Decision trees for regular factorial languages

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Decision trees for regular factorial languages

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Authors Moshkov, Mikhail

Citation Moshkov, M. (2022). Decision trees for regular factorial languages.

Array, 15, 100203. https://doi.org/10.1016/j.array.2022.100203 Eprint version Publisher's Version/PDF

DOI 10.1016/j.array.2022.100203

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Array 15 (2022) 100203

Available online 8 June 2022

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journal homepage:www.elsevier.com/locate/array

Decision trees for regular factorial languages

Mikhail Moshkov

Computer, Electrical and Mathematical Sciences and Engineering Division and Computational Bioscience Research Center, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia

A R T I C L E I N F O

Keywords:

Regular factorial language Recognition problem Membership problem Deterministic decision tree Nondeterministic decision tree

A B S T R A C T

In this paper, we study arbitrary regular factorial languages over a finite alphabet𝛴. For the set of words 𝐿(𝑛)of the length𝑛belonging to a regular factorial language𝐿, we investigate the depth of decision trees solving the recognition and the membership problems deterministically and nondeterministically. In the case of recognition problem, for a given word from𝐿(𝑛), we should recognize it using queries each of which, for some 𝑖∈ {1,…, 𝑛}, returns the𝑖th letter of the word. In the case of membership problem, for a given word over the alphabet𝛴of the length𝑛, we should recognize if it belongs to the set𝐿(𝑛)using the same queries. For a given problem and type of trees, instead of the minimum depthℎ(𝑛)of a decision tree of the considered type solving the problem for𝐿(𝑛), we study the smoothed minimum depth𝐻(𝑛) = max{ℎ(𝑚) ∶𝑚≤𝑛}. With the growth of𝑛, the smoothed minimum depth of decision trees solving the problem of recognition deterministically is either bounded from above by a constant, or grows as a logarithm, or linearly. For other cases (decision trees solving the problem of recognition nondeterministically, and decision trees solving the membership problem deterministically and nondeterministically), with the growth of𝑛, the smoothed minimum depth of decision trees is either bounded from above by a constant or grows linearly. As corollaries of the obtained results, we study joint behavior of smoothed minimum depths of decision trees for the considered four cases and describe five complexity classes of regular factorial languages. We also investigate the class of regular factorial languages over the alphabet{0,1}each of which is given by one forbidden word.

1. Introduction

In this paper, we study arbitrary regular factorial languages over a finite alphabet𝛴. A factorial language satisfies the following condition:

if a word𝑤₁𝑢𝑤₂belongs to the language, then the word𝑢also belongs to it. For the set of words𝐿(𝑛)of the length𝑛belonging to a regular factorial language𝐿, we investigate the depth of decision trees solving the recognition and the membership problems deterministically and nondeterministically. In the case of recognition problem, for a given word from 𝐿(𝑛), we should recognize it using queries each of which, for some𝑖∈ {1,…, 𝑛}, returns the𝑖th letter of the word. In the case of membership problem, for a given word over the alphabet𝛴of the length 𝑛, we should recognize if it belongs to 𝐿(𝑛) using the same queries.

For a given problem (problem of recognition or membership problem) and type of trees (solving the problem deterministically or nondeterministically), instead of the minimum depth ℎ(𝑛)of a decision tree of the considered type solving the problem for𝐿(𝑛), we study the smoothed minimum depth𝐻(𝑛) = max{ℎ(𝑚) ∶𝑚≤𝑛}.

For an arbitrary regular factorial language, with the growth of 𝑛, the smoothed minimum depth of decision trees solving the problem of recognition deterministically is either bounded from above by a

E-mail address: [email protected].

constant, or grows as a logarithm, or linearly. These results follow immediately from more general, obtained in [1] for arbitrary regular languages.

For other cases (decision trees solving the problem of recognition nondeterministically, and decision trees solving the membership problem deterministically and nondeterministically), with the growth of 𝑛, the smoothed minimum depth of decision trees is either bounded from above by a constant, or grows linearly. In the conference paper [2], a classification of arbitrary regular languages depending on the smoothed minimum depth of decision trees solving the problem of recognition nondeterministically was announced without proofs. In the present paper, we consider simpler classification for regular factorial languages with full proof. Results related to the decision trees solving the membership problem are new.

As corollaries of the obtained results, we study joint behavior of smoothed minimum depths of decision trees for the considered four cases and describe five complexity classes of regular factorial languages. We also investigate the class of regular factorial languages over the alphabet𝐸= {0,1}each of which is given by one forbidden word.

A well-known approach to evaluate complexity of an infinite lan- guage𝐿over a finite alphabet𝛴is to study its so-called combinatorial

https://doi.org/10.1016/j.array.2022.100203

Received 7 January 2022; Received in revised form 2 June 2022; Accepted 3 June 2022

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2 complexity (known also as counting function)𝑓_𝐿(𝑛)that is the number of words of the length 𝑛 in 𝐿 [3,4]. The present paper proposes additional ways to evaluate the complexity of the language𝐿based on the study how the depth of decision trees solving the recognition and the membership problems deterministically and nondeterministically depends on the length of words. This way is more complicated, but can give more detailed classification of languages. To show this, we compare languages generated by diagrams𝐼₃and𝐼₄depicted inFigs. 5 and6. For both languages, the counting function grows linearly. For the first language, the minimum depth of decision trees solving the problem of recognition deterministically grows as a logarithm, but for the second language, the minimum depth of decision trees solving the problem of recognition deterministically grows linearly.

We should mention a recent paper [5] in which similar results were obtained for languages over the alphabet 𝐸that are subword-closed:

if a word𝑤₁𝑢₁𝑤₂⋯𝑤_𝑚𝑢_𝑚𝑤_𝑚+1belongs to the language, then the word 𝑢₁⋯𝑢_𝑚also belongs to it.

It is clear that each subword-closed language is a factorial language.

Moreover, each subword-closed language over a finite alphabet is a regular language [6]. One can show that the language 𝐿(00) over the alphabet𝐸given by one forbidden word00is a regular factorial language, which is not subword-closed. Therefore the class of subword- closed languages over the alphabet𝐸is a proper subclass of the class of regular factorial languages over the alphabet𝐸.

The main difference between the present paper and [5] is that, in the latter paper, we do not assume that the subword-closed languages are given by deterministic finite automata. Instead of this, we describe simple criteria (based on the presence in the language of words of special types) for the behavior of the minimum depths of decision trees solving the problem of recognition deterministically and nondeterministically. Differently formulated criteria for the behavior of the minimum depth of decision trees solving the recognition problem require very different proofs. One more difference is that in [5] we directly consider the minimum depth of decision trees.

The rest of the paper is organized as follows. In Section 2, we consider main notions, in Section 3– main results, and in Section4 – two corollaries of these results.

2. Main notions

In this section, we discuss the notions related to regular factorial languages and decision trees solving problems of recognition and membership for these languages.

