Observability of Boolean control networks: A unified ... - CiteSeerX

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Observability of Boolean control networks: A unified approach based on the theories of finite automata and formal languages arXiv:1405.6780v1 [math.OC] 27 May 2014

Kuize Zhang, Lijun Zhang

Abstract—In this paper, we solve a basic problem of whether there are algorithms to determine the observability of Boolean control networks (BCNs). In fact, we give a unified approach to design algorithms to judge whether a given BCN is observable with respect to all different observability. In this work, an algorithm to determine the observability of BCNs is a deterministic mechanical procedure that receives a BCN and after a finite number of processing steps, it returns “Yes”, if the BCN is observable; and returns “No”, otherwise. The observability we mainly investigate are the most general observability that is proposed in this paper, i.e., any two distinct initial states can be distinguished by a designed input sequence, and the observability proposed in [D. Cheng, H. Qi (2009). Controllability and observability of Boolean control networks, Automatica, 45(7), 1659–1667.], [D. Cheng, Y. Zhao (2011). Identification of Boolean control networks, Automatica, 47, 702– 710.] and [E. Fornasini, M. Valcher (2013). Observability, reconstructibility and state observers of Boolean control networks, IEEE Trans. Automat. Control, 58(6), 1390–1401.], respectively. With respect to each of the four observability, we give the following results. First, we give the implication relationships between them, and prove that every two of them are not equivalent. Second, we put forward a concept of weighted pair graphs for BCNs, using which, based on the theories of finite automata and formal languages, we give equivalent test criteria for the observability and design algorithms to determine the observability of BCNs. Lastly, we present methods to determine the initial state, using which we also obtain an upper bound on the smallest length of input/output pair sequences that determine the initial state. Index Terms—Boolean control network, observability, weighted pair graph, finite automaton, formal language, semi-tensor product of matrices

I. I NTRODUCTION The Boolean network, introduced first in [6], and then developed by [7], [8], etc., is a simple and effective model to describe the behavior and relationships of cells, protein, DNA and RNA in a biological system, named genetic regulatory K. Zhang is with College of Automation, Harbin Engineering University, Harbin, 150001, PR China (e-mail: [email protected]) and EXQUISITUS, Centre for E-City, Nanyang Technological University, Singapore 639798, Singapore. L. Zhang is with School of Marine Technology, Northwestern Polytechnical University, Xi’an, 710072, PR China (e-mail: [email protected]) and College of Automation, Harbin Engineering University, Harbin, 150001, PR China. A short version of this paper was submitted to the 33rd Chinese Control Conference, July 28–30, 2014, Nanjing, China. This work is supported by National Natural Science Foundation of China (No. 61174047), Program for New Century Excellent Talents in University of Ministry of Education of China and Basic Research Foundation of Northwestern Polytechnical University (No. JC201230). Manuscript received xxxx xx, xxxx; revised xxxx xx, xxxx.

networks (GRNs) (cf. [4], [5]). Particularly in [4], exogenous perturbation and regulation to biological systems were described as “control”, i.e., the concept of Boolean control networks (BCNs) came up. A BN/BCN is itself simple but reflects the local dynamical interactions of internal nodes (and external nodes). And it was pointed out that “One of the major goals of systems biology is to develop a control theory for complex biological systems” [10]. Hence studying the control problems of BNs/BCNs are of both theoretical and practical importance. The controllability and observability for BCNs are both basic and important control-theoretic problems, which have been paid close attention (cf. [10], [12]– [22], etc.). The concept of controllability for BCNs was first proposed in [10]. The first equivalent test criterion for controllability was given in [12] in the framework of the semi-tensor product (STP) of matrices that was first proposed by Cheng [9]. Moreover, a simple equivalent matrix test criterion for the controllability of BCNs was given in [13], by using which an algorithm to determine the controllability was designed. In [18], a new concept, called controllability constructed path, was proposed by us to derive an equivalent test criterion and to design an algorithm to determine the controllability of BCNs with timevariant delays in states. Compared to the controllability of BCNs, the study for the observability of BCNs seems more challengeable. Due to the nonlinearity of BCNs, several types of observability occurred. To the best of the authors’ knowledge, so far it is still unknown whether there are algorithms to determine most of the observability in spite of some related works. Why do we say “whether there are algorithms ...”? Because there are many problems that cannot be solved by algorithms. We suggest curious readers refer to Appendix A. At present, there are totally four types of observability for BCNs1 . The first was proposed in [12]. In [13], the second was proposed. The second observability has a flaw that it does not characterize a BCN whose any two distinct initial states can be distinguished only by initial outputs. None of the other three observability has this flaw. In this paper, we propose a new observability to make the second observability flawless, that is, any two distinct initial states can be distinguished by an input sequence, and prove that the new observability is the most general one. And we will mainly investigate the new observability instead of 1 In [17], the observability was defined according to a given input sequence. So this observability is essentially for BNs.

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the second one. Until now, it is unknown whether there are algorithms to determine the observability proposed in [12] and [13], because equivalent test criteria have not been obtained. Equivalent test criteria are necessary for designing such algorithms. The third was proposed in [14]. Later we will prove that the observability proposed in [21] is equivalent to the third one. [21] proved that determining the third observability is NPhard, that is, determining this observability is at least as hard as solving any NP problem. An algorithm to determine the third observability was designed in [20]. The fourth, a special case of the third, was proposed in [22], and an algorithm to determine this observability was given. Note that for the definitions of the last two observability, designing input sequences is independent of initial states, which is an essential difference from the first two observability. So compared to the first two observability, the last two are much special. In [21, Remark 6], it was pointed that “This definition (the third one) is different from the one proposed by [15] (the second one). According to their definition, a BCN is observable if for any two initial states a 6= b there exists a control u (that may depend on a, b) for which the corresponding output separates a and b”. In the paragraph just after [22, Definition 3], similar comparison was made. Due to the essential difference between the last two observability and the first two observability, the approaches to designing algorithms to determine the last two observability are not suitable for the first two. Then an issue arises: Are there algorithms to judge whether a given BCN is observable or not with respect to these observability? In this paper, we obtain a breakthrough and will give a positive answer. In fact, we give a unified approach to design algorithms to judge whether a given BCN is observable with respect to any of these observability. We first put forward a concept of weighted pair graphs for BCNs; and then using this concept, based on the theories of finite automata and formal languages, we find equivalent test criteria for the these observability, respectively, and design algorithms to judge whether a BCN is observable with respect to any of the four observability (except for the second), respectively. We will also prove that every two of the four observability are not equivalent via these algorithms. As a remark, we design an algorithm to determine the second observability, since this observability is remarkably different from the other four. In this work, an algorithm to determine the observability of BCNs is a deterministic mechanical procedure that receives a BCN and after a finite number of processing steps, it returns “Yes”, if the BCN is observable; and returns “No”, otherwise. In addition, the implication relationships between the four observability (except for the second) will also be revealed in the present paper. The theories of finite automata and formal languages are among the mathematical foundations of theoretical computer science. Finite automaton theory involves mainly the study of computational problems that can be solved using them. In computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages that can be parsed by machines with limited computational power. The

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details are referred to [2], [3]. In this paper, we first transform BCNs to finite automata, then based on the regular languages recognized by the finite automata, we design algorithms to determine the observability of BCNs by checking the completeness of the finite automata. To this end, the first thing to do is to find a tool to transform a BCN into a finite automaton that determines its observability. The tool we use is a new concept, called weighted pair graph, that is proposed in the present paper. The original idea of pair graph comes from the de Bruijn graph, and was once used to design an algorithm to judge whether a 1-dimensional cellular automaton is reversible or surjective [1]. In this paper, we pay attention to the state pairs that cannot be distinguished by initial outputs. By aggregating these state pairs and the links (inputs) between them, a digraph called weighted pair graph comes up. The results of this paper are in the framework of an intuitive algebraic form based on the STP of matrices. Since such an intuitive algebraic form will help to represent weighted pair graphs and finite automata constructed in the sequel. A comprehensive introduction to the STP of matrices is referred to [11], [15]. The rest of this paper is organized as follows. Section II introduces necessary preliminaries about finite automata, formal languages, STP and BCNs with their algebraic forms. Section III studies the differences and relationships between different observability. Section IV defines weighted pair graphs, and gives a key proposition on deterministic finite automata. In section V, algorithms are designed to determine the observability of BCNs. In Section VI, methods to determine the initial state are proposed. And using the methods, an upper bound on the smallest length of input/output pair sequences that determine the initial state is obtained. The last section is a short conclusion and proposes a challenging problem. II. P RELIMINARIES A. Finite automata and formal languages We use Σ, a nonempty finite set to denote the alphabet2. Elements of Σ are called letters. A word is a finite sequence of letters. The empty word is denoted by ǫ. The set of all words over alphabet Σ is denoted by Σ∗ . For example, {0, 1}∗ = {ǫ, 0, 1, 00, 01, 10, 11, 000, . . .}. | · | denotes the length of word ·. For example, |abc| = 3 over the alphabet {a, b, c}, |ǫ| = 0. A formal language is defined as a subset of Σ∗ . A language over alphabet Σ is called regular, if it is recognized by a deterministic finite automaton (DFA). Next we introduce the concepts of DFAs and regular languages. A DFA is a 5-tuple A = (S, Σ, σ, s0 , F ): • Finite state set S. At all times the internal state is some s ∈ S. • Input alphabet Σ. The automaton only operates on words over the alphabet. 2 A nonempty finite set is an alphabet iff all words consisting of letters of the set has a unique letter sequence. For example, the set {0, 00} is not an alphabet, since word 000 = 0 00 = 00 0.

