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Research ArticleControllability Reachability and Stabilizabilityof Finite Automata A Controllability Matrix Method

Yalu Li1 Wenhui Dou1 Haitao Li 12 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 haitaoli09gmailcom

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

Academic Editor Alessandro Lo Schiavo

Copyright copy 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 automataSecondly 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 computationFinite 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 inputsThe study of finite automata has received many scholarsrsquoresearch interest in the last century [1ndash5] 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 reachabilityand stabilizability of finite automata were defined in [2] byresorting to the classic control theory The controllabilityof a deterministic Rabin automaton was studied in [7] bydefining the ldquocontrollability subsetrdquo Kobayashi et al [8]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 [9] Up to now STP has been successfully applied to manyresearch fields related to finite-valued systems like Booleannetworks [10ndash20] multivalued logical networks [21ndash23]

game theory [24 25] finite automata [5 26] and so on[27ndash35] The main feature of STP is to convert a finite-valuedsystem into an equivalent algebraic form [22] Thus STPprovides a convenient way for the construction and analysisof finite automata [5 26] Xu andHong [5] provided amatrix-based algebraic approach for the reachability analysis of finiteautomata with the help of STP Yan et al [26] studied thecontrollability and stabilizability analysis of finite automatabased on STP and presented some novel results It shouldbe pointed out that although the concepts of controllabilityreachability 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 STPThemain 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 controllabilityreachability 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 pageshttpsdoiorg10115520186719319

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 numbersthe set of natural numbers and the set of positive integersrespectively Δ 119899 fl 120575119896119899 119896 = 1 119899 where 120575119896119899 denotes the119896th column of the 119899 times 119899 identity matrix 119868119899 An 119899 times 119905 matrix119872 is called a logical matrix if119872 = [1205751198941119899 1205751198942119899 sdot sdot sdot 120575119894119905119899 ] whichis briefly denoted by119872 = 120575119899 [1198941 1198942 sdot sdot sdot 119894119905] The set of 119899 times 119905logical matrices is denoted by L119899times119905 Given a real matrix 119860Col119894(119860) Row119895(119860) and (119860)119894119895 denote the 119894th column the 119895throw and the (119894 119895)th element of 119860 respectively 119860 gt 0 if andonly if (119860)119894119895 gt 0holds for any 119894 119895 Blk119894(119860)denote the 119894th blockof an 119899 times 119898119899matrix 1198602 Preliminaries

21 Semitensor Product of Matrices In this part we recallsome necessary preliminaries on STP For details please referto [9]

Definition 1 Given two matrices 119860 isin R119898times119899 and 119861 isin R119901times119902the semitensor product of 119860 and 119861 is defined as

119860 ⋉ 119861 = (119860 otimes 119868120572119899) (119861 otimes 119868120572119901) (1)

where 120572 = lcm(119899 119901) is the least common multiple of 119899 and 119901and otimes is the Kronecker product of matrices

Lemma 2 STP has the following properties

(1) Let119883 isin R119905times1 be a column vector and 119860 isin R119898times119899 Then

119883 ⋉ 119860 = (119868119905 otimes 119860) ⋉ 119883 (2)

(2) Let 119883 isin R119898times1 and 119884 isin R119899times1 be two column vectorsThen

119884 ⋉ 119883 = 119882[119898119899] ⋉ 119883 ⋉ 119884 (3)

where119882[119898119899] isinL119898119899times119898119899 is called the swap matrix

22 Finite Automata In this subsection we recall somedefinitions of finite automata

A finite automaton is a seven-tuple 119860 = (119883119880 119884 119891119892 1199090 119883119898) in which 119883 119880 and 119884 are finite sets of statesinput symbols and outputs respectively 1199090 and 119883119898 sub 119883are the initial state and the set of accepted states 119891 and 119892are transition and output functions which are defined as 119891 119883 times 119880 rarr 2119883 and 119892 119883 times 119880 rarr 2119884 where 2119883 and 2119884 denotethe power set of 119883 and 119884 respectively that is 119891(119909 119906) sub 119883119892(119909 119906) sub 119884 119880lowast represents the finite string set on 119880 whichdoes not include the empty transition Given an initial state1199090 isin 119883 and an input symbol 119906 isin 119880 the function 119891 uniquelydetermines the next subset of states that is 119891(1199090 119906) sub 119883while the function 119892 uniquely determines the next subset ofoutputs that is 119892(1199090 119906) sub 119884

Throughout this paper we only consider the determinis-tic finite automata that is |119891(119909 119906)| le 1 holds for any 119909 isin 119883and 119906 isin 119880 In addition we only investigate the controlla-bility reachability and stabilizability of deterministic finiteautomata and thus we do not use 119884 and 119892 in the seven-tuple119860 = (119883119880 119884 119891 119892 1199090 119883119898)

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

Definition 3 (i) A state 119909119901 isin 119883 is said to be controllable to119909119902 isin 119883 if there exists a control sequence 119906119905 isin 119880lowast such that119891(119909119901 119906119905) = 119909119902(ii) A state 119909119901 isin 119883 is said to be controllable if 119909119901 isin 119883 is

controllable to any state 119909119902 isin 119883Definition 4 (i) A state 119909119902 isin 119883 is said to be reachable from119909119901 isin 119883 if there exists a control sequence 119906119905 isin 119880lowast such that119891(119909119901 119906119905) = 119909119902

