Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 6719319, 6 pages

https://doi.org/10.1155/2018/6719319

## Controllability, Reachability, and Stabilizability of Finite Automata: A Controllability Matrix Method

^{1}School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China^{2}Institute of Data Science and Technology, Shandong Normal University, Jinan 250014, China

Correspondence should be addressed to Haitao Li

Received 15 November 2017; Revised 6 January 2018; Accepted 18 January 2018; Published 28 February 2018

Academic Editor: Alessandro Lo Schiavo

Copyright © 2018 Yalu Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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, , , .