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•

3

0

The transition partial function describes how the automaton changes its internal state. It is a partial function

1 start

σ :S×Σ →S

• •

from (state, input letter)-pairs to states, that is, σ is a function defined on a subset of S ×Σ. If the automaton is in state s, the present input letter is a, then the automaton changes its internal state to σ(s, a) and moves to the next input letter, if σ is well defined at (s, a); and stop, otherwise. Initial state s0 ∈ S is the internal state of the automaton before any letters have been read. Set F ⊂ S of final states specifies which states are accepting and which are rejecting. If the internal state of the automaton, after reading the whole input, is some state of F then the word is accepted, otherwise rejected.

We call a DFA complete if σ is a function from S × Σ to S. In order to represent regular languages, we introduce an extended transition function σ ∗ : S × Σ∗ → S. σ ∗ is recursively defined as • •

σ ∗ (s, ǫ) = s for all s ∈ S. σ ∗ (s, wa) = σ(σ ∗ (s, w), a) for all s ∈ S, w ∈ Σ∗ and a ∈ Σ, if σ is well defined at (σ ∗ (s, w), a) and σ ∗ is well defined at (s, w).

Particularly, one has that for all s ∈ S, all a ∈ Σ, σ ∗ (s, a) = σ(σ ∗ (s, ǫ), a) = σ(s, a), if σ is well defined at (s, a). Hence we will use σ to denote σ ∗ briefly, since no confusion will occur. Hereinafter, if we write “σ(s, a)”, we mean that σ is well defined at (s, a). Given a DFA A = (S, Σ, σ, s0 , F ). A word w ∈ Σ∗ is called accepted by this DFA, if σ(s0 , w) ∈ F . A language L ⊂ Σ∗ is called recognized by this DFA, if L = {w ∈ Σ∗ |σ(s0 , w) ∈ F }, and is denoted by L(A). In order to visualize and represent a DFA and transform a BCN into a DFA related to its observability, we introduce the transition graph of a DFA A = (S, Σ, σ, s0 , F ). A weighted digraph GA = (V, E, W ) is called the transition graph of the DFA A, if the vertex set V = S, the edge set E ⊂ V × V and the weight function W : E → 2Σ , where 2Σ is the power set of Σ, are defined as follows: For all (si , sj ) ∈ V ×V , directed from si to sj , (si , sj ) ∈ E iff there is a letter a ∈ Σ such that σ(si , a) = sj . If (si , sj ) ∈ E, then W ((si , sj )) equals the set of all letters a ∈ Σ such that σ(si , a) = sj , that is, {a ∈ Σ|σ(si , a) = sj }. In a transition graph of a DFA, usually an input arrow is added to the vertex denoting the initial state, double circles are used to denote the final states, the curly bracket “{}” in the weights of edges are not drawn. See the following example. Example 2.1: The graph in Fig. 1 represents the DFA A = ({s0 , s1 , s2 }, {0, 1}, σ, s0, {s0 , s1 }), where σ(s0 , 0) = s0 ,

σ(s1 , 0) = s0 ,

σ(s2 , 0) = s2 ,

σ(s0 , 1) = s1 ,

σ(s1 , 1) = s2 ,

σ(s2 , 1) = s1 .

It is easy to get that a DFA accepts the empty word ǫ iff its initial state is a final state.

0

s0

1 s1

0 Fig. 1.

s2 1

The transition graph of the DFA A in Example 2.1.

B. The semi-tensor product of matrices Since the framework of STP is used in this paper, some notations about logic and STP are introduced. • Z+ : the set of all positive integers • N: the set of all natural integers • D: the set {0, 1} i • δn : the i-th column of the identity matrix In 1 n • ∆n : the set {δn , . . . , δn } (∆2 := ∆) • δn [i1 , . . . , is ]: the logical matrix [δni1 , . . . , δnis ] (i1 , . . . , is ≤ n) • Ln×s : the set of all n × s logical matrices, i.e., {δn [i1 , . . . , is ]|i1 , . . . , is ∈ {1, 2, . . . , n}} • [1, N ]: the consecutive integer set {1, 2, . . . , N } T • A : the transpose of matrix A Pk i • 1k : i=1 δk • |A|: the cardinality of set A Definition 1: [15] Let A ∈ Rm×n , B ∈ Rp×q , and α = lcm(n, p) be the least common multiple of n and p. The STP of A and B is defined as A ⋉ B = (A ⊗ I αn )(B ⊗ I αp ), where ⊗ denotes the Kronecker product. From this definition, one sees that the conventional product of matrices is a particular case of STP. Since STP keeps almost all properties of the conventional product [15], e.g., the associative law, the distributive law, etc., we usually omit the symbol “⋉” hereinafter. C. Boolean control networks and their algebraic forms In this paper, we mainly investigate the following BCNs with n state nodes, m input nodes and q output nodes:  x1 (t + 1) = f1 (u1 (t), . . . , um (t), x1 (t), . . . , xn (t)),     x2 (t + 1) = f2 (u1 (t), . . . , um (t), x1 (t), . . . , xn (t)), ..  (1)  .    xn (t + 1) = fn (u1 (t), . . . , um (t), x1 (t), . . . , xn (t)), yj (t) = hj (x1 (t), . . . , xn (t)), j = 1, . . . , q, where x1 , . . . , xn , u1 , . . . , um , y1 , . . . , yq ∈ D; t = 0, 1, . . . ; fi : Dn+m → D and hj : Dn → D are Boolean functions, i = 1, . . . , n, j = 1, . . . , q. Identifying 1 ∼ δ21 , 0 ∼ δ22 , using the STP of matrices, (1) can be represented equivalently as the following algebraic form [12] x(t + 1) = Lu(t)x(t), y(t) = Hx(t),

(2)

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Fig. 2.

u0

u1

···

up−1

···

x0

x1

x2

···

xp

···

y0

y1

y2

···

yp

···

4

The input-state-output-time transfer graph of BCN (2).

where x ∈ ∆N , u ∈ ∆M and y ∈ ∆Q denote states, inputs and outputs, respectively; t = 0, 1, . . . ; L ∈ LN ×(N M) ; H ∈ LQ×N ; hereinafter, 2n := N , 2m := M and 2q := Q. A swap matrix W[m,n] is the unique (mn) × (mn) matrix i i such that W[m,n] δm ⋉ δnj = δnj ⋉ δm for all 1 ≤ i ≤ m and 1 ≤ j ≤ n. By the definition, W[m,n] ∈ L(mn)×(mn) , W[m,n] −1 T is invertible and W[m,n] = W[m,n] = W[n,m] . The details on the properties of STP, swap matrices, and how to transform a BCN into its equivalent algebraic form are referred to [15]. III. D IFFERENT OBSERVABILITY

OF

B OOLEAN CONTROL

NETWORKS

A. Preliminary notations The input-state-output-time transfer graph of BCN (2) is as shown in Fig. 2. Now we define the following mappings generated by BCN (2): For all x0 ∈ ∆N and all p ∈ Z+ , 1) Lpx0 : (∆M )p → (∆N )p , u0 . . . up−1 7→ x1 . . . xp , N N LN x0 : (∆M ) → (∆N ) , u0 u1 . . . 7→ x1 x2 . . . .

(3)

2) (HL)px0 : (∆M )p → (∆Q )p , u0 . . . up−1 7→ y1 . . . yp , N N (HL)N x0 : (∆M ) → (∆Q ) , u0 u1 . . . 7→ y1 y2 . . . .

(4) N

Given an input sequence U ∈ (∆M ) , indexed by N, and regarding ∆M as the alphabet, for some i, j ∈ N such that i ≤ j, we use U [i, j] to denote the word ui ui+1 . . . uj , and use U [i] to denote U [i, i]. The analogous notations are defined for state sequences and output sequences. We set that the index of an infinite sequence starts at 0. For example, if U ∈ (∆M )N , then U [0] denotes the left most letter of U . Differently we set that the index of a finite sequence starts at 1. For example, if U ∈ (∆M )p for some p ∈ Z+ , then U [1] denotes the left most letter of U .