(ii) A state 119909119902 isin 119883 is said to be reachable if 119909119902 sub 119883 isreachable from any state 119909119901 isin 119883

Given two nonempty sets 1198831 sube 119883 and 1198832 sube 119883 satisfying1198831 cup 1198832 = 119883 and 1198831 cap 1198832 = 0 we have the followingdefinitions

Definition 5 A nonempty set of state 1198831 sube 119883 is said to becontrollable if for any state 119909119902 isin 1198832 there exist an 119909119901 isin 1198831and a control sequence 119906119905 isin 119880lowast such that 119891(119909119901 119906119905) = 119909119902Definition 6 A nonempty set of state 1198832 sube 119883 is said to bereachable if for any state 119909119901 isin 1198831 there exist an 119909119902 isin 1198832and a control sequence 119906119905 isin 119880lowast such that 119891(119909119901 119906119905) = 119909119902Definition 7 A nonempty set of state 1198831 sube 119883 is said to be 1-step returnable if for any state 1199090 = 119909119901 isin 1198831 there exists aninput 119906119894 isin 119880 such that 119891(119909119901 119906119894) = 119909119904 isin 1198831Definition 8 A nonempty set of state 1198831 sube 119883 is said to bestabilizable if1198831 is reachable and 1-step returnable

3 Main Results

In this section we investigate the controllability reachabilityand stabilizability of deterministic finite automata by con-structing a controllability matrix

31 Controllability Matrix For a deterministic finite automa-ton 119860 = (119883119880 119884 119891 119892 1199090 119883119898) where 119883 = 1199091 119909119899 and119880 = 1199061 119906119898 we identify 119909119894 as 120575119894119899 (119894 = 1 119899) andcall 120575119894119899 the vector form of 119909119894 Then 119883 can be denoted as Δ 119899that is 119883 = 1205751119899 120575119899119899 Similarly for 119880 we identify 119906119895 with120575119895119898 (119895 = 1 119898) and call 120575119895119898 the vector form of 119906119895 Then119880 = 1205751119898 120575119898119898

Using the vector form of elements in 119883 and 119880 Yan et al[26] construct the transition structure matrix (TSM) of 119860 =(119883119880 119884 119891 119892 1199090 119883119898) as 119865 = [1198651 sdot sdot sdot 119865119898] One can see that if

Mathematical Problems in Engineering 3

there exists a control 119906119895 isin 119880which moves state 119909119901 to state 119909119902then

120575119902119899 = 119865 ⋉ 120575119895119898 ⋉ 120575119901119899 (4)

In this case (119865119895)119902119901 = 1 Otherwise (119865119895)119902119901 = 0 Thus setting

119872 = 119898sum119895=1

119865119895 isin R119899times119899 (5)

then one can use 119872 to judge whether or not state 119909119901 iscontrollable to state 119909119902 in one step Precisely state 119909119901 iscontrollable to state 119909119902 in one step if and only if (119872)119902119901 gt 0

Now we show that for any 119905 isin Z+ state 119909119901 is controllableto state 119909119902 at the 119905th step if and only if (119872119905)119902119901 gt 0 We proveit by induction Obviously when 119905 = 1 the conclusion holdsAssume that the conclusion holds for some 119905 isin Z+ Thenfor the case of 119905 + 1 state 119909119901 is controllable to state 119909119902 at the(119905+1)th step if and only if there exists some state 119909119903 isin 119883 suchthat state 119909119901 is controllable to state 119909119903 at the 119905th step and state119909119903 is controllable to state 119909119902 in one step Hence

(119872119905+1)119902119901= 119899sum119894=1

(119872)119902119894 (119872119905)119894119901 ge (119872)119902119903 (119872119905)119903119901 gt 0 (6)

By induction for any 119905 isin Z+ state 119909119901 is controllable tostate 119909119902 at the 119905th step if and only if (119872119905)119902119901 gt 0 Thussuminfin119905=1119872119905 contains all the controllability information of thefinite automata Noticing that119872 is an 119899times 119899 square matrix byCayley-Hamilton theorem we only need to consider 119905 le 119899Then we define the controllability matrix for finite automataas follows

Definition 9 Set 119872 = sum119898119895=1 119865119895 isin R119899times119899 The controllabilitymatrix of finite automata is 119862 = sum119899119905=1119872119905

Based on the controllability matrix we have the followingresult

Algorithm 10 Consider the finite automata 119860 = (119883119880 119884119891 119892 1199090 119883119898) Then the controls which force 119909119901 to 119909119902 in theshortest time 119897 can be designed by the following steps

(1) Find the smallest integer 119897 such that for

F1198970 fl119872119897minus1119865= [Blk1 (F1198970) Blk2 (F1198970) sdot sdot sdot Blk119898 (F1198970)] (7)

there exists a block say Blk120579(F1198970) satisfying[Blk120579(F1198970)]119902119901 gt 0(2) Set 119906(0) = 120575120579119898 and 119909119902 = 120575119902119899 If 119897 = 1 stop Otherwise

go to Step (3)(3) Find 119903 and 120578 such that [Blk120578(F10)]119902119903 gt 0 and

