Matrix expression and reachability analysis of finite automata

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1 J Control Theory Appl (2) DOI /s Matrix expression and reachability analysis of finite automata Xiangru XU, Yiguang HONG Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing , China Abstract: In this paper, we propose a matrix-based approach for finite automata and then study the reachability conditions Both the deterministic and nondeterministic automata are expressed in matrix forms, and the necessary and sufficient conditions on reachability are given using semitensor product of matrices Our results show that the matrix expression provides an effective computational way for the reachability analysis of finite automata Keywords: Finite automata; Reachability; Matrix expression; Semitensor product 1 Introduction Automata provide an appropriate model for the design and analysis of discrete-event systems and hybrid dynamics [1 4] Reachability analysis of automata is an interesting and important topic, which is essential to many related problems such as robust stability of hybrid system, blocking detection and safety properties of automata [ 7] Algebraic approach is an elegant branch for automata theory [8], but few efforts have been devoted to the matrixbased methods For example, reference [9] introduced the notion of transition matrix in the study of sequential machines, reference [10] presented a Boolean-matrix-based method for automata to investigate regularity-preserving functions, and reference [11] addressed a state equation-like approach for automata and defines controllability, reachability and stabilizability following the conventional framework of control theory Although some results were provided in the existing literatures, many important problems remain to be solved The objective of the paper is to propose a matrix-based approach for finite automata with the help of semitensor product (STP) First proposed by Cheng [12], STP (or simply called Cheng product) is a generalization of the conventional matrix product, and has been fully developed and widely applied to many areas including Boolean networks [12], Boolean calculus [13], and nonlinear control [14] Compared with the previous models, the STP-based matrix method proposed in this paper provide a very natural way in the modeling of finite automata for both deterministic and nondeterministic cases The state and input of a given automaton can be expressed in column vector forms, and the run of an input string can be considered as straightforward matrix products in the sense of STP What is more, we do not confine the matrix operation in the Boolean domain, which can provide us with more information With the new expression, the analysis on reachability can be carried out uniformly for both deterministic and nondeterministic automata The paper is organized as follows Basic notations and definitions are given and the matrix expression for finite automata is shown in Section 2 Then, a sufficient and necessary condition for the reachability problem of both deterministic and nondeterministic automata are given in Section 3, while further discussion is carried out on the reachability computation and the construction of reachable set in Section 4 Finally, conclusions are given in Section 2 Matrix-based expression of automata In this section, we first give preliminaries and then propose a matrix-based expression for finite automata First of all, we introduce notations for the following usage: δn k is the kth column of the identity matrix I n R n n, and denote Δ n := {δn,,δ 1 n} n col i (A) is the ith column of matrix A, and Col(A) is the set of all the columns of matrix A 0 n R n is the vector with each element equal to 0, while 1 n R n is the vector with each element equal to 1 Given η = (η 1,,η n ) T,ζ = (ζ 1,,ζ n ) T R n, denote η ζ if η i ζ i for i =1,,n M (p,q) denotes the element in the pth row and qth column of a given matrix M STP or Cheng product was first proposed by Cheng [12] Definition 1 For M R m n and N R p q, their STP (or Cheng product), denoted by M N, is defined as follows: M N := (M I s/n )(N I s/p ), (1) where s is the least common multiple of n and p and is the Kronecker product Obviously, the conventional matrix product is the particular case of STP when n = p We assume the matrix product throughout the paper to be STP, so we omit the symbol when there is no confusion For a matrix A, denote A 1 = A, A i+1 = A i A, i 1 For any two column vectors x R m and y R n, there is a unique swap matrix W [m,n] R mn mn only depending on n, m such that W [m,n] xy = yx (2) Received 30 August 2011; revised 6 February 2012 This work was supported by the National Natural Science Foundation of China (No ) c South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2012