Definition 3: BCN (2) is called observable, if for any initial state x0 ∈ ∆N , there exists an input sequence U ∈ (∆M )N such that for all states x0 6= x¯0 ∈ ∆N , Hx0 = H x ¯0 implies 3 N (HL)N ¯0 (U ) . x0 (U ) 6= (HL)x In [12], an equivalent condition for Definition 3 under the assumption of controllability was given. However, there is a BCN that is not controllable but observable (see Eqn. (5)). So there has been no equivalent test criterion for Definition 3 now. On the other hand, it is impossible to design an algorithm to determine Definition 3 directly because it involves infinite sequences, and algorithms can only be executed for finite times. In order to design an algorithm to determine Definition 3, next, we give a new definition equivalent to it by using finite sequences. Definition 4: BCN (2) is called observable, if for any initial state x0 ∈ ∆N , there exists an input sequence U ∈ (∆M )p for some p ∈ Z+ such that for all states x0 6= x ¯0 ∈ ∆N , Hx0 = H x ¯0 implies (HL)px0 (U ) 6= (HL)xp¯0 (U ). Theorem 3.1: Definitions 3 and 4 are equivalent. Proof: Assume that BCN (2) is observable with respect to Definition 3. Arbitrarily given state x0 ∈ ∆N . Then there is an input sequence U ∈ (∆M )N such that for any state x0 6= x ¯0 ∈ ∆N satisfying Hx0 = H x ¯0 , there is px¯0 ∈ N, N such that (HL)N ¯0 ]. Choose ¯0 ] 6= (HL)x ¯0 (U )[0, px x0 (U )[0, px x0 6= x0 , H x ¯0 = Hx0 }. Then for all x0 6= p = max {px¯0 |¯ x ¯0 ∈ ∆N satisfying Hx0 = H x ¯0 , (HL)N x0 (U )[0, p] 6= N (HL)x¯0 (U )[0, p]. Next assume that BCN (2) is observable with respect to Definition 4. Arbitrarily given state x0 ∈ ∆N . Then there exists an input sequence U ∈ (∆M )p for some p ∈ Z+ such that for all states x0 6= x¯0 ∈ ∆N , Hx0 = H x ¯0 implies (HL)px0 (U ) 6= (HL)px¯0 (U ). Define the following new infinite 1 ∞ 1 ∞ ) denotes input sequence U (δM ) ∈ (∆M )N , where (δM 1 the concatenation of infinite copies of δM . Then for all states 1 ∞ x0 6= x ¯0 ∈ ∆N satisfying Hx0 = H x ¯0 , (HL)N x0 (U (δM ) ) 6= 1 ∞ N (HL)x¯0 (U (δM ) ). The second observability was proposed in [13]. Definition 5 ( [13]): A BCN (2) is called observable, if for any distinct states x0 , x ¯0 ∈ ∆N , there is an input sequence U ∈ (∆M )p for some p ∈ Z+ , such that (HL)px0 (U ) 6= (HL)px¯0 (U ). In [13], only sufficient conditions for this observability were given. Now we show that Definition 5 does not characterize a BCN whose any two distinct initial states can be distinguished without using any input sequence. For example, consider the following BCN: x(t + 1) = δ2 [1, 1, 1, 1]u(t)x(t),

B. Different observability of Boolean control networks In this subsection, we study the implication relationships between different observability of BCNs. The first observability of BCNs was proposed in [12]. Definition 2 ( [12]): BCN (2) is called observable, if for any initial state x0 ∈ ∆N , there exists an input sequence such that the initial state can be determined by the output sequence. Or equivalently, Definition 2 can be stated as follows.

y(t) = x(t),

(5)

where t ∈ N, x, u, y ∈ ∆. Obviously BCN (5) is not observable with respect to Definition 5 or controllable by [13, Theorem 3.3], but it is observable with respect to Definition 4 because any initial state can be 3 “Hx = H x N (U )” can be equivalently ¯0 implies (HL)N 0 x0 (U ) 6= (HL)x ¯0 N (U ).” stated as “Hx0 6= H x ¯0 or (HL)N (U ) = 6 (HL) x0 x ¯0

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5

observed by the initial output y0 . These results show that Definition 5 has its own inherent flaw, and Definitions 5 and 4 are not equivalent. In fact, any BCN (2) with H being square and nonsingular is observable with respect to Definitions 4, 6, 9 and 10. In order to fix the disadvantage of Definition 5, we give a more general observability shown in Definition 6. Later, we will prove that Definition 6 is the most general observability, and mainly investigate Definition 6 instead of Definition 5. Definition 6: A BCN (2) is called observable, if for any distinct states x0 , x¯0 ∈ ∆N , there is an input sequence U ∈ (∆M )p for some p ∈ Z+ , such that Hx0 = H x ¯0 implies (HL)px0 (U ) 6= (HL)px¯0 (U ). Using the same procedure of proving Theorem 3.1, one easily sees that Definition 6 is equivalent to the following Definition 7 defined by using infinite sequences, which will be used to prove Theorems 3.3 and 6.2. Definition 7: A BCN (2) is called observable, if for any distinct states x x0 , x ¯0 ∈ ∆N , there is an input sequence U ∈ (∆M )N such that Hx0 = H x ¯0 implies (HL)N x0 (U ) 6= N (HL)x¯0 (U ). Next we give the implication relationships between Definitions 4 and 6. Theorem 3.2: If a BCN (2) is observable with respect to Definition 4, then it is also observable with respect to Definition 6. The converse is not true. Proof: The first part can be got directly using Definitions 4 and 6. To prove the second part, consider the following BCN: x(t + 1) = δ4 [1, 1, 2, 1, 2, 4, 1, 1]x(t)u(t), y(t) = δ2 [1, 2, 2, 2]x(t),

(6)

where t ∈ N, x ∈ ∆4 , y, u ∈ ∆. First, we prove that BCN (6) is not observable with respect to Definition 4. Denote M := δ4[1, 1, 2, 1, 2, 4, 1, 1]W[2,4]12 = 2 102

δ2 [1, 2, 2, 1, 1, 1, 4, 1]12 = 00 10 10 00 . Then for all k ∈ Z+ , 0 010 ∗ ∗ ∗ ∗ ∗∗∗∗ k M = 0 0 0 0 . By [13, Theorem 3.3], BCN (6) is not 0 0 ∗ 0

controllable. So we cannot use the test criteria proposed in [12] to check whether BCN (6) is observable. Next we prove that BCN (6) is not observable by showing that for state δ42 , there is no input sequence such that the corresponding output sequence can determine it. Here we only need consider the states δ42 , δ43 , δ44 , since Hδ41 6= Hδ42 . Given an input sequence U ∈ (∆)N . If U (0) = δ21 , then L1δ2 (δ21 ) = L1δ3 (δ21 ) = δ42 . Then for each such 4 4 U , one has (HL)N (U ) = (HL)N (U ). Else if U (0) = δ22 , δ42 δ43 1 2 1 2 1 then Lδ2 (δ2 ) = Lδ4 (δ2 ) = δ4 . Then for each such U , one has 4 4 (HL)N (U ) = (HL)N (U ). Based on the above analysis, by δ42 δ44 Definition 3 and Theorem 3.1, BCN (6) is not observable with respect to Definition 4. Second, we prove that BCN (6) is observable with respect to Definition 6. We only need check the state pairs (δ42 , δ43 ), (δ42 , δ44 ) and 3 4 (δ4 , δ4 ). For (δ42 , δ43 ), (HL)1δ2 (δ22 ) = δ21 , (HL)1δ3 (δ22 ) = δ22 . 4

4

For (δ42 , δ44 ), (HL)1δ2 (δ21 ) = δ22 , (HL)1δ4 (δ21 ) = δ21 . 4 4 For (δ43 , δ44 ), (HL)1δ3 (δ21 ) = δ22 , (HL)1δ4 (δ21 ) = δ21 . 4 4 Thus, BCN (6) is observable with respect to Definition 6. The third observability was proposed in [14]. Definition 8 ( [14]): A BCN (2) is observable, if there is an input sequence U ∈ (∆M )N such that the initial state x0 can be determined by the corresponding output sequence (HL)N x0 (U ). Definition 8 means that all initial states can be determined by a common input sequence, which is essentially different from Definitions 4 and 6. Similar to the equivalence of Definitions 3 and 4, one easily sees that Definition 8 is equivalent to the following Definition 9 that was proposed in [21]. Definition 9 ( [21]): A BCN (2) is called observable, if there exists an input sequence U ∈ (∆M )p for some p ∈ Z+ , such that for any distinct states x0 , x ¯0 ∈ ∆N , Hx0 = H x ¯0 implies (HL)px0 (U ) 6= (HL)px¯0 (U ). In [21], it was proved that determining this observability is NP-hard; however, it was not clarified whether there is an algorithm to determine this observability. An algorithm to determine this observability was designed in [20] by enumerating all possible input sequences of a common finite length. Obviously for Definitions 4 and 6, one has to consider infinitely many input sequences. So the approach of [20] cannot be used to deal with these two observability. Obviously Definition 9 implies Definitions 4 and 6. The fourth observability was proposed in [22]. Definition 10 ( [22]): A BCN (2) is called observable, if for any distinct states x0 , x ¯0 ∈ ∆N , for any input sequence U ∈ (∆M )N , Hx0 = H x ¯0 implies (HL)N x0 (U ) 6= N (HL)x¯0 (U ). An equivalent condition for this observability was given in [22] mainly by checking every pair of distinct periodic state-input trajectories of the same minimal period and the same length. Although there are infinitely many such pairs, it is enough to check all such pairs of length no greater than N M , the cardinality of the set of state-input pairs. That is, an algorithm to determine this observability was also given. This observability implies Definitions 4, 6 and 9. For the similar reasons related to Definition 9, the approach of [22] cannot be used to deal with Definitions 4 and 6 either. Next we further give the mutual implication relationships between Definitions 6 and 10 and between Definitions 9 and 10. Theorem 3.3: If A BCN (2) is observable with respect to Definition 10, then it is also observable with respect to Definition 6. The converse is not true. Proof: The first part can be got directly by the equivalence of Definitions 6 and 7. To prove the second part, again consider BCN (6). We have proved that BCN (6) is observable with respect to Definition 6 in Theorem 3.2. The fact that BCN (6) is not observable with respect to Definition 10 can be got by Hδ42 = Hδ44 = δ2 [1, 2, 2, 2]δ42 = δ22 and (HL)N (δ 2 (δ 1 )∞ ) = δ42 2 2 (HL)N (δ 2 (δ 1 )∞ ). δ4 2 2 4

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where t ∈ N, N = 2n , x ∈ ∆N , Q = 2q , y ∈ ∆Q , L ∈ LN ×N , H ∈ LQ×N . In [21], a BN (8) is called observable, if there is p ∈ N such that for any distinct states x1 , x2 ∈ ∆N , the corresponding output sequences y(0, x1 ), y(1, x1 ), . . . , y(p, x1 ) and y(0, x2 ), y(1, x2 ), . . . , y(p, x2 ) are different, where y(i, xj ) denotes the output at time step i when the initial state is xj , i = 0, . . . , p, j = 1, 2. Use the matrices L and H in (8) to generate the following special BCN:

+ Def. 9

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Fig. 3. The implication relationships graph between Definitions 4, 6, 9 and 10, where “+” means “implies” and “−” means “does not imply”.