[Blk120579(F119897minus10 )]119903119901 gt 0 where F10 = 119865 and F119897minus10 =119872119897minus2119865 Set 119906(119897 minus 1) = 120575120578119898 and 119909(119897 minus 1) = 120575119903119899

11

2

2

21

x1x2

x3

Figure 1 A finite automata

(4) If 119897 minus 1 = 1 stop Otherwise replace 119897 and 119902 by 119897 minus 1and 119903 respectively and go to Step (3)

Example 11 Consider a finite automaton 119860 = (119883119880 119884 119891119892 1199090 119883119898) given in Figure 1 where 119883 = 1199091 1199092 1199093 119880 =1 2 Suppose that 1199090 = 1199091 and 119883119898 = 1199093 Then 119883 canbe denoted as Δ 3 = 12057513 12057523 12057533 Similarly 119880 = Δ 2 = 12057512 120575221199090 = 12057533 119883119898 = 12057523

The transition structure matrix of the finite automata119860 is

119865 = 1205753 [2 3 1 3 1 3] (8)

Split 119865 = [1198651 1198652] where 1198651 = 1205753 [2 3 1] and 1198652 =1205753 [3 1 3] Then

119872 = [[[0 1 11 0 01 1 1

]]] (9)

Thus the controllability matrix is

119862 = 3sum119905=1

119872119905 = [[[4 5 53 2 27 7 7

]]] (10)

By Algorithm 10 one can obtain that [Blk1(F20)]23 gt 0Setting 119909119901 = 12057533 119906(0) = 12057512 and 119909119902 = 12057523 one can find 119903 = 1and 120578 = 1 such that [Blk1(F10)]21 gt 0 and [Blk1(F10)]13 gt 0Let 119906(1) = 12057512 and 1199091 = 12057513 Hence state 1199093 is controllable tostate 1199092 at the 2nd step

32 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 119909119901 isin 119883 is controllable if and only if

119862119900119897119901 (119862) gt 0 (11)

ProofNecessity Suppose that the state 119909119901 isin 119883 is controllable ByDefinition 3 119909119901 isin 119883 is controllable to any state 119909119902 isin 119883Based on (4) one can see that there exists a control sequence119906119894120572 120572 = 0 119904 minus 1 sube 119880 (1 le 119904 le 119899) satisfying


120575119902119899 = 119865119894119904minus1 sdot sdot sdot 1198651198940 ⋉ 120575119901119899 Thus (119865119894119904minus1 sdot sdot sdot 1198651198940)119902119901 gt 0 which impliesthat

(119862)119902119901 ge (119865119894119904minus1 sdot sdot sdot 1198651198940)119902119901 gt 0 (12)

From the arbitrariness of 119902 we have Col119901(119862) gt 0Sufficiency Suppose that Col119901(119862) gt 0 holds Then for anystate119909119902 isin 119883 one can find some (119865119894119904minus1 sdot sdot sdot 1198651198940)119902119901 gt 0Thereforeunder the control sequence 119906119894120572 120572 = 0 119904 minus 1 sube 119880 thestate 119909119901 sub 119883 is controllable to 119909119902 isin 119883 From the arbitrarinessof 119902 the state 119909119901 isin 119883 is controllable

Theorem 13 The state 119909119902 isin 119883 is reachable if and only if

119877119900119908119902 (119862) gt 0 (13)

ProofNecessity Suppose that the state 119909119902 isin 119883 is reachable ByDefinition 4 119909119902 isin 119883 is reachable to any state 119909119901 isin 119883 One canobtain from (4) that there exists a control sequence 119906119894120572 120572 =0 119904 minus 1 sube 119880 (1 le 119904 le 119899) satisfying 120575119902119899 = 119865119894119904minus1 sdot sdot sdot 1198651198940 ⋉ 120575119901119899 Thus (119865119894119904minus1 sdot sdot sdot 1198651198940)119902119901 gt 0 which shows that


From the arbitrariness of119901 one can conclude that Row119902(119862) gt0Sufficiency Suppose that Row119902(119862) gt 0 Then for any state119909119901 isin 119883 there exists some (119865119894119904minus1 sdot sdot sdot 1198651198940)119902119901 gt 0 Hence underthe control sequence 119906119894120572 120572 = 0 119904 minus 1 sube 119880 the state119909119902 sub 119883 is reachable to 119909119901 isin 119883 By Definition 4 the state119909119902 isin 119883 is reachable

Given two nonempty sets 1198831 = 1205751199011119899 120575119901119904119899 and 1198832 =120575119901119904+1119899 120575119901119899119899 where1198831 cup 1198832 = 119883 and1198831 cap 1198832 = 0 define1198621198831 = [

119904sum120573=1

(119862)119901119904+1119901120573 sdot sdot sdot119904sum120573=1

(119862)119901119899119901120573]

