#### Abstract

This paper investigates the controllability, reachability, and stabilizability of finite automata by using the semitensor product of matrices. Firstly, by expressing the states, inputs, and outputs as vector forms, an algebraic form is obtained for finite automata. Secondly, based on the algebraic form, a controllability matrix is constructed for finite automata. Thirdly, some necessary and sufficient conditions are presented for the controllability, reachability, and stabilizability of finite automata by using the controllability matrix. Finally, an illustrative example is given to support the obtained new results.

#### 1. Introduction

In the research field of theoretical computer science, finite automaton is one of the simplest models of computation. Finite automaton is a device whose states take values from a finite set. It receives a discrete sequence of inputs from the outside world and changes its state according to the inputs. The study of finite automata has received many scholars’ research interest in the last century [1–5] due to its wide applications in engineering, computer science, and so on.

As we all know, controllability and stabilizability analysis of finite automata are fundamental topics, which are important and necessary to the solvability of many related problems [1, 4, 6]. The concepts of controllability, reachability, and stabilizability of finite automata were defined in [2] by resorting to the classic control theory. The controllability of a deterministic Rabin automaton was studied in [7] by defining the “controllability subset.” Kobayashi et al. [8] investigated the state feedback stabilization of a deterministic finite automaton and presented some new results.

Recently, a new matrix product, namely, the semitensor product (STP) of matrices, has been proposed by Cheng et al. [9]. Up to now, STP has been successfully applied to many research fields related to finite-valued systems like Boolean networks [10–20], multivalued logical networks [21–23], game theory [24, 25], finite automata [5, 26], and so on [27–35]. The main feature of STP is to convert a finite-valued system into an equivalent algebraic form [22]. Thus, STP provides a convenient way for the construction and analysis of finite automata [5, 26]. Xu and Hong [5] provided a matrix-based algebraic approach for the reachability analysis of finite automata with the help of STP. Yan et al. [26] studied the controllability and stabilizability analysis of finite automata based on STP and presented some novel results. It should be pointed out that although the concepts of controllability, reachability, and stabilizability of finite automata come from classic control theory, there exist fewer results on the construction of controllability matrix for finite automata.

In this paper, we investigate the controllability, reachability, and stabilizability of deterministic finite automata by using STP. The main contribution of this paper is to construct a controllability matrix for finite automata based on the algebraic form. Using the controllability matrix, we present some necessary and sufficient conditions for the controllability, reachability, and stabilizability of finite automata. Compared with the existing results [5, 26], our results are more easily verified via MATLAB.

The rest of this paper is organized as follows. Section 2 contains some necessary preliminaries on the semitensor product of matrices and finite automata. Section 3 studies the controllability, reachability, and stabilizability of finite automata and presents the main results of this paper. In Section 4, an illustrative example is given to support our new results, which is followed by a brief conclusion in Section 5.

*Notations*. , , and denote the set of real numbers, the set of natural numbers, and the set of positive integers, respectively. , where denotes the th column of the identity matrix . An matrix is called a logical matrix, if , which is briefly denoted by . The set of logical matrices is denoted by . Given a real matrix , , , and denote the th column, the th row, and the th element of , respectively. if and only if holds for any . denote the th block of an matrix .

#### 2. Preliminaries

##### 2.1. Semitensor Product of Matrices

In this part, we recall some necessary preliminaries on STP. For details, please refer to [9].

*Definition 1. *Given two matrices and , the semitensor product of and is defined as where is the least common multiple of and and is the Kronecker product of matrices.

Lemma 2. *STP has the following properties: *(1)*Let be a column vector and . Then *(2)*Let and be two column vectors. Then * *where is called the swap matrix.*

##### 2.2. Finite Automata

In this subsection, we recall some definitions of finite automata.

A finite automaton is a seven-tuple , in which , , and are finite sets of states, input symbols, and outputs, respectively; and are the initial state and the set of accepted states; and are transition and output functions, which are defined as and , where and denote the power set of and , respectively; that is, , . represents the finite string set on , which does not include the empty transition. Given an initial state and an input symbol , the function uniquely determines the next subset of states, that is, , while the function uniquely determines the next subset of outputs; that is, .

