, ψ(λ)}.Using Lemma 1, we conclude that under the condition n ≤ m the considered statement holds.Let it hold for some n, n ≥ m.Let us show that this statement holds for n + 1 too.Let T ∈ C, ψ i (T ) ≤ n + 1 and columns of the table T be labeled with attributes f i1 , . . ., f i k .Since n + 1 > m, we obtain ψ a (T ) ≤ n.Let Γ be a nondeterministic decision tree for the table T and ψ(Γ ) = ψ a (T ).Assume that in Γ there exists a complete path ξ in which there are no worker nodes.In this case, a decision tree, which consists of the root, the terminal node labeled with τ (ξ) and the edge connecting these two nodes is a nondeterministic decision tree for the table T .Therefore ψ a (T ) ≤ ψ(λ) ≤ m 0 .Assume now that each complete path in the decision tree Γ contains a worker node.Let ξ ∈ Path(Γ ), ∆(T π(ξ)) = ∅, ξ = v 0 , d 0 , . . ., v p , d p , v p+1 and, for i = 1, . . ., p, the node v i be labeled with the attribute f i , and the edge d i be labeled with the number δ i .Let the decision table T be obtained from the decision table T by operations of permutation of columns and duplication of columns so that its columns are labeled with attributes f 1 , . . ., f p , f i1 , . . ., f i k .We obtain the decision table T from T by removal the last k columns.Let us denote by T ξ the decision table obtained from T by changing the set of decisions corresponding to the row (δ 1 , . . ., δ p ) with {τ (ξ)}, and for the remaining rows with {τ (ξ) + 1}.It is clear that ψ i (T ξ ) ≤ n.Using the inductive hypothesis, we conclude that there exists a nondeterministic decision tree Γ ξ for the table T ξ such that ψ(Γ ξ ) ≤ m 0 .We denote by Γξ a tree obtained from Γ ξ by removal of all nodes and edges that satisfy the following condition: there is no a complete path ξ in Γ ξ , which contains this node or edge and for which τ (ξ ) = τ (ξ).Let {ξ : ξ ∈ Path(Γ ), ∆(T π(ξ)) = ∅} = {ξ 1 , . . ., ξ r }.Let us identify the roots of the trees Γξ1 , . . ., Γξr .We denote by G the obtained tree.It is not difficult to show that G is a nondeterministic decision tree for the table T and ψ(G) ≤ m 0 .Thus, the considered statement holds.Using Lemma 4, we conclude that typ(U ai Cψ ) = α.The obtained contradiction shows that typ(U ai Cψ ) ∈ {α, γ}.
Let T be a decision table from M k (F ).We now give definitions of parameters N (T ) and M (T ) of the table T .Definition 22.We denote by N (T ) the number of rows in the table T .Definition 23.Let columns of table T be labeled with attributes f 1 , . . ., f n ∈ F .We now define the parameter M (T ).If table T is degenerate, then M (T ) = 0. Let now T be a nondegenerate table and δ = (δ 1 , . . ., δ n ) ∈ E n k .Then M (T, δ) is the minimum natural m such that there exist attributes f i1 , . . ., f im ∈ At(T ) for which The following statement follows immediately from Theorem 3.5 [18].
Lemma 7. Let T be a nonempty decision table from M k (F ) in which each row is labeled with a set containing only one decision.Then Lemma 8. Let (C, ψ) be a limited t-pair and typ(U ai Cψ ) = α.Then typ(U di Cψ ) ∈ {α, β}.
Proof.Using Lemma 4, we conclude that there exists r ∈ N such that the inequality ψ a (T ) ≤ r holds for any table T ∈ C.
Let T be a nonempty table from C in which columns are labeled with attributes f 1 , . . ., f n and δ = (δ 1 , . . ., δ n ) ∈ E n k .We now show that there exist attributes f i1 , . . ., f im ∈ At(T ) such that the subtable Let δ be a row of T .Let us change the set of decisions attached to the row δ with the set {1} and for the remaining rows of T with the set {0}.We denote the obtained table by T .It is clear that T ∈ C. Taking into account that ψ a (T ) ≤ r and the complexity measure ψ has the property (c), it is not difficult to show that there exist attributes f i1 , . . ., f im ∈ At(T ) = At(T ) such that m ≤ r and T (f i1 , δ i1 ) • • • (f im , δ im ) contains only the row δ.From here it follows that Let δ be not a row of T .Let us show that there exist attributes f i1 , . . ., f im ∈ At(T ) such that m ≤ r + 1 and the subtable is empty, then the considered statement holds.Otherwise, there exists q ∈ {1, . . ., n − 1} such that the subtable ) is empty.We denote by T the table obtained from T by removal of attributes f q+1 , . . ., f n .It is clear that T ∈ C and (δ 1 , . . ., δ q ) is a row of T .According to proven above, there exist attributes f i1 , . . ., f ip ∈ {f 1 , . . ., f q } such that and p ≤ r.Using this fact one can show that ) is empty and is equal to T ( δ).