1198771198832 = [119899sum120574=119904+1

(119862)1199011205741199011 sdot sdot sdot119899sum120574=119904+1

(119862)119901120574119901119904] (15)

Based onTheorems 12 and 13 we have the following result

Theorem 14 (i) The nonempty set 1198831 sube 119883 is controllable ifand only if 1198621198831 gt 0

(ii) The nonempty set 1198832 sube 119883 is reachable if and only if1198771198832 gt 0Proof(i) Necessity Suppose that the nonempty set 1198831 sube 119883 iscontrollable By Definition 5 for any state 120575119901120574119899 isin 1198832 thereexist a 120575119901120573119899 isin 1198831 and a control sequence 119906119905 isin 119880lowast such that119891(120575119901120573119899 119906119905) = 120575119901120574119899 Based on Theorems 12 and 13 for a fixed120575119901120574119899 isin 1198832 120574 = 119904 + 1 119904 + 2 119899 at least one of the following

cases is true (119862)1199011205741199011 gt 0 (119862)1199011205741199012 gt 0 (119862)119901120574 119901119904 gt 0Therefore for a fixed 120575119901120574119899 isin 1198832 one can conclude thatsum119904120573=1(119862)119901120574119901120573 gt 0 From the arbitrariness of 119901120574 one can seethat

1198621198831 = [119904sum120573=1


(119862)119901119899119901120573] gt 0 (16)

Sufficiency Suppose that 1198621198831 = [sum119904120573=1(119862)119901119904+1119901120573 sdot sdot sdotsum119904120573=1(119862)119901119899119901120573] gt 0 Then for any 120574 = 119904 + 1 119899 wehave sum119904120573=1(119862)119901120574119901120573 gt 0 It means that for any state 120575119901120574119899 isin 1198832there exist a 120575119901120573119899 isin 1198831 and a control sequence 119906119905 isin 119880lowast suchthat 119891(120575119901120573119899 119906119905) = 120575119901120574119899 By Definition 5 the nonempty set1198831 sube 119883 is controllable

(ii) Necessity Suppose that the nonempty set 1198832 sube 119883 isreachable By Definition 6 for any state 120575119901120573119899 isin 1198831 thereexist a 120575119901120574119899 isin 1198832 and a control sequence 119906119905 isin 119880lowast such that119891(120575119901120573119899 119906119905) = 120575119901120574119899 Based on Theorems 12 and 13 for a fixed120575119901120573119899 isin 1198831 120573 = 1 2 119904 at least one of the following cases istrue (119862)119901119904+1119901120573 gt 0 (119862)119901119904+2119901120573 gt 0 (119862)119901119899119901120573 gt 0 Thereforefor a fixed 120575119901120573119899 isin 1198831 one can see that sum119899120574=119904+1(119862)119901120574119901120573 gt 0From the arbitrariness of 119901120573 we have

[ 119899sum120574=119904+1


(119862)119901120574119901119904] gt 0 (17)

Sufficiency Suppose that [sum119899120574=119904+1(119862)1199011205741199011 sdot sdot sdot sum119899120574=119904+1(119862)119901120574119901119904]gt 0 Then for any 120573 = 1 2 119904 we have sum119899120574=119904+1(119862)119901120574119901120573 gt 0It means that for any state 120575119901120573119899 isin 1198831 there exist a 120575119901120574119899 isin 1198832and a control sequence 119906119905 isin 119880lowast such that119891(120575119901120573119899 119906119905) = 120575119901120574119899 ByDefinition 6 the nonempty set1198832 sube 119883 is reachable

Finally we study the stabilizability of deterministic finiteautomata

For1198831 = 1205751199011119899 120575119901119904119899 and1198832 = 120575119901119904+1119899 120575119901119899119899 define1198721198831 = [

119904sum119894=1

(119872)119901119894 1199011 sdot sdot sdot119904sum119894=1

(119872)119901119894119901119904] (18)

Theorem 15 The nonempty set1198831 sube 119883 is 1-step returnable ifand only if1198721198831 gt 0Proof By Definition 7 one can see that the nonempty set1198831 sube 119883 is 1-step returnable if and only if for any state120575119901120573119899 isin 1198831 there exist an input 119906119894 isin 119880 and some 120575119901119894119899 isin 1198831such that 119891(120575119901120573119899 119906119894) = 120575119901119894119899 that is for a fixed 120575119901120573119899 isin 1198831 120573 =1 2 119904 at least one of the following cases is true (119872)1199011119901120573 gt0 (119872)1199012119901120573 gt 0 (119872)119901119904 119901120573 gt 0 Hence sum119904119894=1(119872)119901119894119901120573 gt 0forall120573 = 1 119904 From the arbitrariness of 119901120573 one can obtainthat1198721198831 gt 0


Based on Theorems 14 and 15 we have the followingresult

Corollary 16 The nonempty set1198831 sube 119883 is stabilizable if andonly if1198721198831 gt 0 and 1198771198831 gt 0Proof By Definition 8 1198831 sube 119883 is stabilizable if and only if1198831 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 automata119860 = (119883119880 119884 119891 119892 1199090 119883119898) givenin Figure 2 where119883 = 1199091 1199092 1199093 1199094 and 119880 = 1 2