Theorem 3.4: If A BCN (2) is observable with respect to Definition 10, then it is also observable with respect to Definition 9. The converse is not true. Proof: Assume that a BCN (2) is observable with respect to Definition 10, then arbitrarily given U ∈ (∆M )N , for j j i i any distinct δN , δN , HδN = HδN implies (HL)N i (U ) 6= δN N (HL)δj (U ). Since N < +∞, there is p ∈ Z+ such that for N

j j i i any distinct δN , δN , HδN = HδN implies (HL)pδi (U [0, p − N p 1]) 6= (HL)δj (U [0, p−1]). That is, the BCN (2) is observable N with respect to Definition 9. To prove the second part, consider the following BCN:

x(t + 1) = δ4 [1, 1, 3, 3, 1, 2, 3, 2]x(t)u(t), y(t) = δ2 [1, 1, 2, 2]x(t),

(7)

where t ∈ N, x ∈ ∆4 , y, u ∈ ∆. Choose U = δ21 ∈ (∆)1 . Hδ41 = Hδ42 = δ21 , (HL)1δ1 (U ) = 4 1 δ2 , (HL)1δ2 (U ) = δ22 . Hδ43 = Hδ44 = δ22 , (HL)1δ3 (U ) = δ21 , 4 4 (HL)1δ4 (U ) = δ22 . Then BCN (7) is observable with respect 4 to Definition 9. Consider any U ∈ (∆)N such that U (0) = δ22 . Then N Lδ3 (U ) = LN (U ) and (HL)N (U ) = (HL)N (U ). That is, δ44 δ43 δ44 4 BCN (7) is not observable with respect to Definition 10. Up to now we can give the implication relationships graph between Definitions 4, 6, 9 and 10 as shown in Fig. 3. The relationships between Definitions 4 and 9 and between Definitions 4 and 10 will be proved later. They are difficult to be proved directly. Based on the above analysis, it is known that only Definitions 9 and 10 can be determined by algorithms. In the sequel, we will prove that all of Definitions 4, 6, 9 and 10 can be determined by algorithms. As a remark, we will show that Definition 5 can also be determined by algorithms. Remark 3.1: [21] proved that determining whether a BCN (2) is observable with respect to Definitions 4 and 9 are both NP-hard. Then is determining whether a BCN (2) is observable with respect to Definitions 6 or 10 also NP-hard? The answer is “Yes”. In fact, one can use the method proposed in [21] to prove these conclusions. The outline is stated as follows. Consider the following BN with output nodes: x(t + 1) = Lx(t), y(t) = H(t),

(8)

x(t + 1) = [L, L]u(t)x(t), y(t) = H(t),

(9)

where t ∈ N, N = 2n , x ∈ ∆N , Q = 2q , y ∈ ∆Q , u ∈ ∆, L ∈ LN ×N , H ∈ LQ×N . It is obvious that a BN (8) is observable, iff the corresponding BCN (9) is observable with respect to Definition 6, iff the corresponding BCN (9) is observable with respect to Definition 10. These equivalences are polynomial-time reductions from the observability of BN (8) to Definitions 6 and 10. [21, Theorem 11] states that determining whether a BN (8) is observable is NP-hard. So by the reductions, determining whether a BCN (2) is observable with respect to Definitions 6 and 10 are both NP-hard. IV. W EIGHTED PAIR

GRAPHS AND A KEY PROPOSITION ON

DETERMINISTIC FINITE AUTOMATA

In this section, we define a weighted digraph for BCN (2), named weighted pair graph, and then give a key proposition on DFAs. Based on the weighted pair graph, in the next section we will construct a DFA with respect to each observability; and then by using the proposition and the DFA obtained, we will design algorithms to determine each of the different observability. A. From Boolean control networks to weighted pair graphs Now we define the weighted pair graph for BCN (2). Definition 11: A weighted digraph G = (V, E, W), where V denotes the vertex set, E ⊂ V × V denotes the edge set, and W : E → 2∆M denotes the weight function, where 2∆M is the power set of the input set of BCN (2), is called a weighted pair graph of BCN (2), if V = {(x, x′ )|x, x′ ∈ ∆N , Hx = Hx′ }; for all ((x1 , x′1 ), (x2 , x′2 )) ∈ V × V, directed from (x1 , x′1 ) to (x2 , x′2 ), ((x1 , x′1 ), (x2 , x′2 )) ∈ E iff there exists u1 ∈ ∆M such that Lu1 x1 = x2 and Lu1 x′1 = x′2 ; for all edges e = ((x1 , x′1 ), (x2 , x′2 )) ∈ E, W(e) = {u1 ∈ ∆M |Lu1 x1 = x2 , Lu1 x′1 = x′2 }. Note that in Definition 11, (x, x′ ) is ordered. That is, (x, x′ ) and (x′ , x) are different vertices if x 6= x′ . On the other hand, one has that if (x, x′ ) ∈ V, then (x′ , x) ∈ V. For any vertex (x, x′ ) ∈ ∆N × ∆N , we call vertex (x, x′ ) the mirror image of vertex (x′ , x), and denote (x, x′ )R = (x′ , x). Similar to the transition graph of a DFA, we do not draw the curly bracket “{}” in the weights of edges either. Hereinafter, we call each vertex (x, x) ∈ ∆N × ∆N a diagonal vertex.

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Fig. 4. The weighted pair graph of BCN (6), where the number ij in each circle denotes the state pair (δ4i , δ4j ), the weight k1 , k2 , . . . beside each edge denotes the weight {δ2k1 , δ2k2 , . . . } of the edge.

From Definition 11, one sees that the weighted pair graph aggregates all state pairs that cannot be distinguished by initial outputs. As a matter of fact, to test whether a BCN is observable, the most important thing to do is to test whether these states can be distinguished by designing input sequences. The algorithms that will be designed are based on this idea. For a graph with vertex set V, given a subset V of V, the subgraph generated by V is the graph with vertex set V , and edge set consisting of each edge starting at some vertex in V and ending at some vertex in V . If there is an edge from vertex v1 ∈ V to vertex v2 ∈ V, then v1 is called the father of v2 and similarly, v2 is called the son of v1 . The weighted pair graph of BCN (6) is as shown in Fig. 4.

B. A key proposition on deterministic finite automata Now we give the key proposition on finite automata. Proposition 4.1: Given a DFA A = (S, Σ, σ, s0 , F ). Assume that F = S and for each s0 6= s ∈ S, there is a path starting at s0 and passing through s in the transition graph of DFA A. Then L(A) = Σ∗ iff A is complete. Proof: “if”: If A is complete and F = S, then ǫ ∈ L(A) and for any nonempty word w ∈ Σ∗ , σ(s0 , w) ∈ F , thus, w ∈ L(A). That is, L(A) = Σ∗ . “only if”: Assume that F = S and A is not complete. Choose an s ∈ S such that σ is not well defined at (s, a) for some a ∈ Σ. There is a path in the transition graph of A starting at s0 and passing through s (if s = s0 , set the path empty). Choose any word w ∈ Σ∗ such that σ(s0 , w) = s (if s = s0 , then w = ǫ), then word wa ∈ / L(A), since A is deterministic. That is, L(A) 6= Σ∗ .

In this subsection, the observability of BCN (2) always means the one with respect to Definition 4 if we do not write “with respect to . . . ”. According to Definition 4, a BCN (2) is not observable iff i there is a state δN in a non-diagonal vertex of its weighted pair graph G = (V, E, W) satisfying that for all p ∈ Z+ , all j j i U ∈ (∆M )p , there is a state δN such that j 6= i, (δN , δN )∈V p p and (HL)δi (U ) = (HL)δj (U ). N