2 X Xu et al / J Control Theory Appl (2) Other useful properties of STP can be found in [12] Before providing the matrix expression, we first introduce the basic concepts of finite automata [1 2]: Definition 2 An automaton is a five-tuple A =(X, E, f,x 0,X m ), where X is a finite set of states; E is a finite set of input symbols called alphabet; x 0 X is the initial state; X m X is the set of accepted states; and f : X E 2 X where 2 X denotes the power set of X (ie, the set of all subsets of X), that is, f(x, e) X whenever it is defined A is a deterministic finite automaton (DFA) if for each x X, e E, f(x, e) 1; otherwise, it is said to be a nondeterministic finite automaton (NFA) Both DFA and NFA can be generally called finite automaton Denote E as the set of finite strings on the alphabet E which excludes the empty transition Given an input string e = e 1 e 2 e t E, we define f(x, e) = f(f(f(f(x, e 1 ),e 2 ) ),e t ) In other words, the automaton starting from the initial state x reads the input e 1 and moves to x f(x, e 1 ), and then reads input e 2 and moves to x f(x,e 2 ), and so on We call x e1 x e 2 x a path of the automaton given input string e and initial state x We call a state x is reachable from y if there is e E such that y f(x, e) It is easy to see that, given y that is reachable from x, there is only one path from x to y if the automaton considered is deterministic, and there might be more than one path if the automaton is nondeterministic Given an automaton A =(X, E, f, x 0,X m ), the reachability problem is to find a string of E that makes A go from the initial state to the target state at least once [1] Then, we will introduce a new matrix expression for automata To do this, we first define the transition structure matrix, which is the key for the modeling and calculation of finite automata Given a finite automaton A = (X, E, f, x 0,X m ) Assume X = {x 1,x 2,,x n } Identify x i with δn i (1 i n) (expressed as x i δn i for simplicity), and call δn i the vector form of x i Now, we can denote Δ n as the set of states X, namely, X = {δn,δ 1 n,,δ 2 n} n Similarly, assume E = {e 1,e 2,,e m }, we identify e j with δm j (1 j m) (denoted as e j δm), j called the vector form of e i, and the set of input symbol E can be expressed as Δ m, namely, E = {δm,δ 1 m,,δ 2 m} m Therefore, x i f(x j,e k ) can be equivalently expressed as δn i f(δn,δ j m) k using the vector form of the input and state Input e i E (1 i m) of A determines a unique matrix F i R n n, called transition structure matrix with respect to e i, where { 1, if δn s f(δ t F i(s,t) = n,δm), i (3) 0, otherwise Actually, F i is the adjacency matrix of the e i -labeled subgraph associated with A [16], and the transition matrix introduced in [1] and [9] is the transpose of F i Therefore, a finite automaton A determines a matrix F = [F 1 F 2 F m ] R n mn, which is called the transition structure matrix of A On the other hand, once the matrix F is given, it is obvious that the transition function of the automaton can be uniquely determined by (3) As a result, there is an equivalent relationship between the transition function f of automaton A and its transition structure matrix F Remark 1 Although each element of matrix F is either 0or1,F is essentially different from the logical matrix discussed in Boolean network [12] If the finite automaton A is deterministic, every column of F contains no more than one 1, while if A is nondeterministic, some columns of F may contain more than one 1 s However, each column of logical matrix in [12] has exactly one 1 If we consider the state of the automaton evolves with a given input e i at every step, a finite automaton can be viewed as a dynamical system We introduce some concepts that will be used