2.1. Regular factorial languages

Let𝜔= {0,1,2,…}be the set of nonnegative integers and𝛴be a finite alphabet with at least two letters. By𝛴^∗, we denote the set of all finite words over the alphabet𝛴, including the empty word𝜆. A word 𝑤∈𝛴^∗is called a factor of a word𝑢∈𝛴^∗if𝑢=𝑣₁𝑤𝑣₂and𝑣₁, 𝑣₂∈𝛴^∗. A language 𝐿 ⊆ 𝛴^∗ is called factorial if it contains all factors of its words. A word 𝑤 ∈ 𝛴^∗is called a minimal forbidden word for𝐿if 𝑤∉𝐿and all proper factors of𝑤belong to𝐿. We denote by𝑀 𝐹(𝐿) the language of minimal forbidden words for𝐿. It is known [7] that a factorial language𝐿is regular if and only if the language𝑀 𝐹(𝐿)is regular. In particular, a factorial language𝐿with a finite set of minimal forbidden words𝑀 𝐹(𝐿)is regular. In this paper, we study arbitrary nonempty regular factorial languages.

It is well known that each regular language can be represented by a deterministic finite automaton (DFA) [8]. As in [8], we will consider not only complete DFA with total transition function but also partial DFA with partial transition function. Such DFA can be represented by its transition diagram (diagram for short) [9].

A diagram over the alphabet𝛴 is a triple𝐼 = (𝐺, 𝑞₀, 𝑄), where𝐺 is a finite directed graph, possibly with multiple edges and loops, in which each edge is labeled with a letter from𝛴and edges leaving each

node are labeled with pairwise different letters,𝑞₀is a node of𝐺called starting, and𝑄is a nonempty set of the graph𝐺nodes called final.

A path of the diagram𝐼is an arbitrary sequence𝜉=𝑣₁, 𝑑₁,…, 𝑣_𝑚, 𝑑_𝑚, 𝑣_𝑚+1of nodes and edges of𝐺such that the edge𝑑_𝑖leaves the node 𝑣_𝑖and enters the node𝑣_𝑖+1for𝑖= 1,…, 𝑚. We now define a word𝑤(𝜉) from𝛴^∗ in the following way: if𝑚 = 0, then𝑤(𝜉) = 𝜆. Let𝑚 > 0 and let𝛿_𝑗 be the letter attached to the edge 𝑑_𝑗, 𝑗 = 1,…, 𝑚. Then 𝑤(𝜉) =𝛿₁⋯𝛿_𝑚. We say that the path𝜉generates the word𝑤(𝜉). Note that different paths which start in the same node generate different words.

We denote by𝛯(𝐼) the set of all paths of the diagram𝐼 each of which starts in the node𝑞₀and finishes in a node from𝑄. Let 𝐿_𝐼 = {𝑤(𝜉) ∶𝜉∈𝛯(𝐼)}.

We say that the diagram𝐼generates the language𝐿_𝐼. It is well known that𝐿_𝐼 is a regular language.

The diagram𝐼 is called complete over the alphabet 𝛴 if exactly

|𝛴| edges leave each node of 𝐺. Note that these edges are labeled with pairwise different letters from𝛴. Such diagram corresponds to a complete DFA [8]. The diagram𝐼is called reduced if, for each node of 𝐺, there exists a path from𝛯(𝐼), which contains this node. Such diagram corresponds to a reduced DFA [8]. It is known [8] that, for each regular language over the alphabet𝛴, there exists a complete over the alphabet𝛴diagram, which generates this language. Therefore, for each nonempty regular language, there exists a reduced diagram, which generates this language.

Let 𝐿 be a regular factorial language and 𝐼 = (𝐺, 𝑞₀, 𝑄) be a reduced diagram that generates the language𝐿. Since the language𝐿is factorial, we can assume additionally that each node of the graph𝐺is final — it will not change the language generated by𝐼since with each word the language𝐿contains each prefix of this word. The diagram 𝐼will be called f-reduced if it is reduced and each node of the graph 𝐺is final. Further we will assume that a considered regular factorial language𝐿is nonempty and it is given by an f-reduced diagram, which generates this language.

We will not consider nondeterministic finite automata (NFA) to rep- resent regular factorial languages since the study of NFA is essentially more complicated task.

2.2. Decision trees for recognition and membership problems

Let 𝐿 be a regular factorial language over the alphabet 𝛴. For any natural𝑛, denote𝐿(𝑛) = 𝐿∩𝛴^𝑛, where 𝛴^𝑛 is the set of words over the alphabet 𝛴, which length is equal to 𝑛. We consider two problems related to the set 𝐿(𝑛). The problem of recognition: for a given word from𝐿(𝑛), we should recognize it using attributes (queries) 𝑙^𝑛

1,…, 𝑙^𝑛_𝑛, where𝑙^𝑛

𝑖,𝑖∈ {1,…, 𝑛}, is a function from𝛴^𝑛to𝛴such that 𝑙^𝑛_𝑖(𝑎₁⋯𝑎_𝑛) =𝑎_𝑖for any word𝑎₁⋯𝑎_𝑛∈𝛴^𝑛. The problem of membership:

for a given word from𝛴^𝑛, we should recognize if this word belongs to the set𝐿(𝑛)using the same attributes. To solve these problems, we use decision trees over𝐿(𝑛).

A decision tree over𝐿(𝑛)is a marked finite directed tree with root, which has the following properties:

• The root and the edges leaving the root are not labeled.

• Each node, which is not the root nor terminal node, is labeled with an attribute from the set{𝑙^𝑛₁,…, 𝑙^𝑛_𝑛}.

• Each edge leaving a node, which is not a root, is labeled with a letter from the alphabet𝛴.

A decision tree over𝐿(𝑛)is called deterministic if it satisfies the following conditions:

• Exactly one edge leaves the root.

• For any node, which is not the root nor terminal node, the edges leaving this node are labeled with pairwise different letters.

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Fig. 1. Decision trees that solve the problem of recognition for the set of words {100,010,001}deterministically and nondeterministically.

Let 𝛤 be a decision tree over 𝐿(𝑛). A complete path in 𝛤 is any sequence𝜉=𝑣₀, 𝑒₀,…, 𝑣_𝑚, 𝑒_𝑚, 𝑣_𝑚+1of nodes and edges of𝛤 such that 𝑣₀ is the root,𝑣_𝑚+1is a terminal node, and𝑣_𝑖is the initial and𝑣_𝑖+1is the terminal node of the edge 𝑒_𝑖 for𝑖= 0,…, 𝑚. We define a subset 𝛴(𝑛, 𝜉)of the set𝛴^𝑛in the following way: if𝑚= 0, then𝛴(𝑛, 𝜉) =𝛴^𝑛. Let𝑚 >0, the attribute𝑙^𝑛

𝑖𝑗 be attached to the node𝑣_𝑗, and𝑏_𝑗 be the letter attached to the edge𝑒_𝑗,𝑗= 1,…, 𝑚. Then

𝛴(𝑛, 𝜉) = {𝑎₁⋯𝑎_𝑛∈𝛴^𝑛∶𝑎_𝑖

1=𝑏₁,…, 𝑎_𝑖

𝑚 =𝑏_𝑚}.