Let T 1 ∈ C. We denote by T 2 the decision table obtained from T 1 by removal of all columns in which all numbers are equal.Let columns of T 2 be labeled with attributes f 1 , . . ., f n .We now consider the decision table T 3 , which is obtained from T 2 by changing decisions so that the decision set attached to each row of table T 3 contains only one decision and, for any two non-equal rows, corresponding decisions are different.It is clear that We now show that the inequality ψ(f ) ≤ r holds for any attribute f ∈ At(T 3 ).Let us denote by T the decision table obtained from T 3 by removal of all columns except the column labeled with the attribute f .If there is more than one column in T 3 , which is labeled with the attribute f , then we keep only one of them.Let the decision table T f be obtained from T by changing the set of decisions for each row (δ) with the set of decisions {δ}.It is clear that T f ∈ C. Let Γ be a nondeterministic decision tree for the table T f and ψ(Γ ) = ψ a (T f ) ≤ r.Since the column f contains different numbers, we have f ∈ At(Γ ).Using the property (b) of the complexity measure ψ, we obtain ψ(Γ ) ≥ ψ(f ).Consequently, ψ(f ) ≤ r.
Consequently, ψ d (T 1 ) ≤ (r + 1) 3 log 2 (kn).Taking into account that the complexity measure ψ has the property (c), we obtain ψ i (T 1 ) ≥ n.Since T 1 is an arbitrary decision  The table typ lu (U, ψ) for the pair (U, ψ) was defined in [17] as follows: this is a table with three rows and three columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices i, d, a.The value typ(U bc U ψ ) is in the intersection of the row with the index b ∈ {i, d, a} and the column with the index c ∈ {i, d, a}.
We now prove the following proposition: Proposition 5. Let U be an information system and ψ be a complexity measure over U .Then typ lu (U, ψ) = typ u (T ab(U ), ψ).
Proof.Let z = (ν, f 1 , . . ., f n ) be a problem over U and T (z) be the decision table corresponding to this problem.It is easy to see that ψ i U (z) = ψ i (T (z)).One can show the set of decision trees solving the problem z nondeterministically and using only attributes from the set {f 1 , . . ., f n } (see corresponding definitions in [17]) is equal to the set of nondeterministic decision trees for the table T (z).From here it follows that ψ a U (z) = ψ a (T (z)) and ψ d U (z) = ψ d (T (z)).Using these equalities, we can show that typ lu (U, ψ) = typ u (T ab(U ), ψ).
This proposition allows us to transfer results obtained for information systems in [17] to the case of closed classes of decision tables.Before each of the following seven lemmas, we define a pair (U, ψ), where U is an information system and ψ is a complexity measure over U .
(a) Let typ(g) = .From here, it follows that Dom(g) is a finite set.Taking into account this fact, we obtain Dom(max(f, g)) is also a finite set.Therefore, typ(max(f, g)) = max(typ(f ), typ(g)) = .Later we will assume that typ(g) = .
(c) Let typ(f ) = β, typ(g) = δ.From here, it follows that Dom − (f ), Dom + (g) are both infinite sets and Dom + (f ), Dom − (g) are both finite sets.Taking into account that both f and g are nondecreasing functions, we obtain that there exists n is an infinite set.Taking into account that Dom − (g) is an infinite set and Dom + (f ) is a finite set, we obtain Dom − (max(f, g)) is also an infinite set.Therefore, typ(max(f, g)) = max(typ(f ), typ(g)) = typ(g) = γ.
The next statement follows immediately from Proposition 1 and Theorem 3.
