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• Research ArticleControllability, Reachability, and Stabilizabilityof Finite Automata: A Controllability Matrix Method

Yalu Li,1 Wenhui Dou,1 Haitao Li ,1,2 and Xin Liu1

1School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China2Institute of Data Science and Technology, Shandong Normal University, Jinan 250014, China

Correspondence should be addressed to Haitao Li; [email protected]

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, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates the controllability, reachability, and stabilizability of finite automata by using the semitensor product ofmatrices. 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 necessaryand sufficient conditions are presented for the controllability, reachability, and stabilizability of finite automata by using thecontrollability matrix. Finally, an illustrative example is given to support the obtained new results.

1. Introduction

In the research field of theoretical computer science, finiteautomaton is one of the simplest models of computation.Finite automaton is a device whose states take values froma finite set. It receives a discrete sequence of inputs from theoutside 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 wideapplications in engineering, computer science, and so on.

As we all know, controllability and stabilizability anal-ysis of finite automata are fundamental topics, which areimportant and necessary to the solvability of many relatedproblems [1, 4, 6].The concepts of controllability, reachability,and stabilizability of finite automata were defined in  byresorting to the classic control theory. The controllabilityof a deterministic Rabin automaton was studied in  bydefining the “controllability subset.” Kobayashi et al. investigated the state feedback stabilization of a deterministicfinite automaton and presented some new results.

Recently, a new matrix product, namely, the semitensorproduct (STP) of matrices, has been proposed by Cheng etal. . Up to now, STP has been successfully applied to manyresearch fields related to finite-valued systems like Booleannetworks [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-valuedsystem into an equivalent algebraic form . Thus, STPprovides a convenient way for the construction and analysisof finite automata [5, 26]. Xu andHong  provided amatrix-based algebraic approach for the reachability analysis of finiteautomata with the help of STP. Yan et al.  studied thecontrollability and stabilizability analysis of finite automatabased on STP and presented some novel results. It shouldbe pointed out that although the concepts of controllability,reachability, and stabilizability of finite automata comefrom classic control theory, there exist fewer results on theconstruction of controllability matrix for finite automata.

In this paper, we investigate the controllability, reacha-bility, and stabilizability of deterministic finite automata byusing STP.Themain contribution of this paper is to constructa controllability matrix for finite automata based on the alge-braic form. Using the controllability matrix, we present somenecessary and sufficient conditions for the controllability,reachability, and stabilizability of finite automata. Comparedwith the existing results [5, 26], our results are more easilyverified via MATLAB.

The rest of this paper is organized as follows. Section 2contains some necessary preliminaries on the semitensorproduct of matrices and finite automata. Section 3 studies

HindawiMathematical Problems in EngineeringVolume 2018, Article ID 6719319, 6 pageshttps://doi.org/10.1155/2018/6719319

http://orcid.org/0000-0001-5322-5375https://doi.org/10.1155/2018/6719319

• 2 Mathematical Problems in Engineering

the controllability, reachability, and stabilizability of finiteautomata and presents the main results of this paper. InSection 4, an illustrative example is given to support our newresults, which is followed by a brief conclusion in Section 5.

Notations. R, N, and Z+ denote the set of real numbers,the set of natural numbers, and the set of positive integers,respectively. Δ 𝑛 fl {𝛿𝑘𝑛 : 𝑘 = 1, . . . , 𝑛}, where 𝛿𝑘𝑛 denotes the𝑘th column of the 𝑛 × 𝑛 identity matrix 𝐼𝑛. An 𝑛 × 𝑡 matrix𝑀 is called a logical matrix, if𝑀 = [𝛿𝑖1𝑛 𝛿𝑖2𝑛 ⋅ ⋅ ⋅ 𝛿𝑖𝑡𝑛 ], whichis briefly denoted by𝑀 = 𝛿𝑛 [𝑖1 𝑖2 ⋅ ⋅ ⋅ 𝑖𝑡]. The set of 𝑛 × 𝑡logical matrices is denoted by L𝑛×𝑡. Given a real matrix 𝐴,Col𝑖(𝐴), Row𝑗(𝐴), and (𝐴)𝑖,𝑗 denote the 𝑖th column, the 𝑗throw, and the (𝑖, 𝑗)th element of 𝐴, respectively. 𝐴 > 0 if andonly if (𝐴)𝑖,𝑗 > 0holds for any 𝑖, 𝑗. Blk𝑖(𝐴)denote the 𝑖th blockof an 𝑛 × 𝑚𝑛matrix 𝐴.2. Preliminaries

2.1. Semitensor Product of Matrices. In this part, we recallsome necessary preliminaries on STP. For details, please referto .