Let 𝐿(𝑛) ≠ ∅. We say that a decision tree 𝛤 over 𝐿(𝑛)solves the problem of recognition for𝐿(𝑛)nondeterministically if𝛤 satisfies the following conditions:

•Each terminal node of𝛤 is labeled with a word from𝐿(𝑛).

•For any word𝑤∈𝐿(𝑛), there exists a complete path𝜉in the tree 𝛤 such that𝑤∈𝛴(𝑛, 𝜉).

•For any word𝑤∈𝐿(𝑛)and for any complete path𝜉in the tree𝛤 such that𝑤∈𝛴(𝑛, 𝜉), the terminal node of the path𝜉is labeled with the word𝑤.

We say that a decision tree𝛤 over𝐿(𝑛)solves the problem of recognition for𝐿(𝑛) deterministically if𝛤 is a deterministic decision tree, which solves the problem of recognition for𝐿(𝑛)nondeterministically.

Examples of decision trees illustrating the considered notions are presented inFig. 1.

We say that a decision tree 𝛤 over 𝐿(𝑛) solves the problem of membership for 𝐿(𝑛)nondeterministically if𝛤 satisfies the following conditions:

•Each terminal node of𝛤 is labeled with a number from the set {0,1}.

•For any word𝑤∈𝛴^𝑛, there exists a complete path𝜉in the tree 𝛤 such that𝑤∈𝛴(𝑛, 𝜉).

•For any word𝑤∈𝛴^𝑛and for any complete path𝜉in the tree𝛤 such that𝑤∈𝛴(𝑛, 𝜉), the terminal node of the path𝜉is labeled with the number1if𝑤∈𝐿(𝑛)and with the number0, otherwise.

We say that a decision tree𝛤over𝐿(𝑛)solves the problem of membership for𝐿(𝑛)deterministically if𝛤 is a deterministic decision tree which solves the problem of membership for𝐿(𝑛)nondeterministically.

Let𝛤be a decision tree over𝐿(𝑛). We denote byℎ(𝛤)the maximum number of nodes in a complete path in 𝛤 that are not the root nor terminal node. The valueℎ(𝛤)is called the depth of the decision tree 𝛤.

We denote byℎ^𝑟𝑎_𝐿(𝑛)(ℎ^𝑟𝑑_𝐿(𝑛)) the minimum depth of a decision tree over 𝐿(𝑛), which solves the problem of recognition for𝐿(𝑛) nondeterministically (deterministically). If𝐿(𝑛) = ∅, thenℎ^𝑟𝑎

𝐿(𝑛) = ℎ^𝑟𝑑

𝐿(𝑛) = 0.

We denote by ℎ^𝑚𝑎

𝐿 (𝑛) (ℎ^𝑚𝑑_𝐿 (𝑛)) the minimum depth of a decision tree over 𝐿(𝑛), which solves the problem of membership for 𝐿(𝑛) nondeterministically (deterministically). If 𝐿(𝑛) = ∅, then ℎ^𝑚𝑎

𝐿 (𝑛) = ℎ^𝑚𝑑

𝐿 (𝑛) = 0.

3. Bounds on decision tree depth

Let 𝐿 be a nonempty factorial regular language. In this section, we consider the behavior of four functions𝐻^𝑟𝑎

𝐿,𝐻^𝑟𝑑

𝐿, 𝐻^𝑚𝑎

𝐿 , and𝐻^𝑚𝑑

𝐿

defined on the set𝜔⧵{0}and with values from𝜔. For any natural𝑛, 𝐻^𝑟𝑎

𝐿(𝑛) = max{ℎ^𝑟𝑎_𝐿(𝑚) ∶ 1≤𝑚≤𝑛}, 𝐻^𝑟𝑑

𝐿(𝑛) = max{ℎ^𝑟𝑑_𝐿(𝑚) ∶ 1≤𝑚≤𝑛}, 𝐻^𝑚𝑎

𝐿 (𝑛) = max{ℎ^𝑚𝑎_𝐿(𝑚) ∶ 1≤𝑚≤𝑛}, 𝐻^𝑚𝑑

𝐿 (𝑛) = max{ℎ^𝑚𝑑_𝐿 (𝑚) ∶ 1≤𝑚≤𝑛}.

For any pair𝑏𝑐∈ {𝑟𝑎, 𝑟𝑑, 𝑚𝑎, 𝑚𝑑}, the function𝐻^𝑏𝑐

𝐿(𝑛)is a smoothed analog of the functionℎ^𝑏𝑐

𝐿(𝑛).

3.1. Decision trees solving recognition problem deterministically

Let𝐼 = (𝐺, 𝑞₀, 𝑄)be a f-reduced diagram over the alphabet 𝛴. A path of the diagram𝐼 is called a cycle of the diagram𝐼 if there is at least one edge in this path, and the first node of this path is equal to the last node of this path. A cycle of the diagram𝐼is called elementary if nodes of this cycle, with the exception of the last node, are pairwise different.

The diagram𝐼 is called simple if every two different elementary cycles of the diagram𝐼do not have common nodes. Let𝐼be a simple diagram and 𝜉 be a path of the diagram𝐼. The number of different elementary cycles of the diagram𝐼, which have common nodes with 𝜉, is denoted by𝑐𝑙(𝜉)and is called the cyclic length of the path𝜉. The value

𝑐𝑙(𝐼) = max{𝑐𝑙(𝜉) ∶𝜉∈𝛯(𝐼)}

is called the cyclic length of the diagram𝐼.

Let𝐼be a simple diagram,𝐶be an elementary cycle of the diagram 𝐼, and𝑣be a node of the cycle𝐶. Beginning with the node𝑣, the cycle 𝐶generates an infinite periodic word over the alphabet𝛴. This word will be denoted by𝑊(𝐼 , 𝐶, 𝑣). We denote by𝑟(𝐼 , 𝐶, 𝑣)the minimum period of the word 𝑊(𝐼 , 𝐶, 𝑣). The diagram 𝐼 is called dependent if there exist two different elementary cycles𝐶₁ and𝐶₂ of the diagram 𝐼, nodes𝑣₁and𝑣₂ of the cycles𝐶₁and𝐶₂, respectively, and a path𝜋 of the diagram𝐼from𝑣₁to𝑣₂, which satisfy the following conditions:

𝑊(𝐼 , 𝐶₁, 𝑣₁) =𝑊(𝐼 , 𝐶₂, 𝑣₂)and the length of the path𝜋is a number divisible by𝑟(𝐼 , 𝐶₁, 𝑣₁). If the diagram𝐼is not dependent, then it is called independent. Next theorem follows immediately from Theorem 2.1 [1], which is a similar statement that holds for all regular languages.