in the sequel The vector form of input of a given automaton A at time t is a column vector u(t) Δ m, and u(t) =δm i with the input e i at time step t An input string e = e i1 e i2 e it E can be identified with t j=1u(j) := u(1)u(2) u(t) Δ tm (that is, e t j=1 u(j)), where e i j is identified with u(j), 1 j t The vector form of state of A at time step t is a column vector x(t) = (x 1 (t),x 2 (t),, x n (t)) T R n, where x i (t) is the number of different paths by which the initial state can reach state δn i with a given input string of length t 1, where x(1) = x 0 For DFA, the target state from which the initial state can move to with a given input string of length t 1 is unique and only one path exists from the initial state to the target state Therefore, there is only one j with 1 j n such that x j (t) =1, namely, x(t) Δ n If no state is reachable with the input string of length t 1, x(t) = 0 n For NFA, given an input string, the states reachable from the initial state forms a set, and there may exist more than one paths from the initial state to the target states Thus, there may exist more than one elements in x(t) greater than 0 The following is a basic result for the matrix expression of automata using notations above Theorem 1 Given a finite input string s j=1u(j) for finite automaton A, the dynamics of A can be described by the following equation: x(t +1)=Fu(t)x(t), 1 t s (4) Proof We prove it by mathematical induction When t =1,wehaveFu(1)x(1) = F k x(1) R n with u(1) = δm, k where 1 k m is arbitrary If u(1) moves x(1) to δn, j 1 j n, then the jth element of F k x(1) equals 1 from the definition of transition structure matrix, while F k x(1) = 0 n if u(1) cannot move x(1) to any other state By the meaning of the vector form of state x(2), itis obvious that (4) holds when t =1 Suppose (4) holds when t = p 1, and u(p) =δm k for any k with 1 k m Then, Fu(p)x(p)=F k (x 1 (p),x 2 (p),,x n (p)) T =( n F k(1,i) x i (p), F k(2,i) x i (p),, F k(n,i) x i (p)) T

3 212 X Xu et al / J Control Theory Appl (2) By (3), we have F k(j,i) x i (p) = i I j x i (p), 1 j n, where I j = {i 1 i n, δn i moves to δn j with input e k } Therefore, x i (p) equals the sum of the number of different paths by which the initial state x 0 can reach state δn i i I j with the input string e = p 1 s=1 u(s), and then from δi n to δn j with input u(p) =δm, k where i can be arbitrary On the other hand, x j (p +1)equals the number of different paths, by which the initial state x 0 can reach δn j with the input string p s=1 u(s) Thus, xj (p +1) = n F k(j,i) x i (p) for each j (double counting), which implies Fu(p)x(p) =x(p +1) By mathematical induction, the proof is completed From equation (4), it is clear that, if x(t) = 0 n, then x(k) = 0 n, k t Remark 2 As mentioned above, matrix-based methods were also investigated in [9 11], where the matrix with respective to every input was defined and considered in a separate manner Our matrix-based formulation, however, handles the automata in a unified way for both DFA and NFA 3 Reachability analysis In this section, we propose matrix-based analysis for the reachability of finite automata Consider a finite automaton A = (X, E, f, x 0,X m ), where X = {x 1,x 2,,x n }, E = {e 1,e 2,,e m } Identify x i with δn(1 i i n), and e j with δm(1 j j m),as described above Then, we have the following main result based on the proposed matrix expression Theorem 2 Suppose F is the transition structure matrix of A Then, target state x = δn q is reachable from x 0 = δn p with input strings of length t if and only if there exists η Col((FW [n,m] ) t δn) p such that δn q η Proof Let x(t) and u(t) be the vector form of state and input of A at time t, respectively x(1) = x 0 For a finite input string e = e i1 e i2 e it E, where e ij u(j), we have x(t +1)=Fu(t)x(t) = FW [n,m] x(t)u(t) =(FW [n,m] ) 2 x(t 1)u(t 1)u(t) =(FW [n,m] ) t x(1) t k=1 u(k) =((FW [n,m] ) t δ p n) t k=1 u(k) only if part: Because (FW [n,m] ) t δn p R n mt and t k=1 u(k) Δ mt, we can let η = x(t +1) Col((FW [n,m] ) t δn) p Since x = δn q is reachable from x 0 = δn p with input string e, it is obvious that x q (t+1) > 0 For DFA, x(t+1) Δ n, we have δn q = x(t +1) = η; and for NFA, we have δn q x(t +1)=η if part: Since there is η such that δn q η and η = col τ ((FW [n,m] ) t δn)(1 p τ m t ),wehavex q (t +1) 1 given input string e t k=1 u(k) =δτ m From the definition of vector form of state, we conclude that x is reachable t from x 0 with input string e Summarizing the above analysis, we obtain the conclusion The proof is completed The vector form of the input string that makes initial state reach the target state can be easily constructed from the proof of Theorem 2 Furthermore, it is not hard to see that if A is DFA, then the necessary and sufficient condition of Theorem 2 becomes δn q Col((FW [n,m] ) t δn) p The following corollary gives a criterion for the language accepted by A Corollary 1 Given finite automaton A =(X, E, f, x 0, X m ) with x 0 = δn p The language of length t accepted by A is the set {e x targ (FW [n,m] ) t δn p t k=1 u(k), where e = t k=1 u(k),xtarg X m } Based on Corollary 1, we can formally obtain the language accepted by A We give the following examples to illustrate the given results and related algorithms Example 1 1) Consider DFA A =(X, E, f, x 0,X m ) shown in Fig 1, where X = {x 1,x 2,x 3,x 4,x }, E = {e 1,e 2,e 3,e 4 }, x 0 = x 1, X m = {x 4,x } Fig 1 DFA of Example 1 The transition structure matrix of A is F = When t =2, (FW [,4] ) 2 δ 1 R 16, where col 2 ((FW [,4] ) 2 δ 1 )=δ 3, col 3 ((FW [,4] ) 2 δ 1 )=δ 4 with all the other columns equal to 0 Thus, by Theorem 2, states δ 3 and δ 4 are reachable from x 0 with input string of length 2 For example, suppose δ 4 is the target state Since δ 4 =(FW [,4] ) 2 δ 1 δ 3 16, the corresponding input string is e = u(1)u(2) = δ 3 16 with u(1) = δ 1 4 e 1,u(2) = δ 3 4 e 3 Therefore, the string e = e 1 e 3 moves the initial state to the target state x 4 When t =3, (FW [,4] ) 3 δ 1 R 64, where col ((FW [,4] ) 3 δ 1 )=δ 4, col 7 ((FW [,4] ) 3 δ 1 )=δ 3, col 8 ((FW [,4] ) 3 δ 1 )=δ, col 12 ((FW [,4] ) 3 δ 1 )=δ with all the other columns equal to 0 Thus, δ,δ 3,δ 4 are reachable from x 0 with input string of length 3 Take δ for example, input string e = u(1)u(2)u(3) = δ64 8 or δ64 12 can move initial state x 0 to target state δ Since δ64 8 = δ4δ 1 4δ e 1 e 2 e 4 and δ64 12 = δ4δ 1 4δ e 1 e 3 e 4, both input strings e = e 1 e 2 e 4 and e = e 1 e 3 e 4 can move δ 1 to the target state x

4 X Xu et al / J Control Theory Appl (2) When t =4, col 20 ((FW [,4] ) 4 δ 1 )=δ, col 2 ((FW [,4] ) 4 δ 1 )=δ 4, col 27 ((FW [,4] ) 4 δ 1 )=δ 3, col 28 ((FW [,4] ) 4 δ 1 )=δ with all the other columns equal to 0 Therefore, δ 3,δ 4,δ are reachable from x 0 with input string of length 4 We have δ 20 δ 2 δ 27 δ = δ4δ 1 4δ 2 4δ e = e 1 e 2 e 1 e 4 moves δ 1 to δ, 26 = δ4δ 1 4δ 2 4δ e = e 1 e 2 e 3 e 1 moves δ 1 to δ, 4 26 = δ4δ 1 4δ 2 4δ e = e 1 e 2 e 3 e 3 moves δ 1 to δ, 3 26 = δ4δ 1 4δ 2 4δ e = e 1 e 2 e 3 e 4 moves δ 1 to δ By Corollary 1, the language of length 4 accepted by A is {e 1 e 2 e 1 e 4,e 1 e 2 e 3 e 1,e 1 e 2 e 3 e 4 } Other arbitrary finite length of input strings can be analyzed likewise 2) Consider NFA A=(X, E, f, x 0,X m ) shown in Fig 2, where X = {x 1,x 2,x 3,x 4,x }, E = {e 1,e 2,e 3,e 4 }, x 0 = x 1, and X m = {x } It can be seen that the state x 2 can move to x 3 or x 4 given input e 2 Fig 2 NFA of Example 1 The transition structure matrix of A is F = When t =2, col 2 ((FW [,4] ) 2 δ 1 )=(0, 0, 1, 1, 0) T, col 3 ((FW [,4] ) 2 δ 1 )=(0, 0, 0, 1, 0) T with all the other columns equal to 0 Since δ 4 (0, 0, 1, 1, 0) T and δ 4 (0, 0, 0, 1, 0) T, δ 4 is reachable from x 0 with input string e = u(1)u(2) = δ16 2 = δ4δ e 1 e 2 or e = u(1)u(2) = δ16 3 = δ4δ e 1 e 3 Since δ 3 (0, 0, 1, 1, 0) T, δ 3 is reachable from x 0 with input string e = u(1)u(2) = δ16 2 = δ4δ e 1 e 2 When t =3,wehave col ((FW [,4] ) 3 δ 1 )=(0, 0, 0, 1, 0) T, col 7 ((FW [,4] ) 3 δ 1 )=(0, 0, 1, 0, 0) T, col 8 ((FW [,4] ) 3 δ 1 )=(0, 0, 0, 0, 2) T, col 12 ((FW [,4] ) 3 δ 1 )=(0, 0, 0, 0, 1) T with all the other columns equal to 0 For instance, since x(3) = (0, 0, 0, 0, 2) T with input string e = u(1)u(2)u(3) = δ 8 64 = δ 1 4δ 2 4δ 4 4 e 1 e 2 e 4, initial state δ 1 can reach the target state δ via two different paths: x 1 e 1 x2 e 2 x3 e 4 x and x 1 e 1 x2 e 2 x4 e 4 x By Corollary 1, the language of length 3 accepted by A is {e 1 e 2 e 4,e 1 e 3 e 4 } 4 Further analysis It is known that the computation is not easy when the size of the automaton is large