Definition 1. Given two matrices 𝐴 ∈ R𝑚×𝑛 and 𝐵 ∈ R𝑝×𝑞,the semitensor product of 𝐴 and 𝐵 is defined as

𝐴 ⋉ 𝐵 = (𝐴 ⊗ 𝐼𝛼/𝑛) (𝐵 ⊗ 𝐼𝛼/𝑝) , (1)where 𝛼 = lcm(𝑛, 𝑝) is the least common multiple of 𝑛 and 𝑝and ⊗ is the Kronecker product of matrices.Lemma 2. STP has the following properties:

(1) Let𝑋 ∈ R𝑡×1 be a column vector and 𝐴 ∈ R𝑚×𝑛. Then𝑋 ⋉ 𝐴 = (𝐼𝑡 ⊗ 𝐴) ⋉ 𝑋. (2)

(2) Let 𝑋 ∈ R𝑚×1 and 𝑌 ∈ R𝑛×1 be two column vectors.Then

𝑌 ⋉ 𝑋 = 𝑊[𝑚,𝑛] ⋉ 𝑋 ⋉ 𝑌, (3)where𝑊[𝑚,𝑛] ∈L𝑚𝑛×𝑚𝑛 is called the swap matrix.

2.2. Finite Automata. In this subsection, we recall somedefinitions of finite automata.

A finite automaton is a seven-tuple 𝐴 = (𝑋,𝑈, 𝑌, 𝑓,𝑔, 𝑥0, 𝑋𝑚), in which 𝑋, 𝑈, and 𝑌 are finite sets of states,input symbols, and outputs, respectively; 𝑥0 and 𝑋𝑚 ⊂ 𝑋are the initial state and the set of accepted states; 𝑓 and 𝑔are transition and output functions, which are defined as 𝑓 :𝑋 × 𝑈 → 2𝑋 and 𝑔 : 𝑋 × 𝑈 → 2𝑌, where 2𝑋 and 2𝑌 denotethe power set of 𝑋 and 𝑌, respectively; that is, 𝑓(𝑥, 𝑢) ⊂ 𝑋,𝑔(𝑥, 𝑢) ⊂ 𝑌. 𝑈∗ represents the finite string set on 𝑈, whichdoes not include the empty transition. Given an initial state𝑥0 ∈ 𝑋 and an input symbol 𝑢 ∈ 𝑈, the function 𝑓 uniquelydetermines the next subset of states, that is, 𝑓(𝑥0, 𝑢) ⊂ 𝑋,while the function 𝑔 uniquely determines the next subset ofoutputs; that is, 𝑔(𝑥0, 𝑢) ⊂ 𝑌.

Throughout this paper, we only consider the determinis-tic finite automata; that is, |𝑓(𝑥, 𝑢)| ≤ 1 holds for any 𝑥 ∈ 𝑋and 𝑢 ∈ 𝑈. In addition, we only investigate the controlla-bility, reachability, and stabilizability of deterministic finiteautomata, and thus we do not use 𝑌 and 𝑔 in the seven-tuple𝐴 = (𝑋,𝑈, 𝑌, 𝑓, 𝑔, 𝑥0, 𝑋𝑚).

In the following, we recall the definitions of controlla-bility, reachability, and stabilizability for deterministic finiteautomata.

Definition 3. (i) A state 𝑥𝑝 ∈ 𝑋 is said to be controllable to𝑥𝑞 ∈ 𝑋, if there exists a control sequence 𝑢𝑡 ∈ 𝑈∗ such that𝑓(𝑥𝑝, 𝑢𝑡) = 𝑥𝑞.(ii) A state 𝑥𝑝 ∈ 𝑋 is said to be controllable, if 𝑥𝑝 ∈ 𝑋 is

controllable to any state 𝑥𝑞 ∈ 𝑋.Definition 4. (i) A state 𝑥𝑞 ∈ 𝑋 is said to be reachable from𝑥𝑝 ∈ 𝑋, if there exists a control sequence 𝑢𝑡 ∈ 𝑈∗ such that𝑓(𝑥𝑝, 𝑢𝑡) = 𝑥𝑞.