Theorem 1. Let 𝐿be a nonempty regular factorial language over the alphabet𝛴and𝐼be a f-reduced diagram, which generates the language𝐿.

Then the following statements hold:

(a) If𝐼is an independent simple diagram and𝑐𝑙(𝐼)≤1, then𝐻^𝑟𝑑

𝐿(𝑛) = 𝑂(1).

(b) If𝐼is an independent simple diagram and𝑐𝑙(𝐼)≥2, then𝐻^𝑟𝑑

𝐿(𝑛) = 𝛩(log𝑛).

(c) If𝐼is not independent simple diagram, then𝐻^𝑟𝑑

𝐿(𝑛) =𝛩(𝑛).

3.2. Decision trees solving recognition problem nondeterministically Let𝐿be a nonempty regular factorial language over the alphabet 𝛴. For any natural𝑛, we define a parameter𝑇_𝐿(𝑛)of the language𝐿.

If𝐿(𝑛) = ∅, then𝑇_𝐿(𝑛) = 0. Let𝐿(𝑛)≠ ∅,𝑤 = 𝑎₁⋯𝑎_𝑛 ∈ 𝐿(𝑛), and 𝐽 ⊆ {1,…, 𝑛}. Denote𝐿(𝑤, 𝐽) = {𝑏₁⋯𝑏_𝑛 ∈ 𝐿(𝑛) ∶ 𝑏_𝑗 = 𝑎_𝑗, 𝑗 ∈ 𝐽} (if 𝐽 = ∅, then 𝐿(𝑤, 𝐽) = 𝐿(𝑛)) and 𝑀_𝐿(𝑛, 𝑤) = min{|𝐽| ∶ 𝐽 ⊆ {1,…, 𝑛},|𝐿(𝑤, 𝐽)|= 1}. Then

𝑇_𝐿(𝑛) = max{𝑀_𝐿(𝑛, 𝑤) ∶𝑤∈𝐿(𝑛)}.

Note that, for any word𝑤∈𝐿(𝑛),𝑀_𝐿(𝑛, 𝑤)is the minimum number of letters of the word𝑤, which allow us to distinguish it from all other words belonging to𝐿(𝑛).

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4 Lemma 2. Let 𝐿 be a nonempty regular factorial language over the alphabet𝛴. Thenℎ^𝑟𝑎

𝐿(𝑛) =𝑇_𝐿(𝑛)for any natural𝑛.

Proof. First, we prove thatℎ^𝑟𝑎

𝐿(𝑛)≥𝑇_𝐿(𝑛). Let𝛤be a decision tree over 𝐿(𝑛), which solves the problem of recognition for𝐿(𝑛)nondeterministically and for whichℎ(𝛤) =ℎ^𝑟𝑎

𝐿(𝑛). Let𝑤be a word from𝐿(𝑛)for which 𝑇_𝐿(𝑛) =𝑀_𝐿(𝑛, 𝑤). Then the decision tree𝛤 contains a complete path𝜉 such that𝑤∈𝛴(𝑛, 𝜉)and the terminal node of the path𝜉is labeled with the word𝑤. It is clear that𝛴(𝑛, 𝜉) ∩𝐿(𝑛) = {𝑤}. Let𝜉contain𝑚nodes that are not the root nor terminal node and 𝑙^𝑛_𝑖

1,…, 𝑙^𝑛_𝑖

𝑚 be attributes attached to these nodes. Denote𝐽 = {𝑖₁,…, 𝑖_𝑚}. Then𝐿(𝑤, 𝐽) = {𝑤}.

Therefore 𝑚 ≥ 𝑀_𝐿(𝑛, 𝑤) = 𝑇_𝐿(𝑛). It is clear that ℎ(𝛤) ≥ 𝑚. Thus, ℎ^𝑟𝑎

𝐿(𝑛) =ℎ(𝛤)≥𝑚≥𝑀_𝐿(𝑛, 𝑤) =𝑇_𝐿(𝑛).

We now prove that ℎ^𝑟𝑎

𝐿(𝑛) ≤ 𝑇_𝐿(𝑛). One can show that, for each 𝑤 ∈ 𝐿(𝑛), we can construct a complete path𝜉_𝑤, which satisfies the following conditions: the number of nodes in𝜉_𝑤that are not the root nor terminal node is equal to𝑀_𝐿(𝑛, 𝑤),𝛴(𝑛, 𝜉_𝑤) ∩𝐿(𝑛) = {𝑤}, and the terminal node of𝜉_𝑤is labeled with the word𝑤. If we merge roots of all paths𝜉_𝑤,𝑤∈𝐿(𝑛), we obtain a decision tree, which solves the problem of recognition for𝐿(𝑛)nondeterministically and which depth is equal to𝑇_𝐿(𝑛). Thus,ℎ^𝑟𝑎

𝐿(𝑛)≤𝑇_𝐿(𝑛)andℎ^𝑟𝑎

𝐿(𝑛) =𝑇_𝐿(𝑛). □

Theorem 3. Let𝐿 be a nonempty regular factorial language over the alphabet𝛴and𝐼= (𝐺, 𝑞₀, 𝑄)be a f-reduced diagram, which generates the language𝐿. Then the following statements hold:

(a) If𝐼is an independent simple diagram, then𝐻^𝑟𝑎

𝐿(𝑛) =𝑂(1).

(b) If𝐼is not independent simple diagram, then𝐻^𝑟𝑎

𝐿(𝑛) =𝛩(𝑛).

Proof. (a) Let𝐼be an independent simple diagram and𝑐𝑙(𝐼)≤1. By Theorem 1,𝐻^𝑟𝑑

𝐿(𝑛) =𝑂(1). It is clear that𝐻^𝑟𝑎

𝐿(𝑛)≤𝐻^𝑟𝑑

𝐿(𝑛). Therefore 𝐻_𝐿^𝑟𝑎(𝑛) =𝑂(1).

Let𝐼 be an independent simple diagram and𝑐𝑙(𝐼)≥2. Let𝑛be a natural number. If𝐿(𝑛) = ∅, then𝑇_𝐿(𝑛) = 0. Let𝐿(𝑛)≠∅. Denote by𝑑 the number of nodes in the graph𝐺. In the proof of Lemma 4.5 [1], it was proved that𝑀_𝐿(𝑛, 𝑤)≤𝑑(4𝑑+ 1)for any word𝑤∈𝐿(𝑛). Therefore 𝑇_𝐿(𝑛)≤𝑑(4𝑑+ 1). Thus, byLemma 2,ℎ^𝑟𝑎_𝐿(𝑛)≤𝑑(4𝑑+ 1)for any natural 𝑛and𝐻^𝑟𝑎

𝐿(𝑛) =𝑂(1).