N

N

N

i Now fix δN , we design an algorithm to construct a DFA for a BCN (2) according to its weighted pair graph. The new i , and accepts exactly all finite input DFA is denoted by AδN i i sequences that do not determine δN . The states of DFA AδN are subsets of V, and the alphabet is ∆M . Algorithm 5.1: 1) Set ∆M as the alphabet of the DFA. Set the subset of V consisting of all the non-diagonal i vertices of V that contain δN as the initial state of the k l k DFA. That is, the set {(δN , δN )|k, l ∈ [1, N ], HδN = l HδN , k 6= l, k or l = i} := s0 is chosen as the initial state of the DFA. j 2) For each letter δM , j = 1, . . . , M , find the value for the j transition partial function of the DFA at (s0 , δM ). The specific procedure is as follows: Fix j ∈ [1, M ]. For each vertex v ∈ s0 , find the son, denoted by vs , of v in the weighted pair graph such that j the weight of the edge from v to vs contains δM . Collect all found vs ’s to form a set sj . If sj 6= ∅, add sj to the state set of the DFA and set sj as the value of the j transition partial function at (s0 , δM ); else, the transition partial function of the DFA is set to be not well defined j at (s0 , δM ). 3) For each new found state s of the DFA in last step, for j , j = 1, . . . , M , find the value for the each letter δM j transition partial function of the DFA at (s, δM ) according to Step 2. 4) Repeat Step 3 step by step until there is no new state of the DFA occurring. Since V is a finite set, so is its power set, this repetition will stop. 5) Set all the states of the DFA obtained as final states. Take Eqn. (6) for example. Choose the state δ42 . Then the DFA Aδ42 generated by Algorithm 5.1 is as shown in Fig. 5. Now we give the key theorem for this observability. Theorem 5.2: A BCN (2) is not observable with respect to i Definition 4 iff there is a state δN in a non-diagonal vertex of i its weighted pair graph such that the DFA AδN generated by Algorithm 5.1 recognizes language (∆M )∗ . Proof: “only if”: Assume that BCN (2) is not observable, i then there is a state δN such that for all p ∈ Z+ , all U ∈ j j p i (∆M ) , there is a state δN such that i 6= j, HδN = HδN , p p and (HL)δi (U ) = (HL)δj (U ). According to Algorithm 5.1, j i i . Denote (δN , δN ) := v0 is in the initial state of DFA AδN the weighted pair graph of BCN (2) by G = (V, E, W). Then ik jk there exist vertices (δN , δN ) := vk ∈ V such that U [k] ∈

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Fig. 5. The DFA Aδ2 with respect to Eqn. (6) generated by Algorithm 5.1, 4

where the number ij in each circle denotes the state pair (δ4i , δ4j ), the weight k beside each edge denotes the input δ2k .

W((vk−1 , vk )), k = 1, . . . , p. That is, for all p ∈ Z+ , every i . It is obvious that U ∈ (∆M )p is accepted by the DFA AδN ∗ i i ǫ ∈ L(AδN ). Then L(AδN ) = (∆M ) . i accepts exactly all finite input “if”: Note that the DFA AδN i ∗ i ) = (∆M ) sequences that do not determine δN . Then L(AδN p implies that for all p ∈ Z+ , all U ∈ (∆M ) , there is a j j i state δN such that i 6= j, HδN = HδN , and (HL)pδi (U ) = N p (HL)δj (U ). That is, BCN (2) is not observable. N Based on Proposition 4.1, Theorem 5.2 and Algorithm 5.1, we design an algorithm to judge whether a given BCN is observable. Instance: A BCN (2) Problem: Is the BCN observable with respect to Definition 4? Algorithm 5.3: 1) Draw the weighted pair graph of a given BCN (2). 2) Find all states of BCN (2) in the non-diagonal vertices of the weighted pair graph. Denote these states by ik i1 i2 δN , δN , . . . , δN . 3) Set j = 1. i 4) Check δNj . Use Algorithm 5.1 to generate DFA Aδij . If N Aδij is complete, return “No”, stop; else, j ← j + 1, if N j ≤ k, repeat this step; else, return “Yes”, stop. Example 5.4: Check whether BCN (6) is observable. According to Algorithm 5.3, we should check δ42 , δ43 , δ44 one by one. First we check δ42 . According to Algorithm 5.1, we calculate the DFA Aδ42 . We have got the transition graph of this DFA as shown in Fig. 5. By Proposition 4.1, this DFA recognizes language (∆)∗ . Then by Theorem 5.2, Eqn. (6) is not observable. B. An algorithm to judge whether a BCN is observable with respect to Definition 6 In this subsection, the observability of BCN (2) always means the one with respect to Definition 6 if we do not write “with respect to . . . ”. According to Definition 6, a BCN (2) is not observable j i iff there is a non-diagonal vertex (δN , δN ) in the weighted pair graph G such that for all p ∈ Z+ , all U ∈ (∆M )p , (HL)pδi (U ) = (HL)pδj (U ). N

N

Fig. 6. The DFA of each non-diagonal vertex of the weighted pair graph of Eqn. (6) generated by Algorithm 5.5, where the number ij in each circle denotes the state pair (δ4i , δ4j ), the weight k beside each edge denotes the input δ2k .

j i Now fix the non-diagonal vertex (δN , δN ), we design an algorithm to construct a DFA for a BCN (2) according to its weighted pair graph. The new DFA is denoted by A(δi ,δj ) , N N and accepts exactly all finite input sequences that do not j i distinguish δN and δN . Algorithm 5.5: 1) Set ∆M as the alphabet of the DFA. j i Set vertex (δN , δN ) as the initial state of the DFA. 2) Find each vertex at which there is a path starting at j i (δN , δN ) and ending. Keep the subgraph generated by j i these found vertices and (δN , δN ), and remove all vertices and edges outside of the subgraph. 3) Set each remained vertex as a final state of the DFA. Again take Eqn. (6) for example. The DFA of each nondiagonal vertex of the weighted pair graph generated by Algorithm 5.5 is shown in Fig. 6. j i , δN ) Besides, Proposition 5.6 holds, since the vertices (δN j i and (δN , δN ) are mirror images of each other, they can be merged. Hence to judge whether a BCN (2) is observable, j i we only need check each non-diagonal vertex (δN , δN ) of its weighted pair graph such that i < j. j i Proposition 5.6: For any non-diagonal vertex (δN , δN ) of the weighted pair graph of BCN (2), it holds that L(A(δi ,δj ) ) = L(A(δj ,δi ) ). N N N N Now we give the key theorem for the observability. Theorem 5.7: A BCN (2) is not observable with respect to j i , δN ) in the Definition 6 iff there is a non-diagonal vertex (δN weighted pair graph of BCN (2) such that the DFA A(δi ,δj ) N N generated by Algorithm 5.5 recognizes language (∆M )∗ . Proof: “only if”: Assume that BCN (2) is not observable, j i then there is a non-diagonal vertex (δN , δN ) in the weighted pair graph of BCN (2) such that for all p ∈ Z+ , all U ∈ (∆M )p , (HL)pδi (U ) = (HL)pδj (U ). Then for all p ∈ Z+ , N N every U ∈ (∆M )p is in the language recognized by the DFA A(δi ,δj ) generated by Algorithm 5.5. It is obvious that ǫ ∈ N N L(A(δi ,δj ) ). Then L(A(δi ,δj ) ) = (∆M )∗ . N N N N “if”: This part can be proved directly by using Definition 6. We omit it. Based on Propositions 4.1 and 5.6, Theorem 5.7 and Algo-

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rithm 5.5, we design an algorithm to judge whether a given BCN is observable. Instance: A BCN (2) Problem: Is the BCN observable with respect to Definition 6? Algorithm 5.8: 1) Draw the weighted pair graph of a given BCN (2). j i 2) Find all non-diagonal vertices (δN , δN ) of the weighted pair graph such that i < j. Denote these vertices by i1 j1 i2 j2 ik jk (δN , δN ), (δN , δN ), . . . , (δN , δN ). 3) Set l = 1. jl il 4) Check (δN , δN ). Use Algorithm 5.5 to generate DFA A(δil ,δjl ) . If A(δil ,δjl ) is complete, return “No”, stop; N N N N else, l ← l + 1, if l ≤ k, repeat this step; else, return “Yes”, stop. Example 5.9: Check whether BCN (6) is observable. According to Algorithm 5.8, one should check (δ42 , δ43 ), (δ42 , δ44 ), (δ43 , δ44 ) one by one. From Fig. 6, one sees that L(A(δ42 ,δ43 ) ) = L(A(δ43 ,δ42 ) ), L(A(δ42 ,δ44 ) ) = L(A(δ44 ,δ42 ) ), and L(A(δ43 ,δ44 ) ) = L(A(δ44 ,δ43 ) ). Besides, one sees that δ22 ∈ / L(A(δ42 ,δ43 ) ), δ21 ∈ / L(A(δ42 ,δ44 ) ), 1 2 and δ2 , δ2 ∈ / L(A(δ43 ,δ44 ) ). Then by Theorem 5.7, BCN (6) is observable. Remark 5.1: Can one use the method proposed in this subsection to determine whether a BCN (2) is observable with resepct to Definition 5? The answer is “No”. However, one can reach this target after modifying the weighted pair graph only a little. With respect to Definition 5, A BCN (2) is not observable j i iff there exist distinct states δN , δN ∈ ∆N such that for all p p ∈ Z+ , all U ∈ (∆M )p , (HL)δi (U ) = (HL)pδj (U ). Hence N N to determine this observability, one should testify every nondiagonal state pair of ∆N × ∆N , which is the only difference between determining Definitions 6 and 5. And because of the unique difference, if the vertex set of weighted pair graph is replaced by ∆N × ∆N , then one can use the conclusions in this subsection to determine Definition 5. The outline is stated as follows. 1) Define a new weighted pair graph for BCN (2): G = (V, E, W), where V = {(x, x′ )|x, x′ ∈ ∆N }; for all ((x1 , x′1 ), (x2 , x′2 )) ∈ V × V, directed from (x1 , x′1 ) to (x2 , x′2 ), ((x1 , x′1 ), (x2 , x′2 )) ∈ E iff there exists u1 ∈ ∆M such that Lu1 x1 = x2 and Lu1 x′1 = x′2 ; for all edges e = ((x1 , x′1 ), (x2 , x′2 )) ∈ E, W(e) = {u1 ∈ ∆M |Lu1 x1 = x2 , Lu1 x′1 = x′2 }. 2) Design an algorithm to determine whether a BCN (2) is observable with respect to Definition 5: a) Draw the new weighted pair graph of a given BCN (2). j i b) Find all non-diagonal vertices (δN , δN ) of the new weighted pair graph such that i < j. Denote these i1 j1 i2 j2 ik jk vertices by (δN , δN ), (δN , δN ), . . . , (δN , δN ). c) Set l = 1. jl il d) Check (δN , δN ). Keep the subgraph of the new jl il weighted pair graph generated by (δN , δN ) and all j j i i vertices (δN , δN ) such that HδN = HδN . Remove the vertices and edges outside of subgraph. Replace the “weighted pair graph” that Algorithm 5.5 is based