(ii) A state 𝑥𝑞 ∈ 𝑋 is said to be reachable, if 𝑥𝑞 ⊂ 𝑋 isreachable from any state 𝑥𝑝 ∈ 𝑋.

Given two nonempty sets 𝑋1 ⊆ 𝑋 and 𝑋2 ⊆ 𝑋 satisfying𝑋1 ∪ 𝑋2 = 𝑋 and 𝑋1 ∩ 𝑋2 = 0, we have the followingdefinitions.

Definition 5. A nonempty set of state 𝑋1 ⊆ 𝑋 is said to becontrollable, if, for any state 𝑥𝑞 ∈ 𝑋2, there exist an 𝑥𝑝 ∈ 𝑋1and a control sequence 𝑢𝑡 ∈ 𝑈∗ such that 𝑓(𝑥𝑝, 𝑢𝑡) = 𝑥𝑞.Definition 6. A nonempty set of state 𝑋2 ⊆ 𝑋 is said to bereachable, if, for any state 𝑥𝑝 ∈ 𝑋1, there exist an 𝑥𝑞 ∈ 𝑋2and a control sequence 𝑢𝑡 ∈ 𝑈∗ such that 𝑓(𝑥𝑝, 𝑢𝑡) = 𝑥𝑞.Definition 7. A nonempty set of state 𝑋1 ⊆ 𝑋 is said to be 1-step returnable, if, for any state 𝑥0 = 𝑥𝑝 ∈ 𝑋1, there exists aninput 𝑢𝑖 ∈ 𝑈 such that 𝑓(𝑥𝑝, 𝑢𝑖) = 𝑥𝑠 ∈ 𝑋1.Definition 8. A nonempty set of state 𝑋1 ⊆ 𝑋 is said to bestabilizable, if𝑋1 is reachable and 1-step returnable.3. Main Results

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

3.1. Controllability Matrix. For a deterministic finite automa-ton 𝐴 = (𝑋,𝑈, 𝑌, 𝑓, 𝑔, 𝑥0, 𝑋𝑚), where 𝑋 = {𝑥1, . . . , 𝑥𝑛} and𝑈 = {𝑢1, . . . , 𝑢𝑚}, we identify 𝑥𝑖 as 𝛿𝑖𝑛 (𝑖 = 1, . . . , 𝑛) andcall 𝛿𝑖𝑛 the vector form of 𝑥𝑖. Then, 𝑋 can be denoted as Δ 𝑛;that is, 𝑋 = {𝛿1𝑛, . . . , 𝛿𝑛𝑛}. Similarly, for 𝑈, we identify 𝑢𝑗 with𝛿𝑗𝑚 (𝑗 = 1, . . . , 𝑚) and call 𝛿𝑗𝑚 the vector form of 𝑢𝑗. Then,𝑈 = {𝛿1𝑚, . . . , 𝛿𝑚𝑚}.

Using the vector form of elements in 𝑋 and 𝑈, Yan et al. construct the transition structure matrix (TSM) of 𝐴 =(𝑋,𝑈, 𝑌, 𝑓, 𝑔, 𝑥0, 𝑋𝑚) as 𝐹 = [𝐹1 ⋅ ⋅ ⋅ 𝐹𝑚]. One can see that if

• Mathematical Problems in Engineering 3

there exists a control 𝑢𝑗 ∈ 𝑈which moves state 𝑥𝑝 to state 𝑥𝑞,then

𝛿𝑞𝑛 = 𝐹 ⋉ 𝛿𝑗𝑚 ⋉ 𝛿𝑝𝑛 . (4)In this case, (𝐹𝑗)𝑞,𝑝 = 1. Otherwise, (𝐹𝑗)𝑞,𝑝 = 0. Thus, setting

𝑀 = 𝑚∑𝑗=1

𝐹𝑗 ∈ R𝑛×𝑛, (5)then one can use 𝑀 to judge whether or not state 𝑥𝑝 iscontrollable to state 𝑥𝑞 in one step. Precisely, state 𝑥𝑝 iscontrollable to state 𝑥𝑞 in one step, if and only if (𝑀)𝑞,𝑝 > 0.