Throughout this paper, we only consider the deterministic finite automata; that is, holds for any and . In addition, we only investigate the controllability, reachability, and stabilizability of deterministic finite automata, and thus we do not use and in the seven-tuple .

In the following, we recall the definitions of controllability, reachability, and stabilizability for deterministic finite automata.

*Definition 3. *(i) A state is said to be controllable to , if there exists a control sequence such that .

(ii) A state is said to be controllable, if is controllable to any state .

*Definition 4. *(i) A state is said to be reachable from , if there exists a control sequence such that .

(ii) A state is said to be reachable, if is reachable from any state .

Given two nonempty sets and satisfying and , we have the following definitions.

*Definition 5. *A nonempty set of state is said to be controllable, if, for any state , there exist an and a control sequence such that .

*Definition 6. *A nonempty set of state is said to be reachable, if, for any state , there exist an and a control sequence such that .

*Definition 7. *A nonempty set of state is said to be 1-step returnable, if, for any state , there exists an input such that .

*Definition 8. *A nonempty set of state is said to be stabilizable, if is reachable and 1-step returnable.

#### 3. Main Results

In this section, we investigate the controllability, reachability, and stabilizability of deterministic finite automata by constructing a controllability matrix.

##### 3.1. Controllability Matrix

For a deterministic finite automaton , where and , we identify as and call the vector form of . Then, can be denoted as ; that is, . Similarly, for , we identify with and call the vector form of . Then, .

Using the vector form of elements in and , Yan et al. [26] construct the transition structure matrix (TSM) of as . One can see that if there exists a control which moves state to state , then In this case, . Otherwise, . Thus, setting then one can use to judge whether or not state is controllable to state in one step. Precisely, state is controllable to state in one step, if and only if .

Now, we show that, for any , state is controllable to state at the th step, if and only if . We prove it by induction. Obviously, when , the conclusion holds. Assume that the conclusion holds for some . Then, for the case of , state is controllable to state at the th step, if and only if there exists some state such that state is controllable to state at the th step and state is controllable to state in one step. Hence, By induction, for any , state is controllable to state at the th step, if and only if . Thus, contains all the controllability information of the finite automata. Noticing that is an square matrix, by Cayley-Hamilton theorem, we only need to consider . Then, we define the controllability matrix for finite automata as follows.

*Definition 9. *Set . The controllability matrix of finite automata is .

Based on the controllability matrix, we have the following result.

*Algorithm 10. *Consider the finite automata . Then, the controls which force to in the shortest time can be designed by the following steps: (1)Find the smallest integer such that, for there exists a block, say, , satisfying .(2)Set and . If , stop. Otherwise, go to Step (3).(3)Find and such that and , where and . Set and .(4)If , stop. Otherwise, replace and by and , respectively, and go to Step (3).

*Example 11. *Consider a finite automaton given in Figure 1, where , . Suppose that and . Then, can be denoted as . Similarly, , , .

The transition structure matrix of the finite automata is Split , where and . Then, Thus, the controllability matrix is By Algorithm 10, one can obtain that . Setting , , and , one can find and such that and . Let and . Hence, state is controllable to state at the step.

##### 3.2. Controllability, Reachability, and Stabilizability

In this part, we study the controllability, reachability, and stabilizability of deterministic finite automata based on the controllability matrix.

According to the meaning of controllability matrix, we have the following results.

Theorem 12. *The state is controllable, if and only if *

*Proof. * *Necessity*. Suppose that the state is controllable. By Definition 3, is controllable to any state . Based on (4), one can see that there exists a control sequence satisfying . Thus, , which implies that From the arbitrariness of , we have .* Sufficiency*. Suppose that holds. Then, for any state , one can find some . Therefore, under the control sequence , the state is controllable to . From the arbitrariness of , the state is controllable.