From Figure 2 we can see that 1199091 1997888rarr 1199092 1997888rarr 1199093 2997888rarr 1199094 and1199091 1997888rarr 1199092 2997888rarr 1199091 Therefore by Definition 3 one can obtainthat 1199091 is controllable Similarly by Definition 3 we concludethat 1199092 1199093 and 1199094 are also controllable By Definition 4 wecan also find that all the states are reachable

Assume that 1198831 = 1199091 1199092 and 1198832 = 1199093 1199094 Since1199091 isin 1198831 1997888rarr 1199092 1997888rarr 1199093 isin 1198832 and 1199091 isin 1198831 2997888rarr 1199094 isin 1198832 byDefinition 5 one can see that1198831 is controllable Similarly wealso obtain that1198832 is controllable Since 1199093 2997888rarr 1199094 1997888rarr 1199092 2997888rarr 1199091and 1199091 1997888rarr 1199092 1997888rarr 1199093 2997888rarr 1199094 by Definition 6 we can obtainthat1198831 and1198832 are reachable From Figure 2 we can see that1199091 1997888rarr 1199092 1199092 2997888rarr 1199091 1199093 2997888rarr 1199094 and 1199094 2997888rarr 1199093 Hence byDefinition 71198831 and1198832 are 1-step returnable ByDefinition 8the sets1198831 and1198832 are stabilizable

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


119865 = 1205754 [2 3 3 2 4 1 4 3] (19)

Split 119865 = [1198651 1198652] where 1198651 = 1205754 [2 3 3 2] and 1198652 =1205754 [4 1 4 3] Then

119872 = [[[[[

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

]]]]] (20)


119862 = 4sum119905=1

119872119905 = [[[[[[

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

]]]]]] (21)

1

2

1

21

2

1

2 x1x2

x3

x4


Since all rows and columns of119862 are positive byTheorems12 and 13 any state 119909119894 isin 119883 is controllable and reachable 119894 =1 2 3 4

A simple calculation gives 1198621198831 = [25 16] 1198771198831 = [7 9]1198621198832 = [5 11] 1198771198832 = [20 21]1198721198831 = [10 9] and1198721198832 =[23 21] ByTheorems 14 and 15 and Corollary 161198831 and1198832are controllable reachable 1-step returnable and stabilizablerespectively

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 61503225the Natural Science Foundation of Shandong Province underGrant ZR2015FQ003 and the Natural Science Fund forDistinguished Young Scholars of Shandong Province underGrant JQ201613

References

[1] S Abdelwahed and W M Wonham ldquoBlocking Detection inDiscrete Event Systemsrdquo in Proceedings of the American ControlConference pp 1673ndash1678 USA 2003

[2] M Dogruel and U Ozguner ldquoControllability reachabilitystabilizability and state reduction in automatardquo in Proceedingsof the IEEE International Symposium on Intelligent Control pp192ndash197 Glasgow UK 1992


[3] Y Gang ldquoDecomposing a kind of weakly invertible finiteautomata with delay 2rdquo Journal of Computer Science andTechnology vol 18 no 3 pp 354ndash360 2003

[4] J Lygeros C Tomlin and S Sastry ldquoControllers for reachabilityspecifications for hybrid systemsrdquoAutomatica vol 35 no 3 pp349ndash370 1999

[5] X Xu and Y Hong ldquoMatrix expression and reachability analysisof finite automatardquo Control Theory and Technology vol 10 no2 pp 210ndash215 2012

[6] A Casagrande A Balluchi L Benvenuti A Policriti TVilla and A Sangiovanni-Vincentelli ldquoImproving reachabilityanalysis of hybrid automata for engine controlrdquo in Proceedingsof the 43rd IEEE Conference on Decision and Control (CDC) pp2322ndash2327 2004

[7] JThistle andWWonham ldquoControl of infinite behavior of finiteautomatardquo SIAM Journal on Control and Optimization vol 32no 4 pp 1075ndash1097 1994

[8] K Kobayashi J Imura and K Hiraishi ldquoStabilization of finiteautomata with application to hybrid systems controlrdquo DiscreteEvent Dynamic Systems vol 21 no 4 pp 519ndash545 2011

[9] D Cheng H Qi and Z Li Analysis and Control of BooleanNetwork A Semi-Tensor Product Approach Communicationsand Control Engineering Series Springer London UK 2011

[10] E Fornasini and M E Valcher ldquoOn the periodic trajectories ofBoolean control networksrdquoAutomatica vol 49 no 5 pp 1506ndash1509 2013

[11] Y Guo P Wang W Gui and C Yang ldquoSet stability and setstabilization of Boolean control networks based on invariantsubsetsrdquo Automatica vol 61 pp 106ndash112 2015

[12] D Laschov and M Margaliot ldquoMinimum-time control ofBoolean networksrdquo SIAM Journal on Control and Optimizationvol 51 no 4 pp 2869ndash2892 2013

[13] F Li and J Sun ldquoControllability and optimal control of atemporal Boolean networkrdquoNeural Networks vol 34 pp 10ndash172012