Now, we show that, for any 𝑡 ∈ Z+, state 𝑥𝑝 is controllableto state 𝑥𝑞 at the 𝑡th step, if and only if (𝑀𝑡)𝑞,𝑝 > 0. We proveit by induction. Obviously, when 𝑡 = 1, the conclusion holds.Assume that the conclusion holds for some 𝑡 ∈ Z+. Then,for the case of 𝑡 + 1, state 𝑥𝑝 is controllable to state 𝑥𝑞 at the(𝑡+1)th step, if and only if there exists some state 𝑥𝑟 ∈ 𝑋 suchthat state 𝑥𝑝 is controllable to state 𝑥𝑟 at the 𝑡th step and state𝑥𝑟 is controllable to state 𝑥𝑞 in one step. Hence,(𝑀𝑡+1)

𝑞,𝑝= 𝑛∑𝑖=1

(𝑀)𝑞,𝑖 (𝑀𝑡)𝑖,𝑝 ≥ (𝑀)𝑞,𝑟 (𝑀𝑡)𝑟,𝑝 > 0. (6)By induction, for any 𝑡 ∈ Z+, state 𝑥𝑝 is controllable tostate 𝑥𝑞 at the 𝑡th step, if and only if (𝑀𝑡)𝑞,𝑝 > 0. Thus,∑∞𝑡=1𝑀𝑡 contains all the controllability information of thefinite automata. Noticing that𝑀 is an 𝑛× 𝑛 square matrix, byCayley-Hamilton theorem, we only need to consider 𝑡 ≤ 𝑛.Then, we define the controllability matrix for finite automataas follows.

Definition 9. Set 𝑀 = ∑𝑚𝑗=1 𝐹𝑗 ∈ R𝑛×𝑛. The controllabilitymatrix of finite automata is 𝐶 = ∑𝑛𝑡=1𝑀𝑡.

Based on the controllability matrix, we have the followingresult.

Algorithm 10. Consider the finite automata 𝐴 = (𝑋,𝑈, 𝑌,𝑓, 𝑔, 𝑥0, 𝑋𝑚). Then, the controls which force 𝑥𝑝 to 𝑥𝑞 in theshortest time 𝑙 can be designed by the following steps:

(1) Find the smallest integer 𝑙 such that, forF𝑙0 fl𝑀𝑙−1𝐹= [Blk1 (F𝑙0) Blk2 (F𝑙0) ⋅ ⋅ ⋅ Blk𝑚 (F𝑙0)] , (7)

there exists a block, say, Blk𝜃(F𝑙0), satisfying[Blk𝜃(F𝑙0)]𝑞,𝑝 > 0.(2) Set 𝑢(0) = 𝛿𝜃𝑚 and 𝑥𝑞 = 𝛿𝑞𝑛. If 𝑙 = 1, stop. Otherwise,

go to Step (3).(3) Find 𝑟 and 𝜂 such that [Blk𝜂(F10)]𝑞,𝑟 > 0 and[Blk𝜃(F𝑙−10 )]𝑟,𝑝 > 0, where F10 = 𝐹 and F𝑙−10 =𝑀𝑙−2𝐹. Set 𝑢(𝑙 − 1) = 𝛿𝜂𝑚 and 𝑥(𝑙 − 1) = 𝛿𝑟𝑛.

11

2

2

21

x1x2

x3

Figure 1: A finite automata.

(4) If 𝑙 − 1 = 1, stop. Otherwise, replace 𝑙 and 𝑞 by 𝑙 − 1and 𝑟, respectively, and go to Step (3).

Example 11. Consider a finite automaton 𝐴 = (𝑋,𝑈, 𝑌, 𝑓,𝑔, 𝑥0, 𝑋𝑚) given in Figure 1, where 𝑋 = {𝑥1, 𝑥2, 𝑥3}, 𝑈 ={1, 2}. Suppose that 𝑥0 = 𝑥1 and 𝑋𝑚 = {𝑥3}. Then, 𝑋 canbe denoted as Δ 3 = {𝛿13 , 𝛿23 , 𝛿33}. Similarly, 𝑈 = Δ 2 = {𝛿12 , 𝛿22},𝑥0 = 𝛿33 ,𝑋𝑚 = {𝛿23}.