(b) Let𝐼be not simple diagram and𝐶₁, 𝐶₂be different elementary cycles of the diagram𝐼, which have a common node𝑣. Since 𝐼 is a f-reduced diagram, it contains a path𝜉from the node𝑞₀to the node 𝑣, and 𝑣is a final node. Let the length of the path 𝜉 be equal to 𝑎, the length of the cycle𝐶₁ be equal to𝑏, and the length of the cycle 𝐶₂ be equal to𝑐. Let𝛼be the word generated by the path𝜉,𝛽be the word generated by a path from𝑣to𝑣obtained by the passage𝑐times along the cycle𝐶₁, and𝛾be the word generated by a path from𝑣to𝑣 obtained by the passage𝑏times along the cycle𝐶₂. The words𝛽and𝛾 are different and they have the same length𝑏𝑐.

Consider the sequence of numbers 𝑛_𝑖 = 𝑎+𝑖𝑏𝑐, 𝑖 = 1,2,…. Let 𝑖∈𝜔⧵{0}. The set𝐿(𝑛_𝑖)contains the word𝛼𝛾^𝑖and the words𝛼𝛾^𝑗𝛽𝛾^{𝑖−𝑗−1} for𝑗= 0,…, 𝑖− 1. It is easy to show that𝑀_𝐿(𝑛_𝑖, 𝛼𝛾^𝑖)≥𝑖: to distinguish the word𝛼𝛾^𝑖from the words𝛼𝛾^𝑗𝛽𝛾^{𝑖−𝑗−1},𝑗= 0,…, 𝑖− 1, we need to use at least one letter from each of𝑖words𝛾appearing in𝛼𝛾^𝑖. Therefore 𝑇_𝐿(𝑛_𝑖) ≥ 𝑖and, byLemma 2, ℎ^𝑟𝑎

𝐿(𝑛_𝑖) ≥ 𝑖 = (𝑛_𝑖−𝑎)∕(𝑏𝑐). Let𝑛 ≥ 𝑛₁ and let𝑖be the maximum natural number such that𝑛≥𝑛_𝑖. Evidently, 𝑛−𝑛_𝑖 ≤ 𝑏𝑐. Hence 𝐻^𝑟𝑎

𝐿(𝑛) ≥ ℎ^𝑟𝑎

𝐿(𝑛_𝑖) ≥ (𝑛−𝑏𝑐−𝑎)∕(𝑏𝑐). Therefore 𝐻^𝑟𝑎

𝐿(𝑛) ≥ 𝑛∕(2𝑏𝑐) for large enough 𝑛. The inequality 𝐻^𝑟𝑎

𝐿(𝑛) ≤ 𝑛 is obvious. Thus,𝐻^𝑟𝑎

𝐿(𝑛) =𝛩(𝑛).

Let𝐼be a dependent simple diagram. Then there exist two different elementary cycles𝐶₁and𝐶₂of the diagram𝐼, nodes𝑣₁and𝑣₂ of the cycles𝐶₁and𝐶₂, respectively, and a path𝜋of the diagram𝐼from𝑣₁to 𝑣₂, which satisfy the following conditions:𝑊(𝐼 , 𝐶₁, 𝑣₁) =𝑊(𝐼 , 𝐶₂, 𝑣₂) and the length of the path𝜋is a number divisible by𝑟(𝐼 , 𝐶₁, 𝑣₁). Let us remind that, for𝑖= 1,2,𝑊(𝐼 , 𝐶_𝑖, 𝑣_𝑖)is the infinite periodic word over the alphabet𝛴generated by the cycle𝐶_𝑖beginning with the node𝑣_𝑖, and𝑟(𝐼 , 𝐶₁, 𝑣₁)is the minimum period of the word𝑊(𝐼 , 𝐶₁, 𝑣₁). Since

Fig. 2.Diagram𝐼₀.

𝐼is a f-reduced diagram, it contains a path𝜉from the node𝑞₀ to the node𝑣₁, and all nodes of the graph𝐺are final. Let the path𝜉generate the word𝛼of the length𝑎. Denote𝑟=𝑟(𝐼 , 𝐶₁, 𝑣₁). Let the length of the cycle𝐶₁ be equal to𝑏𝑟, the length of the path𝜋be equal to𝑐𝑟, and the path𝜋generate the word𝛽. Denote by𝛾 the prefix of the length𝑟 of the word𝑊(𝐼 , 𝐶₁, 𝑣₁). We now define two words of the length𝑟𝑏𝑐:

𝑢=𝛾^𝑏𝑐 and𝑤=𝛽𝛾^{𝑐(𝑏−1)}. It is clear that𝑢≠𝑤.

Consider the sequence of numbers𝑛_𝑖 = 𝑎+𝑖𝑟𝑏𝑐, 𝑖 = 1,2,…. Let 𝑖∈𝜔⧵{0}. The set𝐿(𝑛_𝑖)contains the word𝛼𝑢^𝑖and the words𝛼𝑢^𝑗𝑤𝑢^{𝑖−𝑗−1} for𝑗= 0,…, 𝑖− 1. It is easy to show that𝑀_𝐿(𝑛, 𝛼𝑢^𝑖)≥𝑖: to distinguish the word𝛼𝑢^𝑖from the words𝛼𝑢^𝑗𝑤𝑢^{𝑖−𝑗−1},𝑗= 0,…, 𝑖− 1, we need to use at least one letter from each of𝑖words𝑢appearing in𝛼𝑢^𝑖. Therefore 𝑇_𝐿(𝑛_𝑖) ≥ 𝑖and, byLemma 2,ℎ^𝑟𝑎

𝐿(𝑛_𝑖)≥ 𝑖 = (𝑛_𝑖−𝑎)∕(𝑟𝑏𝑐). Let𝑛≥ 𝑛₁ and let𝑖be the maximum natural number such that𝑛≥𝑛_𝑖. Evidently, 𝑛−𝑛_𝑖 ≤ 𝑟𝑏𝑐. Hence 𝐻^𝑟𝑎

𝐿(𝑛) ≥ ℎ^𝑟𝑎

𝐿(𝑛_𝑖) ≥ (𝑛−𝑟𝑏𝑐−𝑎)∕(𝑟𝑏𝑐). Therefore 𝐻^𝑟𝑎

𝐿(𝑛) ≥ 𝑛∕(2𝑟𝑏𝑐) for large enough𝑛. The inequality𝐻^𝑟𝑎

𝐿(𝑛) ≤ 𝑛is obvious. Thus,𝐻^𝑟𝑎

𝐿(𝑛) =𝛩(𝑛). □

Note that in general case (when we consider not only factorial languages) the classification of reduced diagrams depending on the minimum depth of decision trees solving the problem of recognition nondeterministically is more complicated [2]. In particular, there exists a dependent simple reduced diagram𝐼₀ (seeFig. 2) with the starting node labeled with the symbol+and the unique final node labeled with the symbol∗that generates the regular language𝐿₀= {0^𝑖10^𝑗∶𝑖, 𝑗∈𝜔}

over the alphabet{0,1}, which is not factorial and for which𝐻^𝑟𝑎

𝐿₀(𝑛) = 𝑂(1).