[14] F Li ldquoPinning control design for the synchronization of twocoupled boolean networksrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 63 no 3 pp 309ndash313 2016

[15] H Li L Xie and Y Wang ldquoOn robust control invariance ofBoolean control networksrdquo Automatica vol 68 pp 392ndash3962016

[16] H Li Y Wang and L Xie ldquoOutput tracking control of Booleancontrol networks via state feedback Constant reference signalcaserdquo Automatica vol 59 article 6422 pp 54ndash59 2015

[17] H Li and Y Wang ldquoControllability analysis and control designfor switchedBooleannetworkswith state and input constraintsrdquoSIAM Journal on Control and Optimization vol 53 no 5 pp2955ndash2979 2015

[18] H Li L Xie and Y Wang ldquoOutput regulation of Boolean con-trol networksrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 62 no 6 pp 2993ndash2998 2017

[19] J Lu J Zhong CHuang and J Cao ldquoOnpinning controllabilityof Boolean control networksrdquo Institute of Electrical andElectron-ics Engineers Transactions on Automatic Control vol 61 no 6pp 1658ndash1663 2016

[20] M Meng L Liu and G Feng ldquoStability and 1198971 gain analysis ofBoolean networks with Markovian jump parametersrdquo Instituteof Electrical andElectronics Engineers Transactions onAutomaticControl vol 62 no 8 pp 4222ndash4228 2017

[21] Z Liu Y Wang and H Li ldquoNew approach to derivativecalculation of multi-valued logical functions with applicationto fault detection of digital circuitsrdquo IET Control Theory ampApplications vol 8 no 8 pp 554ndash560 2014

[22] J Lu H Li Y Liu and F Li ldquoSurvey on semi-tensor productmethod with its applications in logical networks and otherfinite-valued systemsrdquo IET Control Theory amp Applications vol11 no 13 pp 2040ndash2047 2017

[23] Y Wu and T Shen ldquoAn algebraic expression of finite horizonoptimal control algorithm for stochastic logical dynamicalsystemsrdquo SystemsampControl Letters vol 82 article 3915 pp 108ndash114 2015

[24] D Cheng F He H Qi and T Xu ldquoModeling analysis andcontrol of networked evolutionary gamesrdquo Institute of Electricaland Electronics Engineers Transactions on Automatic Controlvol 60 no 9 pp 2402ndash2415 2015

[25] P Guo H Zhang F E Alsaadi and T Hayat ldquoSemi-tensorproduct method to a class of event-triggered control for finiteevolutionary networked gamesrdquo IET Control Theory amp Applica-tions vol 11 no 13 pp 2140ndash2145 2017

[26] Y Yan Z Chen and Z Liu ldquoSemi-tensor product approach tocontrollability and stabilizability of finite automatardquo Journal ofSystems Engineering and Electronics vol 26 no 1 pp 134ndash1412015

[27] D Cheng and H Qi ldquoNon-regular feedback linearization ofnonlinear systems via a normal form algoithmrdquo Automaticavol 40 pp 439ndash447 2004

[28] H Li G Zhao M Meng and J Feng ldquoA survey on applicationsof semi-tensor product method in engineeringrdquo Science ChinaInformation Sciences vol 61 no 1 Article ID 010202 2018

[29] Z Li Y Qiao H Qi and D Cheng ldquoStability of switchedpolynomial systemsrdquo Journal of Systems Science andComplexityvol 21 no 3 pp 362ndash377 2008

[30] Y Liu H Chen J Lu and BWu ldquoControllability of probabilis-tic Boolean control networks based on transition probabilitymatricesrdquo Automatica vol 52 pp 340ndash345 2015

[31] Y Wang C Zhang and Z Liu ldquoA matrix approach to graphmaximum stable set and coloring problems with application tomulti-agent systemsrdquo Automatica vol 48 no 7 pp 1227ndash12362012

[32] Y Yan Z Chen and Z Liu ldquoSolving type-2 fuzzy relationequations via semi-tensor product of matricesrdquo Control Theoryand Technology vol 12 no 2 pp 173ndash186 2014

[33] K Zhang L Zhang andLXie ldquoInvertibility andnonsingularityof Boolean control networksrdquo Automatica vol 60 article 6475pp 155ndash164 2015

[34] J Zhong J Lu Y Liu and J Cao ldquoSynchronization in an arrayof output-coupled boolean networks with time delayrdquo IEEETransactions on Neural Networks and Learning Systems vol 25no 12 pp 2288ndash2294 2014

[35] Y Zou and J Zhu ldquoSystemdecompositionwith respect to inputsfor Boolean control networksrdquo Automatica vol 50 no 4 pp1304ndash1309 2014

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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 numbersthe set of natural numbers and the set of positive integersrespectively Δ 119899 fl 120575119896119899 119896 = 1 119899 where 120575119896119899 denotes the119896th column of the 119899 times 119899 identity matrix 119868119899 An 119899 times 119905 matrix119872 is called a logical matrix if119872 = [1205751198941119899 1205751198942119899 sdot sdot sdot 120575119894119905119899 ] whichis briefly denoted by119872 = 120575119899 [1198941 1198942 sdot sdot sdot 119894119905] The set of 119899 times 119905logical matrices is denoted by L119899times119905 Given a real matrix 119860Col119894(119860) Row119895(119860) and (119860)119894119895 denote the 119894th column the 119895throw and the (119894 119895)th element of 119860 respectively 119860 gt 0 if andonly if (119860)119894119895 gt 0holds for any 119894 119895 Blk119894(119860)denote the 119894th blockof an 119899 times 119898119899matrix 1198602 Preliminaries