The transition structure matrix of the finite automata𝐴 is𝐹 = 𝛿3 [2 3 1 3 1 3] . (8)

Split 𝐹 = [𝐹1 𝐹2], where 𝐹1 = 𝛿3 [2 3 1] and 𝐹2 =𝛿3 [3 1 3]. Then,

𝑀 = [[[0 1 11 0 01 1 1

]]]. (9)

Thus, the controllability matrix is

𝐶 = 3∑𝑡=1

𝑀𝑡 = [[[4 5 53 2 27 7 7

]]]. (10)

By Algorithm 10, one can obtain that [Blk1(F20)]2,3 > 0.Setting 𝑥𝑝 = 𝛿33 , 𝑢(0) = 𝛿12 , and 𝑥𝑞 = 𝛿23 , one can find 𝑟 = 1and 𝜂 = 1 such that [Blk1(F10)]2,1 > 0 and [Blk1(F10)]1,3 > 0.Let 𝑢(1) = 𝛿12 and 𝑥1 = 𝛿13 . Hence, state 𝑥3 is controllable tostate 𝑥2 at the 2nd step.3.2. Controllability, Reachability, and Stabilizability. In thispart, we study the controllability, reachability, and stabiliz-ability of deterministic finite automata based on the control-lability matrix.

According to the meaning of controllability matrix, wehave the following results.

Theorem 12. The state 𝑥𝑝 ∈ 𝑋 is controllable, if and only if𝐶𝑜𝑙𝑝 (𝐶) > 0. (11)

Proof.Necessity. Suppose that the state 𝑥𝑝 ∈ 𝑋 is controllable. ByDefinition 3, 𝑥𝑝 ∈ 𝑋 is controllable to any state 𝑥𝑞 ∈ 𝑋.Based on (4), one can see that there exists a control sequence{𝑢𝑖𝛼 : 𝛼 = 0, . . . , 𝑠 − 1} ⊆ 𝑈 (1 ≤ 𝑠 ≤ 𝑛) satisfying

• 4 Mathematical Problems in Engineering

𝛿𝑞𝑛 = 𝐹𝑖𝑠−1 ⋅ ⋅ ⋅ 𝐹𝑖0 ⋉ 𝛿𝑝𝑛 . Thus, (𝐹𝑖𝑠−1 ⋅ ⋅ ⋅ 𝐹𝑖0)𝑞,𝑝 > 0, which impliesthat

(𝐶)𝑞,𝑝 ≥ (𝐹𝑖𝑠−1 ⋅ ⋅ ⋅ 𝐹𝑖0)𝑞,𝑝 > 0. (12)From the arbitrariness of 𝑞, we have Col𝑝(𝐶) > 0.Sufficiency. Suppose that Col𝑝(𝐶) > 0 holds. Then, for anystate𝑥𝑞 ∈ 𝑋, one can find some (𝐹𝑖𝑠−1 ⋅ ⋅ ⋅ 𝐹𝑖0)𝑞,𝑝 > 0.Therefore,under the control sequence {𝑢𝑖𝛼 : 𝛼 = 0, . . . , 𝑠 − 1} ⊆ 𝑈, thestate 𝑥𝑝 ⊂ 𝑋 is controllable to 𝑥𝑞 ∈ 𝑋. From the arbitrarinessof 𝑞, the state 𝑥𝑝 ∈ 𝑋 is controllable.Theorem 13. The state 𝑥𝑞 ∈ 𝑋 is reachable, if and only if

𝑅𝑜𝑤𝑞 (𝐶) > 0. (13)Proof.Necessity. Suppose that the state 𝑥𝑞 ∈ 𝑋 is reachable. ByDefinition 4, 𝑥𝑞 ∈ 𝑋 is reachable to any state 𝑥𝑝 ∈ 𝑋. One canobtain from (4) that there exists a control sequence {𝑢𝑖𝛼 : 𝛼 =0, . . . , 𝑠 − 1} ⊆ 𝑈 (1 ≤ 𝑠 ≤ 𝑛) satisfying 𝛿𝑞𝑛 = 𝐹𝑖𝑠−1 ⋅ ⋅ ⋅ 𝐹𝑖0 ⋉ 𝛿𝑝𝑛 .Thus, (𝐹𝑖𝑠−1 ⋅ ⋅ ⋅ 𝐹𝑖0)𝑞,𝑝 > 0, which shows that

(𝐶)𝑞,𝑝 ≥ (𝐹𝑖𝑠−1 ⋅ ⋅ ⋅ 𝐹𝑖0)𝑞,𝑝 > 0. (14)From the arbitrariness of𝑝, one can conclude that Row𝑞(𝐶) >0.Sufficiency. Suppose that Row𝑞(𝐶) > 0. Then, for any state𝑥𝑝 ∈ 𝑋, there exists some (𝐹𝑖𝑠−1 ⋅ ⋅ ⋅ 𝐹𝑖0)𝑞,𝑝 > 0. Hence, underthe control sequence {𝑢𝑖𝛼 : 𝛼 = 0, . . . , 𝑠 − 1} ⊆ 𝑈, the state𝑥𝑞 ⊂ 𝑋 is reachable to 𝑥𝑝 ∈ 𝑋. By Definition 4, the state𝑥𝑞 ∈ 𝑋 is reachable.