With our matrix method, (FW [n,m] ) t in Theorem 1 is of dimension n m t n When the length of input string t is not short enough, the size of (FW [n,m] ) t will be too large to calculate However, if we are only interested in the reachability relation between states of a finite automaton and do not care much about what the paths are with input strings of a given length, the computation may become relatively easier The following result is given to show the number of different paths from state δn p to state δn q Theorem 3 Consider a finite automaton A =(X, E, f,x 0,X m ) with transition structure matrix F Then, M t (q,p) equals the number of different paths that state δn p can reach state δn q with input string of length t, where matrix M t =(δm) 1 T ( 1 m F ) t 1 m () Proof Set matrix K =( 1 m F ) R mn mn Since F =[F 1 F 2 F m ] R n mn,wehave F 1 F 2 F m F 1 F 2 F m K = F 1 F 2 F m Obviously, K is row-periodic, and so is K t for t 1 Partition K t into equal block matrices as follows: K1 t K2 t Km t K K t 1 t K2 t K t m =, K1 t K2 t Km t where Ks t R n n, 1 s m We claim that, if Ks(i,j) t = c, there are c different paths to make x j reach x i, with input string of length t which begins with e s In fact, the claim can be proved by mathematical induction When t =1, the conclusion follows immediately Suppose the claim holds when t = l Then, K l+1 = KK l and Ks l+1 (i,j) = mn K (i,a) K l (a,j) = m F b(i,k) Ks(k,j) l a=1 b=1 k=1 Given an input string e = e α1 e α2 e αt+1 with e α1 = e s, a path from x j to reach x i can be considered as follows: the state first moves from x j to x k with input string e α1 e α2 e αt, and then from x k to x i with input string e αt+1 Since Ks(k,j) l is the number of paths by which x j can reach x k, with input string of length l that begins with e s, and F b(i,k) =1if and only if input e b can move x k to x i, we can find that the rightmost hand side of (6) exactly shows the number of paths that x j can reach x i with input string e of length l +1which begins with e s The claim is thus proved Since (δm) 1 T K t 1 m = m Ki t, it is clear that M t (q,p) = m (6) Ki t (q,p) The conclusion is obtained imme-

5 214 X Xu et al / J Control Theory Appl (2) diately by the claim we have just proved Remark 3 Reference [17] proposed a so-called inputstate incidence matrix to reduce the size of the matrices in the computation for Boolean networks In some sense, matrix 1 m F in () serves the same purpose, since it is a square matrix, and for any t the size of ( 1 m F ) t will not increase In some sense, transition structure matrix F of automaton A and matrix 1 m F correspond to the logical structure matrix L and the input-state incidence matrix J (Σ) in [17], respectively However, Theorem 3 cannot be obtained immediately from [17], because matrix in our paper is in general not a logical matrix The following result is quite straightforward Corollary 2 With notations used in Theorem 3, the target state x = δn q is reachable from x 0 = δn p with an input string of length t if and only if M t (q,p) > 0 Moreover, we consider the construction of the reachable set Corollary 3 For automaton A with F as its transition structure matrix, the set of states reachable from x 0 = δn p is R(x 0 )={δn k M j (k,p) > 0, 1 k n} j=1 with matrix M j defined in () If τ ( n) is the the length of the longest simple path (that is, a path without repeated state) originating from initial state x 0, then τ R(x 0 )={δn k M j (k,p) > 0, 1 k n} j=1 Proof If a state x is reachable from x 0, there must be a string with its length no more than τ ( n) that can move x 0 to x δn q is reachable from x 0 if and only if there exist at least one string of length s (1 s τ) that moves x 0 s to δn, q namely, M j (q,p) > 0 by Corollary 2 Thus, the j=1 conclusion follows The proof is completed The automaton A is called accessible if every state x X is reachable from x 0 [1] It is clear that A is accessible if and only if R(x 0 )=X Generally, define the reachability matrix M(A) of automaton A as follows: M(A) (i,j) =1if M k (i,j) > 0 and M(A) (i,j) =0otherwise k=1 Remark 4 Boolean algebra can be used in the STP calculation and matrix-based approach for automata [10, 16 17] If elements in F are taken as Boolean values with addition and multiplication between