9

on by the subgraph, then use the new Algorithm 5.5 to generate a new DFA, also denoted by A(δil ,δjl ) . N N Replace the “weighted pair graph” and “Algorithm 5.5” in Theorem 5.7 by the new weighted pair graph and the new Algorithm 5.5, respectively. Then the new Theorem 5.7 holds for Definition 5. If A(δil ,δjl ) is N N complete, return “No”, stop; else, l ← l + 1, if l ≤ k, repeat this step; else, return “Yes”, stop. Hereinafter, “weighted pair graph” always means Definition 11. C. An algorithm to judge whether a BCN is observable with respect to Definition 9 In this subsection, the observability of BCN (2) always means the one with respect to Definition 9 if we do not write “with respect to . . . ”. According to Definition 9, to judge whether a BCN (2) is observable, we need check the set of all non-diagonal vertices, denoted by Vn , of its weighted pair graph (V, E, W). Now we design an algorithm to construct a DFA for a BCN (2) according to its weighted pair graph. The new DFA is denoted by AVn , and accepts exactly any finite input sequence by which not all non-diagonal state pairs can be distinguished. The states of the DFA AVn are subsets of V. Algorithm 5.10: 1) Set ∆M as the alphabet of the DFA. Set the set of all non-diagonal vertices of V, denoted by Vn , as the initial state of the DFA. j 2) For each letter δM , j = 1, . . . , M , find the value for the j transition partial function of the DFA at (Vn , δM ). The specific procedure is as follows: Fix j ∈ [1, M ]. For each vertex v ∈ Vn , find the son, denoted by vs , of v in the weighted pair graph such that j the weight of the edge from v to vs contains δM . Collect all found vs ’s to form a set sj . If sj 6= ∅, add sj to the state set of the DFA and set sj as the value of the j transition partial function at (Vn , δM ); else, the transition partial function of the DFA is set to be not well defined j at (Vn , δM ). 3) For each new found state s of the DFA in last step, for j each letter δM , j = 1, . . . , M , find the value for the j transition partial function of the DFA at (s, δM ) according to Step 2. 4) Repeat Step 3 step by step until there is no new state of the DFA occurring. Since V is a finite set, so is its power set, this repetition will stop. 5) Set all the states of the DFA obtained as final states. Then with respect to Definition 9, the following theorem holds. Theorem 5.11: A BCN (2) is not observable with respect to Definition 9 iff the DFA AVn generated by Algorithm 5.10 recognizes language (∆M )∗ . Proof: A BCN (2) is not observable, iff each finite input sequence cannot distinguish all state pairs of Vn , that is, L(AVn ) = (∆M )∗ . Based on Proposition 4.1, Theorem 5.11 and Algorithm 5.10, we design an algorithm to judge whether a BCN (2) is observable.

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Fig. 9. The DFA AVn with respect to Eqn. (10) generated by Algorithm 5.10, where the number ij in each circle denotes the state pair (δ4i , δ4j ), the weight k of each edge denotes the input δ2k .

1, 2

12

1 2

22

1

12, 21 34, 43

2 start

34, 43

2

1 33

Fig. 10. The DFAs Aδ1 and Aδ3 with respect to Eqn. (10) generated by 4 4 Algorithm 5.1, where the number ij in each circle denotes the state pair (δ4i , δ4j ), the weight k beside each edge denotes the input δ2k .

2 43

Fig. 8. The weighted pair graph of BCN (10), where the number ij in each circle denotes the state pair (δ4i , δ4j ), the weight k1 , k2 , . . . beside each edge denotes the weight {δ2k1 , δ2k2 , . . . } of the edge.

Instance: A BCN (2) Problem: Is the BCN observable with respect to Definition

The DFA A{(δ41 ,δ42 ),(δ42 ,δ41 ),(δ43 ,δ44 ),(δ44 ,δ43 )} generated by Algorithm 5.10 (see Fig. 9) is complete. Then by Theorem 5.11, BCN (6) is not observable with respect to Definition 9. The DFAs Aδ41 and Aδ43 generated by Algorithm 5.1 (see / L(Aδ41 ) and δ21 ∈ / L(Aδ43 ). Then Fig. 10) satisfy δ22 ∈ by Theorem 5.2, BCN (10) is observable with respect to Definition 4.

9? Algorithm 5.12: 1) Draw the weighted pair graph of a given BCN (2). 2) Use Algorithm 5.10 to obtain DFA AVn . If the DFA is complete, return “No”, stop; else, return “Yes”, stop. Example 5.13: Check whether BCN (6) is observable. According to Algorithm 5.12, we should check whether the DFA A{(δ42 ,δ43 ),(δ43 ,δ42 ),(δ42 ,δ44 )(δ44 ,δ42 ),(δ43 ,δ44 ),(δ44 ,δ43 )} is complete. From Fig. 7, one sees that this DFA is complete. Then by Theorem 5.11, BCN (6) is not observable. Besides, without using Algorithm 5.12, BCN (6) is not observable with respect to Definition 4 (see Example 5.4) implies it is not observable with respect to Definition 9 either by the following Theorem 5.14, which indicates that Definition 9 implies Definition 4 . Theorem 5.14: If A BCN (2) is observable with respect to Definition 9, then it is also observable with respect to Definition 4. The converse is not true. Proof: The first part holds naturally. To prove the second part, consider the following BCN: x(t + 1) = δ4 [1, 1, 1, 3, 1, 2, 3, 2]x(t)u(t), y(t) = δ2 [1, 1, 2, 2]x(t),

(10)

where t ∈ N, x ∈ ∆4 , y, u ∈ ∆. The weighted pair graph of BCN (10) is as shown in Fig. 8.

Until now, we have finished the implication relationships between Definitions 4 and 9 as shown in Fig. 3. D. An algorithm to judge whether a BCN is observable with respect to Definition 10 In this subsection, the observability of BCN (2) always means the one with respect to Definition 10 if we do not write “with respect to . . . ”. According to Definition 10, a BCN (2) is not observable j i iff there are two distinct states δN , δN and an input sequence j N i U ∈ (∆M ) such that HδN = HδN and (HL)N i (U ) = δN N (HL)δj (U ). Then the following theorem holds. N Theorem 5.15: A BCN (2) is not observable with respect j i , δN ) in to Definition 10 iff there is a non-diagonal vertex (δN the weighted pair graph of BCN (2) such that the transition graph of the DFA A(δi ,δj ) generated by Algorithm 5.5 has a N N cycle. Proof: Since the transition graph has a finite number of vertices, the graph has a cycle iff there is an input sequence N U ∈ (∆M )N such that (HL)N i (U ) = (HL) j (U ). δN δN Theorem 5.15 itself is an algorithm to determine the observability of BCN (2). Example 5.16: Check whether BCN (6) is observable. By Theorem 5.15 and Fig. 6, BCN (6) is not observable.

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Next we present Theorem 5.17 to reveal the implication relationships between Definitions 4 and 10 so as to finish Fig. 3. Theorem 5.17: If a BCN (2) is observable with respect to Definition 10, then it is also observable with respect to Definition 4. The converse is not true. Proof: The first part holds by the definitions. To prove the second part, consider Eqn. (10) again. We have proved that Eqn. (10) is observable with respect to Definition 4 in Theorem 5.14. Note that in Fig. 10, if one removes 21 and 43, then the DFAs are the corresponding ones generated by Algorithm 5.5. By Theorem 5.15 and Fig. 10, it is not observable with respect to Definition 10. VI. D ETERMINING THE INITIAL STATE The final target of investigating observability is using input/output sequences to determine the initial state. In this section, we attempt to obtain an upper bound on the smallest length of input/output pair sequences that determine the initial state, and then investigate how to determine the initial state. We investigate four observability: Definitions 4, 6, 9 and 10. We will not consider Definition 5 because of its inherent flaw. For Definition 9, in which case designing input sequences is independent of initial states, using our approach, one can directly find the shortest input/output sequences that determine the initial state. However, for Definitions 4, 6 and 10, searching for the shortest input/output sequences that determine the initial state may lead to remarkable increase of computational complexity due to the nondeterminism of algorithms. Note that “nondeterminism” is the noun of the “N” in “NP-hard”. The specific reasons are shown later. Different from obtaining algorithms to determine observability, the results of this section are based on a new concept of reduced weighted pair graph of BCN (2). A reduced weighted pair graph is derived directly from the weighted pair graph, and will help to obtain a tight upper bound on the smallest length of input/output pair sequences that determine the initial state. First, we give the definition for reduced weighted pair graph. Definition 12: For a BCN (2) and its weighted pair graph G = (V, E, W). A digraph G r = (V r , E r , W r ) is called the reduced weighted pair graph of BCN (2), if the vertex set V r , the edge set E r and the weight function W r are defined j i i as follows: V r = {(δN , δN )|i, j ∈ [1, N ], i ≤ j, HδN = j r r HδN } ⊂ V; for all v1 , v2 ∈ V , (v1 , v2 ) ∈ E iff either (v1 , v2 ) ∈ E or (v1 , v2R ) ∈ E; for all (v1 , v2 ) ∈ E r , W r ((v1 , v2 )) = W((v1 , v2 )) ∪ W((v1 , v2R )) 6= ∅, where if W is not well defined at (v1 , v2 ) (or (v1 , v2R )), we set that W((v1 , v2 )) = ∅ (or W((v1 , v2R )) = ∅). Second, we give some properties on the subgraph generated by all diagonal vertices of the reduced weighted pair graph, which we let alone call diagonal subgraph. Proposition 6.1: Consider a BCN (2). The diagonal subgraph, denoted by Gd , of its reduced weighted pair graph has the following properties: (i) Gd has a cycle. (ii) For each vertex of Gd , the union of the weights of all edges that start at this vertex equal ∆M .