Given two nonempty sets 𝑋1 = {𝛿𝑝1𝑛 , . . . , 𝛿𝑝𝑠𝑛 } and 𝑋2 ={𝛿𝑝𝑠+1𝑛 , . . . , 𝛿𝑝𝑛𝑛 }, where𝑋1 ∪ 𝑋2 = 𝑋 and𝑋1 ∩ 𝑋2 = 0, define𝐶𝑋1 = [

𝑠∑𝛽=1

(𝐶)𝑝𝑠+1,𝑝𝛽 ⋅ ⋅ ⋅𝑠∑𝛽=1

(𝐶)𝑝𝑛,𝑝𝛽] ,

𝑅𝑋2 = [𝑛∑𝛾=𝑠+1

(𝐶)𝑝𝛾,𝑝1 ⋅ ⋅ ⋅𝑛∑𝛾=𝑠+1

(𝐶)𝑝𝛾,𝑝𝑠] .(15)

Based onTheorems 12 and 13, we have the following result.

Theorem 14. (i) The nonempty set 𝑋1 ⊆ 𝑋 is controllable, ifand only if 𝐶𝑋1 > 0.

(ii) The nonempty set 𝑋2 ⊆ 𝑋 is reachable, if and only if𝑅𝑋2 > 0.Proof.(i) Necessity. Suppose that the nonempty set 𝑋1 ⊆ 𝑋 iscontrollable. By Definition 5, for any state 𝛿𝑝𝛾𝑛 ∈ 𝑋2, thereexist a 𝛿𝑝𝛽𝑛 ∈ 𝑋1 and a control sequence 𝑢𝑡 ∈ 𝑈∗ such that𝑓(𝛿𝑝𝛽𝑛 , 𝑢𝑡) = 𝛿𝑝𝛾𝑛 . Based on Theorems 12 and 13, for a fixed𝛿𝑝𝛾𝑛 ∈ 𝑋2, 𝛾 = 𝑠 + 1, 𝑠 + 2, . . . , 𝑛, at least one of the following

cases is true: (𝐶)𝑝𝛾,𝑝1 > 0, (𝐶)𝑝𝛾,𝑝2 > 0, . . . , (𝐶)𝑝𝛾 ,𝑝𝑠 > 0.Therefore, for a fixed 𝛿𝑝𝛾𝑛 ∈ 𝑋2, one can conclude that∑𝑠𝛽=1(𝐶)𝑝𝛾,𝑝𝛽 > 0. From the arbitrariness of 𝑝𝛾, one can seethat

𝐶𝑋1 = [𝑠∑𝛽=1

(𝐶)𝑝𝑠+1,𝑝𝛽 ⋅ ⋅ ⋅𝑠∑𝛽=1

(𝐶)𝑝𝑛,𝑝𝛽] > 0. (16)

Sufficiency. Suppose that 𝐶𝑋1 = [∑𝑠𝛽=1(𝐶)𝑝𝑠+1,𝑝𝛽 ⋅ ⋅ ⋅∑𝑠𝛽=1(𝐶)𝑝𝑛,𝑝𝛽] > 0. Then, for any 𝛾 = 𝑠 + 1, . . . , 𝑛, wehave ∑𝑠𝛽=1(𝐶)𝑝𝛾,𝑝𝛽 > 0. It means that, for any state 𝛿𝑝𝛾𝑛 ∈ 𝑋2,there exist a 𝛿𝑝𝛽𝑛 ∈ 𝑋1 and a control sequence 𝑢𝑡 ∈ 𝑈∗ suchthat 𝑓(𝛿𝑝𝛽𝑛 , 𝑢𝑡) = 𝛿𝑝𝛾𝑛 . By Definition 5, the nonempty set𝑋1 ⊆ 𝑋 is controllable.(ii) Necessity. Suppose that the nonempty set 𝑋2 ⊆ 𝑋 isreachable. By Definition 6, for any state 𝛿𝑝𝛽𝑛 ∈ 𝑋1, thereexist a 𝛿𝑝𝛾𝑛 ∈ 𝑋2 and a control sequence 𝑢𝑡 ∈ 𝑈∗ such that𝑓(𝛿𝑝𝛽𝑛 , 𝑢𝑡) = 𝛿𝑝𝛾𝑛 . Based on Theorems 12 and 13, for a fixed𝛿𝑝𝛽𝑛 ∈ 𝑋1, 𝛽 = 1, 2, . . . , 𝑠, at least one of the following cases istrue: (𝐶)𝑝𝑠+1,𝑝𝛽 > 0, (𝐶)𝑝𝑠+2,𝑝𝛽 > 0, . . . , (𝐶)𝑝𝑛,𝑝𝛽 > 0. Therefore,for a fixed 𝛿𝑝𝛽𝑛 ∈ 𝑋1, one can see that ∑𝑛𝛾=𝑠+1(𝐶)𝑝𝛾,𝑝𝛽 > 0.From the arbitrariness of 𝑝𝛽, we have