3.3. Decision trees solving membership problem

For a regular factorial language𝐿, the notation|𝐿|= ∞means that 𝐿is an infinite language, and the notation|𝐿|<∞means that𝐿is a finite language.

Theorem 4. Let𝐿be a regular factorial language over the alphabet𝛴.

(a) If|𝐿|= ∞and𝐿≠𝛴^∗, then𝐻^𝑚𝑑

𝐿 (𝑛) =𝛩(𝑛)and𝐻^𝑚𝑎

𝐿 (𝑛) =𝛩(𝑛).

(b) If|𝐿|<∞or𝐿=𝛴^∗, then𝐻^𝑚𝑑

𝐿 (𝑛) =𝑂(1)and𝐻^𝑚𝑎

𝐿 (𝑛) =𝑂(1).

Proof. It is clear thatℎ^𝑚𝑎_𝐿 (𝑛)≤ℎ^𝑚𝑑

𝐿 (𝑛)for any natural𝑛.

(a) Let|𝐿|= ∞,𝐿≠𝛴^∗, and𝑤₀be a word with the minimum length from𝛴^∗⧵𝐿. Denote by𝑡the length of𝑤₀. Since|𝐿|= ∞,𝐿(𝑛)≠∅for any natural𝑛. Let𝑛be a natural number such that𝑛 > 𝑡and𝛤 be a decision tree over𝐿(𝑛)that solves the problem of membership for𝐿(𝑛) nondeterministically and has the minimum depth. Let𝑤∈𝐿(𝑛)and𝜉 be a complete path in𝛤 such that𝑤∈𝛴(𝑛, 𝜉). Then the terminal node of𝜉 is labeled with the number1. Beginning with the first letter, we divide the word𝑤into⌊𝑛∕𝑡⌋blocks with𝑡letters in each and the suffix of the length𝑛−𝑡⌊𝑛∕𝑡⌋. Let us assume that the number of nodes labeled with attributes in𝜉is less than⌊𝑛∕𝑡⌋. Then there is a block such that queries (attributes) attached to nodes of𝜉 does not ask about letters from the block. We replace this block in the word𝑤with the word 𝑤₀ and denote by𝑤^′the obtained word. It is clear that 𝑤^′ ∉ 𝐿and 𝑤^′∈𝛴(𝑛, 𝜉), but this is impossible since the terminal node of the path 𝜉 is labeled with the number 1. Therefore the depth of𝛤 is greater than or equal to⌊𝑛∕𝑡⌋. Thus,ℎ^𝑚𝑎

𝐿 (𝑛)≥⌊𝑛∕𝑡⌋. It is easy to construct a decision tree over𝐿(𝑛)that solves the problem of membership for𝐿(𝑛) deterministically and has the depth equals to𝑛. Thereforeℎ^𝑚𝑑

𝐿 (𝑛)≤𝑛.

Thus,𝐻^𝑚𝑑

𝐿 (𝑛) =𝛩(𝑛)and𝐻^𝑚𝑎

𝐿 (𝑛) =𝛩(𝑛).

(6)

Table 1

Complexity classes1,…,5.

𝐼is independent 𝑐𝑙(𝐼) 𝐿_𝐼 𝐻^𝑟𝑑

𝐿𝐼

𝐻^𝑟𝑎

𝐿𝐼

𝐻^𝑚𝑑

𝐿𝐼

𝐻^𝑚𝑎

𝐿𝐼

simple diagram

1 Yes = 0 𝑂(1) 𝑂(1) 𝑂(1) 𝑂(1)

2 Yes = 1 𝑂(1) 𝑂(1) 𝛩(𝑛) 𝛩(𝑛)

3 Yes ≥2 𝛩(log𝑛) 𝑂(1) 𝛩(𝑛) 𝛩(𝑛)

4 No ≠𝛴^∗ 𝛩(𝑛) 𝛩(𝑛) 𝛩(𝑛) 𝛩(𝑛)

5 No =𝛴^∗ 𝛩(𝑛) 𝛩(𝑛) 𝑂(1) 𝑂(1)

(b) Let|𝐿|<∞. Then there exists natural𝑚such that𝐿(𝑛) = ∅for any natural𝑛≥𝑚. Therefore, for each natural𝑛≥𝑚,ℎ^𝑚𝑑_𝐿 (𝑛) = 0and ℎ^𝑚𝑎

𝐿 (𝑛) = 0. Thus,𝐻^𝑚𝑑

𝐿 (𝑛) =𝑂(1)and𝐻^𝑚𝑎

𝐿 (𝑛) =𝑂(1).

Let𝐿=𝛴^∗,𝑛be a natural number, and𝛤 be a decision tree over 𝐿(𝑛), which consists of the root, a terminal node labeled with1, and an edge that leaves the root and enters the terminal node. One can show that𝛤solves the problem of membership for𝐿(𝑛)deterministically and has the depth equals to0. Thereforeℎ^𝑚𝑑

𝐿 (𝑛) = 0andℎ^𝑚𝑎

𝐿 (𝑛) = 0. Thus, 𝐻_𝐿^𝑚𝑑(𝑛) =𝑂(1)and𝐻_𝐿^𝑚𝑎(𝑛) =𝑂(1). □

4. Corollaries

In this section, we consider two corollaries ofTheorems 1,3, and4.

4.1. Joint behavior of functions𝐻^𝑟𝑎

𝐿,𝐻^𝑟𝑑

𝐿,𝐻^𝑚𝑎

𝐿 , and𝐻^𝑚𝑑

𝐿

In this section, we assume that each regular factorial language over the alphabet𝛴is given by a f-reduced diagram𝐼, which generates the considered language denoted by𝐿_𝐼. To study all possible types of joint behavior of functions𝐻^𝑟𝑑

𝐿𝐼,𝐻^𝑟𝑎

𝐿𝐼,𝐻^𝑚𝑑

𝐿𝐼, and𝐻^𝑚𝑎

𝐿𝐼, we consider five classes of regular factorial languages1,…,5 described in the columns 2–4 ofTable 1. In particular,1 consists of all regular factorial languages 𝐿_𝐼 for which the diagram 𝐼 is an independent simple diagram and 𝑐𝑙(𝐼) = 0. It is easy to show that the complexity classes1,…,5are pairwise disjoint, and each regular factorial language𝐿_𝐼belongs to one of these classes. The behavior of functions𝐻^𝑟𝑑

𝐿𝐼,𝐻^𝑟𝑎

𝐿𝐼,𝐻^𝑚𝑑

𝐿𝐼, and𝐻^𝑚𝑎

𝐿𝐼

for languages from these classes is described in the last four columns of Table 1. For each class, the results considered inTable 1 for the functions𝐻^𝑟𝑑

𝐿𝐼 and𝐻^𝑟𝑎

𝐿𝐼 follow directly fromTheorems 1and3.