11

(iii) For each vertex of Gd , there is a path in Gd that starts at this vertex and goes into a cycle of Gd . (iv) Each son of each vertex of Gd is a vertex of Gd . Proof: Since for every vertex of Gd , there is an edge that starts at it, and there are a finite number of vertices, (i) and (iii) hold naturally. (ii) and (iv) can be got directly from Definition 12. Third, we characterize non-diagonal vertices of the reduced weighted pair graph of a BCN (2). j i For i = 1, . . . , Q, denote ri = |{j|HδN = δQ }|, that is, i the number of columns of H equaling δQ . Then ri ≥ 0 and PQ i=1 ri = N . The number of non-diagonal vertices of the reduced weighted pair graph is Q

Nnd =

1X ri (ri − 1). 2 i=1

It is easy to get that 2Nnd is the number of non-diagonal vertices of the weighted pair graph, and 2Nnd ≤ N (N − 1). Note that 2Nnd = N (N − 1) iff all columns of H are the same, in which case the BCN (2) cannot be observable with respect to any observability. So if a BCN (2) is observable, then 2Nnd ≤ (N − 1)(N − 2). 2Nnd = (N − 1)(N − 2) corresponds to the case that H has N − 1 same columns and 1 column that is different from the others. Particularly, if r1 = · · · = rQ = N/Q, then 2Nnd = N (N/Q − 1). For a BCN (2) and its reduced weighted pair graph G r = (V r , E r , W r ), given a vertex v ∈ V r , we call the cardinality of unions of weights of all edges that start at v the outdegree of v, and denote the outdegree of v by deg(v). That is, deg(v) = |∪w∈V r ,(v,w)∈E r W r ((v, w))|. From (ii) of Proposition 6.1, the outdegree of any diagonal vertex equals M . The outdegree of each vertex is no greater than M . A. For Definition 6 We first investigate Definition 6. The results on this observability will help to investigate other observability. Now we give the key theorem that plays a central role of obtaining an upper bound on the smallest length of input/output pair sequences that determine the initial state. In fact, this theorem provides a new equivalent definition for Definition 6. Theorem 6.2: Consider a BCN (2). Denote the number of non-diagonal vertices of its reduced weighted pair graph by Nnd . The following two items are equivalent. (i) There exist distinct states x, x′ ∈ ∆N such that Hx = Hx′ , and for any input sequence U ∈ (∆M )Nnd , it holds nd nd that (HL)N (U ) = (HL)N x x′ (U ). (ii) This BCN is not observable. Proof: (ii) ⇒ (i): If (ii) holds, then from the equivalence of Definitions 6 and 7, there exist distinct states x, x′ ∈ ∆N such that Hx = Hx′ , and for any input sequence U ∈ (∆M )N , it holds that N (HL)N x (U ) = (HL)x′ (U ). That is, (i) holds naturally. (i) ⇒ (ii): Use Algorithm 5.5 to generate DFA A(x,x′ ) . From Proposition 5.6, L(A(x,x′ ) ) = L(A(x′ ,x) ). Denote the reduced

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weighted pair graph of BCN (2) by G r = (V r , E r , W r ). Without loss of generality, we assume that (x, x′ ) ∈ V r . Note that the instance of Algorithm 5.5 is the weighted pair graph of BCN (2). Now we replace the instance by G r , and use Algorithm 5.5 to generate DFA Ar(x,x′ ) . From the definitions of weighted pair graph and reduced weighted pair graph, one has that L(A(x,x′ ) ) = L(Ar(x,x′ ) ). Since for any input sequence U ∈ (∆M )Nnd , Nnd Nnd (HL)x (U ) = (HL)x′ (U ), one has that any input sequence U ∈ (∆M )Nnd belongs to L(Ar(x,x′ ) ). Then any input sequence of length less than Nnd also belongs to L(Ar(x,x′ ) ). We claim that each non-diagonal vertex of the transition graph of DFA Ar(x,x′ ) has outdegree M . Suppose that there is a non-diagonal vertex v of the transition graph of Ar(x,x′ ) having outdegree less than M . Then the transition partial function of Ar(x,x′ ) is not well i defined at (v, δM ) for some i ∈ [1, M ]. If v = (x, x′ ), i r then δM ∈ / L(A(x,x′ ) ), which is a contradiction. Next assume that v 6= (x, x′ ). Since there are at most Nnd non-diagonal vertices in the transition graph, there is a path of length 0 < p ≤ Nnd that starts at (x, x′ ) and passes through v. Then there is an input sequence U1 ∈ (∆M )p−1 such p−1 R that (Lp−1 x (U1 ), Lx′ (U1 ))[p − 1] = v or v . So the input i r sequence U1 δM ∈ / L(A(x,x′ ) ), which is also a contradiction. From (ii) of Proposition 6.1, each diagonal vertex of the transition graph of Ar(x,x′ ) has outdegree M . Then DFA Ar(x,x′ ) is complete. That is, L(Ar(x,x′ ) ) = L(A(x,x′ ) ) = (∆M )∗ . Based on the above analysis, by Theorem 5.7, this BCN is not observable. From Theorem 6.2, if a BCN (2) is observable with respect to Definition 6, then any two distinct states can be distinguished by an input sequence of length no greater than Nnd . Denote the initial state of an observable BCN (2) by x0 ∈ ∆N . Note that now we do not know the value of x0 . But we know y0 = Hx0 . Find the set {x ∈ ∆N |Hx = Hx0 } := Sx0 . If |Sx0 | > 1, choose any two distinct elements of Sx0 , say x1 and x2 , use the DFA A(x1 ,x2 ) generated by Algorithm 5.5 to generate one of the shortest input sequences, denoted by |U| U , of length no greater than Nnd such that (HL)x1 (U ) 6= |U| |U| |U| (HL)x2 (U ); for all i = 1, 2, if (HL)xi (U ) 6= (HL)x0 (U ), remove xi from Sx0 . Repeat this step time and again until |Sx0 | = 1, then the unique element of Sx0 is x0 . Hence an upper bound on the smallest length of input/output pair sequences that determine the initial state x0 is (Rx0 − 1)Nnd , where Rx0 = |{x ∈ ∆N |Hx = Hx0 }|. B. For Definition 10 Assume that a BCN (2) is observable with respect to Definition 10. Then by Theorem 5.15, the transition graph of every DFA A(δi ,δj ) generated by Algorithm 5.5 has no cycle, N

N

j i where (δN , δN ) is a non-diagonal vertex of the weighted pair graph. Furthermore, from (i), (iii) and (iv) of Proposition 6.1, each state of each DFA A(δi ,δj ) is a non-diagonal vertex of N N the weighted pair graph. Hence there is a non-diagonal vertex v of the transition graph of each DFA A(δi ,δj ) such that N

N

12

deg(v) = 0. This is because, if every vertex of the transition graph has a positive outdegree, then A(δi ,δj ) must have a N N cycle. That is, any input sequence (i.e., word) U ∈ (∆M )Nnd does not belong to any L(A(δi ,δj ) ). N N Note that there are at most Nnd non-diagonal vertices in the reduced weighted pair graph. We then have that a BCN (2) is observable with respect to Definition 10 iff for any distinct states x, x′ ∈ ∆N such that Hx = Hx′ , for any nd input sequence U ∈ (∆M )Nnd , it holds that (HL)N (U ) 6= x nd (HL)N (U ). ′ x Denote the initial state of an observable BCN (2) by x0 ∈ ∆N . Note that now we do not know the value of x0 . Find the set {x ∈ ∆N |Hx = Hx0 } := Sx0 . Arbitrarily given an input sequence U of length Nnd . Then for all x ∈ Sx0 , x = x0 |U| |U| iff (HL)x (U ) = (HL)x0 (U ). One can use this property to determine x0 . That is, Nnd is an upper bound on the smallest length of input/output pair sequences that determine the initial state. Furthermore, when judging whether a given BCN is observable, that is, judging whether the transition graph of each DFA A(δi ,δj ) generated by Algorithm 5.5 has no cycle, N N we can obtain the length of the shortest input/output pair sequences that determine the initial state. Remark 6.1: The upper bound on the smallest length of input/output pair sequences that determine the initial state given in [22, Remark 6] is (N + 1)(N − 2)/2. We have shown that Nnd ≤ (N − 1)(N − 2)/2 < (N + 1)(N − 2)/2, and in the best case, Nnd = N (N/Q − 1)/2. Our upper bound is tighter. Particularly when the number of output nodes is almost the same as that of state nodes and the number of different columns of H are almost the same, our upper bound is much tighter. C. For Definition 4 Assume that a BCN (2) is observable with respect to Definition 4, then by Fig. 3, it is also observable with respect to Definition 6. Then one can use the approach to determine the initial state with respect to Definition 6 to determine the initial state. That is, (Rx0 − 1)Nnd is also an upper bound on the smallest length of input/output pair sequences that determine the initial state x0 , where Rx0 = |{x ∈ ∆N |Hx = Hx0 }|. One can also use the algorithms to determine the observability defined in Definition 4 to determine the initial state. However, this approach is usually not more efficient than the former one. Next we introduce briefly the new approach. i From (ii) of Proposition 6.1, one sees that in each AδN generated by Algorithm 5.1, there is a state s that contains a non-diagonal vertex of the weighted pair graph such that the j i is not well defined at (s, δ transition partial function of AδN M) for some j ∈ [1, M ]. Then one can find a finite input sequence j p i ), where U ∈ (∆M ) U δM that does not belong to L(AδN for j i some p ∈ N. Then U δM separates δN and all other states. Denote the initial state of an observable BCN (2) by x0 ∈ ∆N . Note that now we do not know the value of x0 . But we know y0 = Hx0 . For any i ∈ [1, N ] such that i i , use the DFA AδN generated by Algorithm Hx0 = HδN 5.1 to generate one of the shortest finite input sequences, |U | |U | denoted by Ui , such that (HL)δi i (Ui ) 6= (HL)δj i (Ui ) for N