[ 𝑛∑𝛾=𝑠+1

(𝐶)𝑝𝛾,𝑝1 ⋅ ⋅ ⋅𝑛∑𝛾=𝑠+1

(𝐶)𝑝𝛾,𝑝𝑠] > 0. (17)

Sufficiency. Suppose that [∑𝑛𝛾=𝑠+1(𝐶)𝑝𝛾,𝑝1 ⋅ ⋅ ⋅ ∑𝑛𝛾=𝑠+1(𝐶)𝑝𝛾,𝑝𝑠]> 0. Then, for any 𝛽 = 1, 2, . . . , 𝑠, we have ∑𝑛𝛾=𝑠+1(𝐶)𝑝𝛾,𝑝𝛽 > 0.It means that, for any state 𝛿𝑝𝛽𝑛 ∈ 𝑋1, there exist a 𝛿𝑝𝛾𝑛 ∈ 𝑋2and a control sequence 𝑢𝑡 ∈ 𝑈∗ such that𝑓(𝛿𝑝𝛽𝑛 , 𝑢𝑡) = 𝛿𝑝𝛾𝑛 . ByDefinition 6, the nonempty set𝑋2 ⊆ 𝑋 is reachable.

Finally, we study the stabilizability of deterministic finiteautomata.

For𝑋1 = {𝛿𝑝1𝑛 , . . . , 𝛿𝑝𝑠𝑛 } and𝑋2 = {𝛿𝑝𝑠+1𝑛 , . . . , 𝛿𝑝𝑛𝑛 }, define𝑀𝑋1 = [

𝑠∑𝑖=1

(𝑀)𝑝𝑖 ,𝑝1 ⋅ ⋅ ⋅𝑠∑𝑖=1

(𝑀)𝑝𝑖,𝑝𝑠] . (18)Theorem 15. The nonempty set𝑋1 ⊆ 𝑋 is 1-step returnable, ifand only if𝑀𝑋1 > 0.Proof. By Definition 7, one can see that the nonempty set𝑋1 ⊆ 𝑋 is 1-step returnable, if and only if, for any state𝛿𝑝𝛽𝑛 ∈ 𝑋1, there exist an input 𝑢𝑖 ∈ 𝑈 and some 𝛿𝑝𝑖𝑛 ∈ 𝑋1such that 𝑓(𝛿𝑝𝛽𝑛 , 𝑢𝑖) = 𝛿𝑝𝑖𝑛 , that is, for a fixed 𝛿𝑝𝛽𝑛 ∈ 𝑋1, 𝛽 =1, 2, . . . , 𝑠, at least one of the following cases is true: (𝑀)𝑝1,𝑝𝛽 >0, (𝑀)𝑝2,𝑝𝛽 > 0, . . . , (𝑀)𝑝𝑠 ,𝑝𝛽 > 0. Hence, ∑𝑠𝑖=1(𝑀)𝑝𝑖,𝑝𝛽 > 0,∀𝛽 = 1, . . . , 𝑠. From the arbitrariness of 𝑝𝛽, one can obtainthat𝑀𝑋1 > 0.

• Mathematical Problems in Engineering 5

Based on Theorems 14 and 15, we have the followingresult.