21 Semitensor Product of Matrices In this part we recallsome necessary preliminaries on STP For details please referto [9]

Definition 1 Given two matrices 119860 isin R119898times119899 and 119861 isin R119901times119902the semitensor product of 119860 and 119861 is defined as

119860 ⋉ 119861 = (119860 otimes 119868120572119899) (119861 otimes 119868120572119901) (1)

where 120572 = lcm(119899 119901) is the least common multiple of 119899 and 119901and otimes is the Kronecker product of matrices

Lemma 2 STP has the following properties

(1) Let119883 isin R119905times1 be a column vector and 119860 isin R119898times119899 Then

119883 ⋉ 119860 = (119868119905 otimes 119860) ⋉ 119883 (2)

(2) Let 119883 isin R119898times1 and 119884 isin R119899times1 be two column vectorsThen

119884 ⋉ 119883 = 119882[119898119899] ⋉ 119883 ⋉ 119884 (3)

where119882[119898119899] isinL119898119899times119898119899 is called the swap matrix

22 Finite Automata In this subsection we recall somedefinitions of finite automata

A finite automaton is a seven-tuple 119860 = (119883119880 119884 119891119892 1199090 119883119898) in which 119883 119880 and 119884 are finite sets of statesinput symbols and outputs respectively 1199090 and 119883119898 sub 119883are the initial state and the set of accepted states 119891 and 119892are transition and output functions which are defined as 119891 119883 times 119880 rarr 2119883 and 119892 119883 times 119880 rarr 2119884 where 2119883 and 2119884 denotethe power set of 119883 and 119884 respectively that is 119891(119909 119906) sub 119883119892(119909 119906) sub 119884 119880lowast represents the finite string set on 119880 whichdoes not include the empty transition Given an initial state1199090 isin 119883 and an input symbol 119906 isin 119880 the function 119891 uniquelydetermines the next subset of states that is 119891(1199090 119906) sub 119883while the function 119892 uniquely determines the next subset ofoutputs that is 119892(1199090 119906) sub 119884

Throughout this paper we only consider the determinis-tic finite automata that is |119891(119909 119906)| le 1 holds for any 119909 isin 119883and 119906 isin 119880 In addition we only investigate the controlla-bility reachability and stabilizability of deterministic finiteautomata and thus we do not use 119884 and 119892 in the seven-tuple119860 = (119883119880 119884 119891 119892 1199090 119883119898)

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

Definition 3 (i) A state 119909119901 isin 119883 is said to be controllable to119909119902 isin 119883 if there exists a control sequence 119906119905 isin 119880lowast such that119891(119909119901 119906119905) = 119909119902(ii) A state 119909119901 isin 119883 is said to be controllable if 119909119901 isin 119883 is

controllable to any state 119909119902 isin 119883Definition 4 (i) A state 119909119902 isin 119883 is said to be reachable from119909119901 isin 119883 if there exists a control sequence 119906119905 isin 119880lowast such that119891(119909119901 119906119905) = 119909119902

(ii) A state 119909119902 isin 119883 is said to be reachable if 119909119902 sub 119883 isreachable from any state 119909119901 isin 119883

Given two nonempty sets 1198831 sube 119883 and 1198832 sube 119883 satisfying1198831 cup 1198832 = 119883 and 1198831 cap 1198832 = 0 we have the followingdefinitions

Definition 5 A nonempty set of state 1198831 sube 119883 is said to becontrollable if for any state 119909119902 isin 1198832 there exist an 119909119901 isin 1198831and a control sequence 119906119905 isin 119880lowast such that 119891(119909119901 119906119905) = 119909119902Definition 6 A nonempty set of state 1198832 sube 119883 is said to bereachable if for any state 119909119901 isin 1198831 there exist an 119909119902 isin 1198832and a control sequence 119906119905 isin 119880lowast such that 119891(119909119901 119906119905) = 119909119902Definition 7 A nonempty set of state 1198831 sube 119883 is said to be 1-step returnable if for any state 1199090 = 119909119901 isin 1198831 there exists aninput 119906119894 isin 119880 such that 119891(119909119901 119906119894) = 119909119904 isin 1198831Definition 8 A nonempty set of state 1198831 sube 119883 is said to bestabilizable if1198831 is reachable and 1-step returnable

3 Main Results

In this section we investigate the controllability reachabilityand stabilizability of deterministic finite automata by con-structing a controllability matrix