We now consider the behavior of the functions 𝐻^𝑚𝑑

𝐿𝐼 and𝐻^𝑚𝑎

𝐿𝐼 for each of the classes1,…,5. Let𝐼= (𝐺, 𝑞₀, 𝑄)be a f-reduced diagram over the alphabet𝛴, which generates a regular factorial language.

Let𝐿_𝐼 ∈1. Since𝑐𝑙(𝐼) = 0,𝐺is a directed acyclic graph, and the language𝐿_𝐼 is finite. UsingTheorem 4we obtain𝐻_𝐿^𝑚𝑑

𝐼(𝑛) =𝑂(1)and 𝐻^𝑚𝑎

𝐿𝐼(𝑛) =𝑂(1).

Let𝐿_𝐼 ∈2. Since𝑐𝑙(𝐼) = 1,𝐺is a graph containing a cycle, and the language𝐿_𝐼is infinite. By Lemma 4.2 [1],|𝐿_𝐼(𝑛)|=𝑂(1). Therefore 𝐿_𝐼≠𝛴^∗. UsingTheorem 4we obtain𝐻^𝑚𝑑

𝐿_𝐼(𝑛) =𝛩(𝑛)and𝐻^𝑚𝑎

𝐿_𝐼(𝑛) =𝛩(𝑛).

Let𝐿_𝐼∈3. Since𝑐𝑙(𝐼)≥2,𝐺is a graph containing a cycle, and the language𝐿_𝐼is infinite. By Lemma 4.2 [1],|𝐿_𝐼(𝑛)|=𝑂(𝑛^{𝑐𝑙(𝐼)}). Therefore 𝐿_𝐼≠𝛴^∗. UsingTheorem 4we obtain𝐻_𝐿^𝑚𝑑

𝐼(𝑛) =𝛩(𝑛)and𝐻_𝐿^𝑚𝑎

𝐼(𝑛) =𝛩(𝑛).

Let𝐿_𝐼 ∈4. Since𝐼is not an independent simple diagram,𝐺is a graph containing a cycle, and the language𝐿_𝐼is infinite. We know that 𝐿_𝐼≠𝛴^∗. UsingTheorem 4we obtain𝐻^𝑚𝑑

𝐿𝐼(𝑛) =𝛩(𝑛)and𝐻^𝑚𝑎

𝐿𝐼(𝑛) =𝛩(𝑛).

Let𝐿_𝐼 ∈5. Then𝐿_𝐼 =𝛴^∗. UsingTheorem 4we obtain𝐻^𝑚𝑑

𝐿𝐼(𝑛) = 𝑂(1)and𝐻^𝑚𝑎

𝐿𝐼(𝑛) =𝑂(1).

We now show that the classes1,…,5are nonempty. For simplic- ity, we assume that𝛴=𝐸, where𝐸= {0,1}. It is easy to generalize the considered examples to the case of an arbitrary finite alphabet𝛴 with at least two letters. In the examples of diagrams, the starting node is labeled with the symbol+, and all nodes are final.

Denote by𝐼₁ the diagram over the alphabet𝐸depicted inFig. 3.

One can show that𝐼₁is an independent simple f-reduced diagram and 𝑐𝑙(𝐼₁) = 0. This diagram generates the language𝐿_𝐼

1= {𝜆,0}, which is factorial. Therefore𝐿_𝐼

1∈1.

Fig. 3.Diagram𝐼₁.

Fig. 4.Diagram𝐼₂.

Fig. 5.Diagram𝐼₃.

Fig. 6.Diagram𝐼₄.

Fig. 7.Diagram𝐼₅.

Denote by𝐼₂ the diagram over the alphabet𝐸depicted inFig. 4.

One can show that𝐼₂ is an independent simple f-reduced diagram and 𝑐𝑙(𝐼₂) = 1. This diagram generates the language𝐿_𝐼

2 = {0^𝑖 ∶ 𝑖∈𝜔}, which is factorial. Therefore𝐿_𝐼

2∈2.

Denote by𝐼₃ the diagram over the alphabet𝐸depicted inFig. 5.

One can show that𝐼₃ is an independent simple f-reduced diagram and 𝑐𝑙(𝐼₁) = 2. This diagram generates the language𝐿_𝐼

3= {0^𝑖1^𝑗 ∶𝑖, 𝑗∈𝜔}, which is factorial. Therefore𝐿_𝐼

3∈3.

Denote by𝐼₄ the diagram over the alphabet𝐸depicted inFig. 6.

One can show that𝐼₄is a dependent simple f-reduced diagram generat- ing the language𝐿_𝐼

4= {0^𝑖1^𝑗0^𝑘∶𝑖, 𝑘∈𝜔, 𝑗∈ {0,1}}, which is factorial.

It is clear that𝐿_𝐼

4≠𝐸^∗. Therefore𝐿_𝐼

4∈4.

Denote by𝐼₅ the diagram over the alphabet𝐸depicted inFig. 7.

One can show that𝐼₅ is a f-reduced diagram that is not simple. This diagram generates the language𝐿_𝐼

5=𝐸^∗, which is factorial. It is clear that𝐿_𝐼

5=𝐸^∗. Therefore𝐿_𝐼

5∈5.

A regular factorial language 𝐿 can have different f-reduced diagrams, which generate it. However, for each of such diagrams𝐼, the language𝐿_𝐼 = 𝐿 will belong to the same complexity class. Let us assume the contrary: there exist a regular factorial language 𝐿 and two f-reduced diagrams 𝐼₁ and 𝐼₂, which generate it and for which languages 𝐿_𝐼

1 and 𝐿_𝐼

2 belong to different complexity classes. Then, for some pair𝑏𝑐∈ {𝑟𝑑, 𝑟𝑎, 𝑚𝑑, 𝑚𝑎}, the functions𝐻^𝑏𝑐

𝐿𝐼1

and𝐻^𝑏𝑐

𝐿𝐼2

have different behavior, but this is impossible since𝐻^𝑏𝑐

𝐿𝐼1

(𝑛) =𝐻^𝑏𝑐

𝐿𝐼2

(𝑛)for any natural𝑛.

(7)

6 Fig. 8. Diagram𝐼(0).

4.2. Languages over alphabet{0,1}given by one forbidden word Let 𝐸 = {0,1}, 𝛼 ∈ 𝐸^∗, and 𝛼 ≠ 𝜆. We denote by 𝐿(𝛼) the language over the alphabet𝐸, which consists of all words from𝐸^∗that do not contain𝛼as a factor. This is a regular factorial language with 𝑀 𝐹(𝐿(𝛼)) = {𝛼}. The following theorem indicates for each nonempty word 𝛼 ∈ 𝐸^∗ the complexity class 𝑖 to which the language 𝐿(𝛼) belongs.