Corollary 16. The nonempty set𝑋1 ⊆ 𝑋 is stabilizable, if andonly if𝑀𝑋1 > 0 and 𝑅𝑋1 > 0.Proof. By Definition 8, 𝑋1 ⊆ 𝑋 is stabilizable, if and only if𝑋1 is reachable and 1-step returnable. Based on Theorems 14and 15, the conclusion follows.

Remark 17. Compared with the existing results on the con-trollability and stabilizability of deterministic finite automata[5, 26], the main advantage of our results is to propose aunified tool, that is, controllability matrix, for the study ofdeterministic finite automata. The new conditions are moreeasily verified via MATLAB.

4. An Illustrative Example

Consider the finite automata𝐴 = (𝑋,𝑈, 𝑌, 𝑓, 𝑔, 𝑥0, 𝑋𝑚) givenin Figure 2, where𝑋 = {𝑥1, 𝑥2, 𝑥3, 𝑥4} and 𝑈 = {1, 2}.

From Figure 2, we can see that 𝑥1 1→ 𝑥2 1→ 𝑥3 2→ 𝑥4 and𝑥1 1→ 𝑥2 2→ 𝑥1. Therefore, by Definition 3, one can obtainthat 𝑥1 is controllable. Similarly, by Definition 3, we concludethat 𝑥2, 𝑥3, and 𝑥4 are also controllable. By Definition 4, wecan also find that all the states are reachable.

Assume that 𝑋1 = {𝑥1, 𝑥2} and 𝑋2 = {𝑥3, 𝑥4}. Since𝑥1 ∈ 𝑋1 1→ 𝑥2 1→ 𝑥3 ∈ 𝑋2 and 𝑥1 ∈ 𝑋1 2→ 𝑥4 ∈ 𝑋2, byDefinition 5, one can see that𝑋1 is controllable. Similarly, wealso obtain that𝑋2 is controllable. Since 𝑥3 2→ 𝑥4 1→ 𝑥2 2→ 𝑥1and 𝑥1 1→ 𝑥2 1→ 𝑥3 2→ 𝑥4, by Definition 6, we can obtainthat𝑋1 and𝑋2 are reachable. From Figure 2, we can see that𝑥1 1→ 𝑥2, 𝑥2 2→ 𝑥1, 𝑥3 2→ 𝑥4, and 𝑥4 2→ 𝑥3. Hence, byDefinition 7,𝑋1 and𝑋2 are 1-step returnable. ByDefinition 8,the sets𝑋1 and𝑋2 are stabilizable.

Now, we check the above properties based on the control-lability matrix.

The transition structure matrix of the finite automata𝐴 is𝐹 = 𝛿4 [2 3 3 2 4 1 4 3] . (19)

Split 𝐹 = [𝐹1 𝐹2], where 𝐹1 = 𝛿4 [2 3 3 2] and 𝐹2 =𝛿4 [4 1 4 3]. Then,

𝑀 = [[[[[

0 1 0 01 0 0 10 1 1 11 0 1 0

]]]]]. (20)

Thus, the controllability matrix is

𝐶 = 4∑𝑡=1

𝑀𝑡 = [[[[[[

3 4 2 37 5 5 612 13 14 148 8 9 7

]]]]]]. (21)

1

2

1

21

2

1

2 x1 x2

x3

x4

Figure 2: A finite automata.

Since all rows and columns of𝐶 are positive, byTheorems12 and 13, any state 𝑥𝑖 ∈ 𝑋 is controllable and reachable, 𝑖 =1, 2, 3, 4.

A simple calculation gives 𝐶𝑋1 = [25 16], 𝑅𝑋1 = [7 9],𝐶𝑋2 = [5 11], 𝑅𝑋2 = [20 21],𝑀𝑋1 = [10 9], and𝑀𝑋2 =[23 21]. ByTheorems 14 and 15 and Corollary 16,𝑋1 and𝑋2are controllable, reachable, 1-step returnable, and stabilizable,respectively.

5. Conclusion

In this paper, we have investigated the controllability, reach-ability, and stabilizability of deterministic finite automata byusing the semitensor product of matrices. We have obtainedthe algebraic form of finite automata by expressing thestates, inputs, and outputs as vector forms. Based on thealgebraic form, we have defined the controllability matrixfor deterministic finite automata. In addition, using the con-trollability matrix, we have presented several necessary andsufficient conditions for the controllability, reachability, andstabilizability of finite automata. The study of an illustrativeexample has shown that the obtained new results are effective.

Conflicts of Interest

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

Acknowledgments

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

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