Theorem 13. *The state is reachable, if and only if *

*Proof. * *Necessity*. Suppose that the state is reachable. By Definition 4, is reachable to any state . One can obtain from (4) that there exists a control sequence satisfying . Thus, , which shows that From the arbitrariness of , one can conclude that .*Sufficiency*. Suppose that . Then, for any state , there exists some . Hence, under the control sequence , the state is reachable to . By Definition 4, the state is reachable.

Given two nonempty sets and , where and , define Based on Theorems 12 and 13, we have the following result.

Theorem 14. *(i) The nonempty set is controllable, if and only if .**(ii) The nonempty set is reachable, if and only if .*

*Proof. * *(i) Necessity*. Suppose that the nonempty set is controllable. By Definition 5, for any state , there exist a and a control sequence such that . Based on Theorems 12 and 13, for a fixed , , at least one of the following cases is true: . Therefore, for a fixed , one can conclude that . From the arbitrariness of , one can see that *Sufficiency*. Suppose that . Then, for any , we have . It means that, for any state , there exist a and a control sequence such that . By Definition 5, the nonempty set is controllable.*(ii) Necessity*. Suppose that the nonempty set is reachable. By Definition 6, for any state , there exist a and a control sequence such that . Based on Theorems 12 and 13, for a fixed , , at least one of the following cases is true: . Therefore, for a fixed , one can see that . From the arbitrariness of , we have *Sufficiency*. Suppose that . Then, for any , we have . It means that, for any state , there exist a and a control sequence such that . By Definition 6, the nonempty set is reachable.

Finally, we study the stabilizability of deterministic finite automata.

For and , define

Theorem 15. *The nonempty set is 1-step returnable, if and only if .*

*Proof. *By Definition 7, one can see that the nonempty set is 1-step returnable, if and only if, for any state , there exist an input and some such that , that is, for a fixed , , at least one of the following cases is true: . Hence, , . From the arbitrariness of , one can obtain that .

Based on Theorems 14 and 15, we have the following result.

Corollary 16. *The nonempty set is stabilizable, if and only if and .*

*Proof. *By Definition 8, is stabilizable, if and only if is reachable and 1-step returnable. Based on Theorems 14 and 15, the conclusion follows.

*Remark 17. *Compared with the existing results on the controllability and stabilizability of deterministic finite automata [5, 26], the main advantage of our results is to propose a unified tool, that is, controllability matrix, for the study of deterministic finite automata. The new conditions are more easily verified via MATLAB.

#### 4. An Illustrative Example

Consider the finite automata given in Figure 2, where and .

From Figure 2, we can see that and . Therefore, by Definition 3, one can obtain that is controllable. Similarly, by Definition 3, we conclude that , , and are also controllable. By Definition 4, we can also find that all the states are reachable.

Assume that and . Since and , by Definition 5, one can see that is controllable. Similarly, we also obtain that is controllable. Since and , by Definition 6, we can obtain that and are reachable. From Figure 2, we can see that , , , and . Hence, by Definition 7, and are 1-step returnable. By Definition 8, the sets and are stabilizable.

Now, we check the above properties based on the controllability matrix.

The transition structure matrix of the finite automata is Split , where and . Then, Thus, the controllability matrix is

Since all rows and columns of are positive, by Theorems 12 and 13, any state is controllable and reachable, .

A simple calculation gives , , , , , and . By Theorems 14 and 15 and Corollary 16, and are controllable, reachable, 1-step returnable, and stabilizable, respectively.

#### 5. Conclusion

In this paper, we have investigated the controllability, reachability, and stabilizability of deterministic finite automata by using the semitensor product of matrices. We have obtained the algebraic form of finite automata by expressing the states, inputs, and outputs as vector forms. Based on the algebraic form, we have defined the controllability matrix for deterministic finite automata. In addition, using the controllability matrix, we have presented several necessary and sufficient conditions for the controllability, reachability, and stabilizability of finite automata. The study of an illustrative example has shown that the obtained new results are effective.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The research was supported by the National Natural Science Foundation of China under Grants 61374065 and 61503225, the Natural Science Foundation of Shandong Province under Grant ZR2015FQ003, and the Natural Science Fund for Distinguished Young Scholars of Shandong Province under Grant JQ201613.