By these operations, decision tables from T i can be obtained from G i in three ways: a) only changing of decisions, b) removing one column and changing of decisions, and c) removing two columns and changing of decisions.Figure 11 demonstrates examples of decision tables from T i for each case.Without loss of generality, we can restrict ourselves to considering these three tables H 1 , H 2 , and H 3 .i (H 1 ) = i and ψ d i (H 1 ) = i + 1.In the second case, ψ a i (H 1 ) = i and ψ d i (H 1 ) = i.In the third case, ψ a i (H 1 ) = 0 and ψ d i (H 1 ) = 0. (b) There are three different cases for the table H 2 : (i) the sets of decisions d 4 , d 5 , d 6 are pairwise disjoint, (ii) there are l, t ∈ {4, 5, 6} such that l = t, d l ∩d t = ∅ and d 4 ∩d 5 ∩d 6 = ∅, and (iii) d 4 ∩d 5 ∩d 6 = ∅.In the first case, ψ a i (H 2 ) = i+1 and ψ d i (H 2 ) = i + 1.In the second case, we have either ψ a i (H 2 ) = ψ d i (H 2 ) = i + 1 or ψ a i (H 2 ) = ψ d i (H 2 ) = i depending on the intersecting decision sets.In the third case, ψ a i (H 2 ) = 0 and ψ d i (H 2 ) = 0. (c) There are two different cases for the table H 3 : (i) d 7 ∩ d 8 = ∅ and (ii) d 7 ∩ d 8 = ∅.In the first case, ψ a i (H 3 ) = i and ψ d i (H 3 ) = i.In the second case, ψ a i (H 3 ) = 0 and ψ d i (H 3 ) = 0.As a result, we obtain that, for any n ∈ N, Let K be an infinite subset of the set N. Denote F K = ∪ i∈K F i and T K = ∪ i∈K [G i ].It is clear that T K is a closed class of decision tables from M 2 (F K ).We now define a complexity measure ψ K over M 2 (F K ).Let α ∈ F * K .If α ∈ F * i for some i ∈ K, then ψ K (α) = ψ i (α).If α contains letters both from F i and F j , i = j, then ψ K (α) = 0. Let K = {n j : j ∈ N} and n j < n j+1 for any j ∈ N. We define a function ϕ K : N → N as follows.Let n ∈ N. If n < n 0 , then ϕ K (n) = 0. Let, for some j ∈ N, n j ≤ n < n j+1 .Then ϕ K (n) = n j .Using (3), one can show that, for any n ∈ N, U da T K ψ K (n) = ϕ K (n).Using this equality, one can prove that typ(U da T K ψ K ) = γ if the set N\K is infinite and typ(U da T K ψ K ) = δ if the set N \ K is finite.Denote K 1 1 = {3j : j ∈ N}, K 1 2 = {3j +1 : j ∈ N} and K 1 = K 1 1 ∪K 1 2 .Denote τ 1 1 = (T K 1 1 , ψ K 1 1 ), τ 1 2 = (T K 1 2 , ψ K 1 2 ) and τ 1 = (T K 1 , ψ K 1 ).One can show that tpairs τ 1 1 and τ 1 2 are compatible and τ 1 is a union of τ 1 1 and τ 1 2 .It is easy to prove that typ(U da T K 1 ψ K 1 ) = γ.Using Proposition 2, we obtain typ u (τ 1 1 ) = typ u (τ 1 2 ) = typ u (τ 1 ) = t 5 .Denote K 2 1 = {2j : j ∈ N}, K 2 2 = {2j + 1 : j ∈ N} and ) and τ 2 = (T K 2 , ψ K 2 ).One can show that t-pairs τ 2 1 and τ 2 2 are compatible and τ 2 is a union of τ 2 1 and τ 2 2 .It is easy to prove that typ(U da ) = γ and typ(U da T K 2 ψ K 2 ) = δ.Using Proposition 2, we obtain typ u (τ 2 1 ) = typ u (τ 2 2 ) = t 5 and typ u (τ 2 ) = t 6 .

Fig. 4 .
Fig. 4. Decision tables T 1 ,T 2 ,T 1 , and T 2 obtained from tables T1 and T2 shown in Fig. 1 by operations of changing the decisions, removal of columns, permutation of columns and duplication of columns, respectively

Lemma 4 .
Let (C, ψ) be a t-pair and b, c ∈ {i, d, a}.Then (a) typ(U bc Cψ ) = α if and only if the function ψ b is bounded from above on the closed class C. (b) If the function ψ b is unbounded from above on C, then typ(U bb Cψ ) = γ.Proof.The statement (a) is obvious.(b) Let the function ψ b be unbounded from above on C. One can show that in this case the equality U bb Cψ (n) = n holds for infinitely many n ∈ N. Therefore typ(U bb Cψ ) = γ.Corollary 1.Let (C, ψ) be a t-pair and b ∈ {i, d, a}.Then typ(U bb Cψ ) ∈ {α, γ}.Lemma 5. Let (C, ψ) be a t-pair and typ(U ii Cψ ) = α.Then typ(U id Cψ ) = typ(U ia Cψ ) = .

Fig. 9 .Fig. 10 .
Fig. 9. Possible upper types of a union of two compatible t-pairs
Definition 4. Changing of decisions.In a given table T , we can change in an arbitrary way sets of decisions attached to rows.
Definition 5. Permutation of columns.We can swap any two columns in a table T , including the attached attribute names.Definition 6. Duplication of columns.For any column in a table T , we can add its duplicate next to that column.
From the definition of the functions U bc Cψ , b, c ∈ {i, d, a}, and from Lemma 1 it follows that U bi Cψ U bd Cψ U ba Cψ and U ab Cψ U db Cψ U ib Cψ for any b ∈ {i, d, a}.Using these relations and Lemma 2 we obtain the statement of the lemma.
Example 9. Let us consider a t-pair (C 0 , h), where C 0 is closed class described in Example 5.It is clear that the function h i is unbounded from above on C 0 and the functions h a and h d are bounded from above on C 0 .