31 Controllability Matrix For a deterministic finite automa-ton 119860 = (119883119880 119884 119891 119892 1199090 119883119898) where 119883 = 1199091 119909119899 and119880 = 1199061 119906119898 we identify 119909119894 as 120575119894119899 (119894 = 1 119899) andcall 120575119894119899 the vector form of 119909119894 Then 119883 can be denoted as Δ 119899that is 119883 = 1205751119899 120575119899119899 Similarly for 119880 we identify 119906119895 with120575119895119898 (119895 = 1 119898) and call 120575119895119898 the vector form of 119906119895 Then119880 = 1205751119898 120575119898119898

Using the vector form of elements in 119883 and 119880 Yan et al[26] construct the transition structure matrix (TSM) of 119860 =(119883119880 119884 119891 119892 1199090 119883119898) as 119865 = [1198651 sdot sdot sdot 119865119898] One can see that if



120575119902119899 = 119865 ⋉ 120575119895119898 ⋉ 120575119901119899 (4)


119872 = 119898sum119895=1

119865119895 isin R119899times119899 (5)



(119872119905+1)119902119901= 119899sum119894=1

(119872)119902119894 (119872119905)119894119901 ge (119872)119902119903 (119872119905)119903119901 gt 0 (6)










11

2

2

21

x1x2

x3





119865 = 1205753 [2 3 1 3 1 3] (8)


119872 = [[[0 1 11 0 01 1 1

]]] (9)


119862 = 3sum119905=1

119872119905 = [[[4 5 53 2 27 7 7

]]] (10)





119862119900119897119901 (119862) gt 0 (11)







119877119900119908119902 (119862) gt 0 (13)





119904sum120573=1


(119862)119901119899119901120573]

1198771198832 = [119899sum120574=119904+1


(119862)119901120574119901119904] (15)





1198621198831 = [119904sum120573=1


(119862)119901119899119901120573] gt 0 (16)



[ 119899sum120574=119904+1


(119862)119901120574119901119904] gt 0 (17)



For1198831 = 1205751199011119899 120575119901119904119899 and1198832 = 120575119901119904+1119899 120575119901119899119899 define1198721198831 = [

119904sum119894=1


(119872)119901119894119901119904] (18)












119865 = 1205754 [2 3 3 2 4 1 4 3] (19)


119872 = [[[[[

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

]]]]] (20)


119862 = 4sum119905=1

119872119905 = [[[[[[

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

]]]]]] (21)

1

2

1

21

2

1

2 x1x2

x3

x4




5 Conclusion




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018










120575119902119899 = 119865 ⋉ 120575119895119898 ⋉ 120575119901119899 (4)


119872 = 119898sum119895=1

119865119895 isin R119899times119899 (5)



(119872119905+1)119902119901= 119899sum119894=1

(119872)119902119894 (119872119905)119894119901 ge (119872)119902119903 (119872119905)119903119901 gt 0 (6)










11

2

2

21

x1x2

x3





119865 = 1205753 [2 3 1 3 1 3] (8)


119872 = [[[0 1 11 0 01 1 1

]]] (9)


119862 = 3sum119905=1

119872119905 = [[[4 5 53 2 27 7 7

]]] (10)





119862119900119897119901 (119862) gt 0 (11)







119877119900119908119902 (119862) gt 0 (13)





119904sum120573=1


(119862)119901119899119901120573]

1198771198832 = [119899sum120574=119904+1


(119862)119901120574119901119904] (15)





1198621198831 = [119904sum120573=1


(119862)119901119899119901120573] gt 0 (16)



[ 119899sum120574=119904+1


(119862)119901120574119901119904] gt 0 (17)



For1198831 = 1205751199011119899 120575119901119904119899 and1198832 = 120575119901119904+1119899 120575119901119899119899 define1198721198831 = [

119904sum119894=1


(119872)119901119894119901119904] (18)












119865 = 1205754 [2 3 3 2 4 1 4 3] (19)


119872 = [[[[[

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

]]]]] (20)


119862 = 4sum119905=1

119872119905 = [[[[[[

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

]]]]]] (21)

1

2

1

21

2

1

2 x1x2

x3

x4




5 Conclusion




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018













119877119900119908119902 (119862) gt 0 (13)





119904sum120573=1


(119862)119901119899119901120573]

1198771198832 = [119899sum120574=119904+1


(119862)119901120574119901119904] (15)





1198621198831 = [119904sum120573=1


(119862)119901119899119901120573] gt 0 (16)



[ 119899sum120574=119904+1


(119862)119901120574119901119904] gt 0 (17)



For1198831 = 1205751199011119899 120575119901119904119899 and1198832 = 120575119901119904+1119899 120575119901119899119899 define1198721198831 = [

119904sum119894=1


(119872)119901119894119901119904] (18)












119865 = 1205754 [2 3 3 2 4 1 4 3] (19)


119872 = [[[[[

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

]]]]] (20)


119862 = 4sum119905=1

119872119905 = [[[[[[

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

]]]]]] (21)

1

2

1

21

2

1

2 x1x2

x3

x4




5 Conclusion




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018


















119865 = 1205754 [2 3 3 2 4 1 4 3] (19)


119872 = [[[[[

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

]]]]] (20)


119862 = 4sum119905=1

119872119905 = [[[[[[

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

]]]]]] (21)

1

2

1

21

2

1

2 x1x2

x3

x4




5 Conclusion




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018

















































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018