Theorem 5. Let𝛼∈𝐸^∗and𝛼≠𝜆.

(a) If𝛼∈ {0,1}, then𝐿(𝛼) ∈2. (b) If𝛼∈ {01,10}, then𝐿(𝛼) ∈3. (c) If𝛼∉ {0,1,01,10}, then𝐿(𝛼) ∈4.

We now describe a f-reduced diagram𝐼(𝛼)that generates the lan- guage𝐿(𝛼)for a nonempty word𝛼∈𝐸^∗. Let𝛼=𝑎₁⋯𝑎_𝑛,𝛼₀=𝜆, and 𝛼_𝑖=𝑎₁⋯𝑎_𝑖for𝑖= 1,…, 𝑛− 1. The set𝑃(𝛼) = {𝛼₀, 𝛼₁,…, 𝛼_𝑛−1}is the set of all proper prefixes of the word𝛼. Then𝐼(𝛼) = (𝐺, 𝑞₀, 𝑄), where the set of nodes of the graph𝐺is equal to𝑃(𝛼),𝑞₀=𝛼₀, and𝑄=𝑃(𝛼).

For𝑖= 0,…, 𝑛− 2, an edge leaves the node𝛼_𝑖and enters the node𝛼_𝑖+1. This edge is labeled with the letter 𝑎_𝑖+1. For𝑖= 0,…, 𝑛− 1, an edge leaves the node𝛼_𝑖 and enters the node𝛼_𝑗 ∈𝑃(𝛼)such that𝛼_𝑗 is the longest suffix of the word𝛼_𝑖𝑎̄_𝑖+1, where𝑎̄_𝑖+1= 0if𝑎_𝑖+1= 1and𝑎̄_𝑖+1= 1 if𝑎_𝑖+1= 0. This edge is labeled with the letter𝑎̄_𝑖+1. It is easy to show that𝐼(𝛼)is a f-reduced diagram over the alphabet𝐸. From Theorem 10 [7] it follows that the diagram𝐼(𝛼)generates the language𝐿(𝛼).

Let𝛼∈𝐸^∗⧵{𝜆}and𝛼=𝑎₁⋯𝑎_𝑛. We denote by𝛼̄the word𝑎̄₁⋯𝑎̄_𝑛. It is easy to prove the following statement.

Lemma 6. Let𝛼∈𝐸^∗and𝛼≠𝜆. Then𝐻_𝐿(̄^𝑏𝑐_𝛼)(𝑛) =𝐻_𝐿(𝛼)^𝑏𝑐 (𝑛)for any pair 𝑏𝑐∈ {𝑟𝑑, 𝑟𝑎, 𝑚𝑑, 𝑚𝑎}and any natural𝑛.

Lemma 7. Let𝛼∈𝐸^∗⧵{𝜆},𝛽∈𝐸^∗, and𝐿(𝛼) ∈4. Then𝐿(𝛼𝛽) ∈4. Proof. Since𝐿(𝛼) ∈4,𝐻^𝑟𝑑

𝐿(𝛼)(𝑛) =𝛩(𝑛)and𝐻^𝑟𝑎

𝐿(𝛼)(𝑛) =𝛩(𝑛). One can show that𝐿(𝛼)⊆ 𝐿(𝛼𝛽). Using this fact it is not difficult to prove that 𝐻_𝐿(𝛼)^𝑟𝑑 (𝑛)≤𝐻_{𝐿(𝛼𝛽)}^𝑟𝑑 (𝑛)and𝐻_𝐿(𝛼)^𝑟𝑎 (𝑛)≤𝐻_{𝐿(𝛼𝛽)}^𝑟𝑎 (𝑛)for any natural𝑛. From here and fromTheorems 1and3it follows that𝐻^𝑟𝑑

𝐿(𝛼𝛽)(𝑛) =𝛩(𝑛)and 𝐻_{𝐿(𝛼𝛽)}^𝑟𝑎 (𝑛) =𝛩(𝑛).

Since𝛼𝛽∉𝐿(𝛼𝛽),𝐿(𝛼𝛽)≠𝐸^∗. The diagram𝐼(𝛼𝛽)contains at least one circle formed by the edge that leaves and enters the node𝜆and is labeled with the letter𝑎̄₁, where 𝑎₁ is the first letter of the word 𝛼. Therefore the language𝐿(𝛼𝛽)is infinite. ByTheorem 4,𝐻^𝑚𝑑

𝐿(𝛼𝛽)(𝑛) = 𝛩(𝑛)and𝐻^𝑚𝑎

𝐿(𝛼𝛽)(𝑛) =𝛩(𝑛). Thus,𝐿(𝛼𝛽) ∈4. □

Proof of Theorem 5. In each figure depicting a diagram 𝐼(𝛼), 𝛼 ∈ 𝐸^∗⧵{𝜆}, we label each node with a corresponding prefix of the word 𝛼.

(a) The diagram𝐼(0)is depicted inFig. 8. This is an independent simple f-reduced diagram with𝑐𝑙(𝐼(0)) = 1. Therefore𝐿(0) ∈2. By Lemma 6,𝐿(1) ∈2.

(b) The diagram𝐼(01)is depicted inFig. 9. This is an independent simple f-reduced diagram with𝑐𝑙(𝐼(01)) = 2. Therefore𝐿(01) ∈3. By Lemma 6,𝐿(10) ∈3.

Fig. 9. Diagram𝐼(01).

(c) The diagram𝐼(00)is depicted inFig. 10. This is not a simple diagram. It is clear that𝐿(00)≠𝐸^∗. Therefore𝐿(00) ∈4. ByLemma 6, 𝐿(11) ∈4. UsingLemma 7we obtain𝐿(000), 𝐿(001), 𝐿(110), 𝐿(111) ∈

4.

The diagram 𝐼(010) is depicted in Fig. 11. This is not a simple diagram. It is clear that 𝐿(010) ≠ 𝐸^∗. Therefore 𝐿(010) ∈ 4. By Lemma 6,𝐿(101) ∈4.

The diagram 𝐼(011) is depicted in Fig. 12. This is not a simple diagram. It is clear that 𝐿(011) ≠ 𝐸^∗. Therefore 𝐿(011) ∈ 4. By Lemma 6,𝐿(100) ∈4.

We proved that, for any word𝛼∈𝐸^∗of the length three,𝐿(𝛼) ∈4. UsingLemma 7we obtain that, for any word 𝛼 ∈ 𝐸^∗of the length greater than or equal to four,𝐿(𝛼) ∈4. □

Declaration of competing interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Research reported in this publication was supported by King Abdul- lah University of Science and Technology (KAUST), Saudi Arabia. The