matrices as Boolean operations, Corollaries 1, 2 and 3 still hold after small modifications In fact, it is easy to find that Lemmas 21 and 22 in [10] are consistent with Corollaries 2 and 1, respectively Nevertheless, the lemmas in [10] is basically descriptive, but our results are constructive Before the end of this section, we show an illustrative example for Theorem 3 Example 2 Consider Example 1 again For the DFA case, we obtain M 2 =, M 3 =, M 4 = Take M 4 for example M 4 (,2) =2means that there are two different paths for δ 2 to reach δ with input strings of length 4, which are e = e 2 e 3 e 3 e 4 and e = e 2 e 3 e 1 e 4 (as shown in Fig 1) For the NFA case, we obtain M 2 =, M 3 =, M 4 = The conclusion can be checked similarly Furthermore, the set of states that are reachable from an initial state δ 3 is obtained as R(δ)={δ 3,δ 3,δ 4 } Conclusions We proposed a matrix-based approach for finite automata using semitensor product in this paper We expressed deterministic and nondeterministic finite automata in matrix forms, based on which we studied the reachability problem and obtained necessary and sufficient conditions We believe that the matrix framework provides an effective way in the analysis and computation for finite automata Many interesting related problems such as hierarchical decomposition and task assignment using automata are under investigation References [1] S Eilenberg Automata, Languages, and Machines New York: Academic Press, 1976 [2] C Cassandras, S Lafortune Introduction to Discrete Event Systems New York: Springer-Verlag, 2008 [3] M Lamego Automata control systems IET Control Theory & Applications, 2007, 1(1): [4] W Womham, P Ramadge On the supremal controllable sublanguage of a given language SIAM Journal on Control and Optimization, 1987, 2(3): [] S Abdelwahed, W Wonham Blocking detection in discrete event systems Proceedings of American Control Conference New York: IEEE, 2003: [6] J Lygeros, C Tomlin, S Sastry Controllers for reachability

6 X Xu et al / J Control Theory Appl (2) specifications for hybrid systems Automatica, 1999, 3(3): [7] A Casagrande, A Balluchi, L Benvenuti, et al Improving reachability analysis of hybrid automata for engine control IEEE Conference on Decision and Control New York: IEEE, 200: [8] W Kuich, A Salomaa Semirings, Automata, Languages Berlin: Speringer-Verlag, 1986 [9] S Seshu, R Miller, G Metze Transition matrices of sequential machines IRE Transactions on Circuit Theory, 199, 6(1): 12 [10] G Zhang Automata, Boolean matrices, and ultimate periodicity Information and Computatioin, 1999, 12(1): [11] M Dogruel, U Ozguner Controllability, reachability, stabilizability and state reduction in automata Proceedings of the IEEE International Symposium on Intelligent Control New York: IEEE, 1992: [12] D Cheng, H Qi, Z Li Analysis and Control of Boolean Networks: A Semi-tensor Product Approach London: Springer-Verlag, 2010 [13] D Cheng, Y Zhao, X Xu From Boolean algebra to Boolean calculus Control Theory & Applications, 2011, 28(10): (in Chinese) [14] A Xue, F Wu, Q Lu, et al Power system dynamic security region and its approximations IEEE Transactions on Circuits and Systems I, 2006, 3(12): [1] J Delvenne, V Blondel Complexity of control on finite automata IEEE Transactions on Automatic Control, 2006, 1(6): [16] K Kim Boolean Matrix Theory and Applications New York: Dekker, 1982 [17] Y Zhao, H Qi, D Cheng Input-state incidence matrix of Boolean control networks and its applications Systems & Control Letters, 2010, 9(12): Xiangru XU received his BS degree in Applied Mathematics from Beijing Normal University in 2009 Currently, he is a PhD candidate of Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China His research interests include automata theory and multi-agent systems xuxiangru@amssaccn Yiguang HONG received his BS and MS degrees from Peking University in 1987 and 1990, respectively, and PhD degree from Chinese Academy of Sciences (CAS) in 1993 He is currently a professor in Academy of Mathematics and Systems Science, CAS He is serving as Deputy Editor-in-Chief of Acta Automatica Sinca, and Associate Editors for several journals including IEEE TAC His current research interests include nonlinear control, multi-agent systems, and complex systems yghong@issaccn