N

JOURNAL OF LATEX CLASS FILES, VOL. X, NO. X, XXXX XXXX j i i all j satisfying j 6= i and HδN = HδN . Then x0 = δN iff |Ui | |Ui | (HL)δi (Ui ) = (HL)x0 (Ui ). One can use this property to N determine x0 .

13

1, 2

1 2

11

2

22

33

12

55

21

1

14

2

24

35

42

53

1

D. For Definition 9

2

Assume that a BCN (2) is observable with respect to Definition 9, then by Fig. 3, it is also observable with respect to Definition 6. Then similar to Definition 4, one can also use the approach to determine the initial state with respect to Definition 6 to determine the initial state. That is, (Rx0 − 1)Nnd is also an upper bound on the smallest length of input/output pair sequences that determine the initial state x0 , where Rx0 = |{x ∈ ∆N |Hx = Hx0 }|. Next we show how to use algorithms to determine the observability defined in Definition 9 to determine the initial state. Usually, using this approach is more efficient than using the former approach. For example, in Example 6.3, the upper bound on the smallest length of input/output pair sequences that determine the initial state is no greater than Nnd . Denote the initial state of an observable BCN (2) by x0 ∈ ∆N . Note that now we do not know the value of x0 . Find the set {x ∈ ∆N |Hx = Hx0 } := Sx0 . Use the DFA AVn generated by Algorithm 5.10 to generate one of the shortest finite input sequences, denoted by U , such |U| |U| that (HL)x1 (U ) 6= (HL)x2 (U ) for all distinct states x1 , x2 satisfying Hx1 = Hx2 . Then for all x ∈ Sx0 , x = x0 iff |U| |U| (HL)x (U ) = (HL)x0 (U ). One can use this property to determine x0 . By using this approach, one can directly find all shortest input/output pair sequences that determine the initial state. Remark 6.2: The upper bound on the smallest length of input/output pair sequences that determine the initial state given in [20, Theorem 2] is s∗ Nnd , where s∗ is the minimal length of input sequences that drive any initial state to all its possible reachable states. Particularly, if a BCN (2) is controllable, then for any two states, there is an input sequence of length no greater than s∗ driving one state to the other. s∗ is formally defined as follows: For k = 1, 2, . . . , let Γk = ∗

j1 {LδM

jk · · · LδM |j1 , . . . , jk

∈ [1, M ]}.

Let s be the smallest positive integer such that Γs+1 ⊂ ∪sk=1 Γk . Although [20] gave an upper bound on the smallest length of input/output pair sequences that determine the initial state, it did not give how to find the shortest input/output pair sequences that determine the initial state. Usually, s∗ Nnd is much larger than the smallest length (see Example 6.3). Remark 6.3: From the results obtained in this section, one can use every one of the four observability to determine the initial state. The difference lies in that for Definitions 9 and 10, there is a common input sequence that separates any two distinct initial states. While for the other two observability, designing input sequences is dependent of the initial state. So the computational complexity of using Definitions 9 and 10 to determine the initial state may be lower than that of using the other two.

44

1 1, 2

1

41

2

Fig. 11. The weighted pair graph of BCN (11), where the number ij in each circle denotes the state pair (δ5i , δ5j ), the weight k1 , k2 , . . . beside each edge denotes the weight {δ2k1 , δ2k2 , . . . } of the edge. 14 41

14 41

1

1 start

12 21 14 41

2

start

2

12 21 24 42

2

24 42

start

14 41 24 42

2

24 42

24 42

Fig. 12. The DFAs Aδ1 , Aδ2 and Aδ4 with respect to Eqn. (11) generated 5 5 5 by Algorithm 5.1, where the number ij in each circle denotes the state pair j i (δ5 , δ5 ), the weight k beside each edge denotes the input δ2k .

On the other hand, if the final target is determining the initial state, using the observability defined in Definition 6 proposed in this paper is enough. Definition 6 is the most general known observability now. Then an interesting problem can be proposed: “Is Definition 6 necessary as for determining the initial state?” Or say, “Is there a more general observability that can be used to determine the initial state?” E. An illustrated example Now we give an illustrated example. Example 6.3 ( [22]): Consider the following logical dynamical network: x(t + 1) = δ5 [1, 4, 3, 5, 4, 2, 3, 3, 4, 4]u(t)x(t), = δ5 [1, 2, 4, 3, 3, 3, 5, 4, 4, 4]x(t)u(t),

(11)

y(t) = δ2 [1, 1, 2, 1, 2]x(t), where t ∈ N, x ∈ ∆5 , y, u ∈ ∆. Note that although 5 is not a power of 2, the analysis is not affected by this fact. The weighted pair graph of BCN (11) is shown in Fig. 11. From Fig. 11, one sees that for any non-diagonal vertex v, the transition graph of the DFA Av generated by Algorithm 5.5 has no cycle. By Theorem 5.15, BCN (11) is observable with respect to Definition 10. Then it is also observable with respect to Definitions 4, 6 and 9 by Fig. 3. Now we assume that the initial state of BCN (11) is δ54 and we do not know what the initial state is. Then we illustrate how to determine the initial state. Denote the initial state by x0 ∈ ∆5 , we know that Hx0 = δ21 . Denote Sx0 = {δ51 , δ52 , δ54 }. Then x0 ∈ Sx0 . For Definition 4: The transition graphs of DFAs Aδ51 , Aδ52 and Aδ54 generated by Algorithm 5.1 is as shown in Fig. 12. From Fig. 12, for δ51 , choose U1 = δ21 δ21 ∈ / L(Aδ51 ) to ensure that (HL)2δ1 (U1 ) 6= (HL)2δ2 (U1 ) and 5

5

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start

12

1

14

2

24

start

14

14

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24

Fig. 13. The DFAs A(δ1 ,δ2 ) and A(δ1 ,δ4 ) with respect to Eqn. (11) 5 5 5 5 generated by Algorithm 5.5, where the number ij in each circle denotes the j i state pair (δ5 , δ5 ), the weight k beside each edge denotes the input δ2k . 14 41

1

start

12, 21, 14, 41 24, 42, 35, 53

2 2 24 42

Fig. 14. The DFA AVn with respect to Eqn. (11) generated by Algorithm 5.10, where the number ij in each circle denotes the state pair (δ5i , δ5j ), the weight k beside each edge denotes the input δ2k .

(HL)2δ1 (U1 ) 6= (HL)2δ4 (U1 ). One has that (HL)2δ1 (U1 ) = 5 5 5 δ21 δ21 , (HL)2x0 (U1 ) = δ22 δ21 , so x0 6= δ51 . From Fig. 12, for δ52 , choose U2 = δ22 ∈ / L(Aδ52 ) to ensure that (HL)1δ2 (U2 ) 6= (HL)1δ1 (U2 ) and (HL)1δ2 (U2 ) 6= 5 5 5 (HL)1δ4 (U2 ). One has that (HL)1δ2 (U2 ) = δ22 , (HL)1x0 (U2 ) = 5 5 δ21 , so x0 6= δ52 . / L(Aδ54 ) to From Fig. 12, for δ54 , choose U4 = δ21 ∈ ensure that (HL)1δ4 (U4 ) 6= (HL)1δ1 (U4 ) and (HL)1δ4 (U4 ) 6= 5 5 5 (HL)1δ2 (U4 ). One has that (HL)1δ4 (U4 ) = δ22 , (HL)1x0 (U4 ) = 5 5 δ22 , so x0 = δ54 . For Definition 6: The transition graphs of DFAs A(δ51 ,δ52 ) and A(δ51 ,δ54 ) generated by Algorithm 5.5 is as shown in Fig. 13. First consider δ51 and δ52 . From A(δ51 ,δ52 ) of Fig. 13, choose U1 = δ22 ∈ / L(A(δ51 ,δ52 ) ) to ensure that (HL)1δ1 (U1 ) 6= 5 1 (HL)δ2 (U1 ). One has that (HL)1δ1 (U1 ) = δ21 , (HL)1δ2 (U1 ) = 5 5 5 δ22 , (HL)1x0 (U1 ) = δ21 , so x0 6= δ52 . Remove δ52 from Sx0 , then Sx0 becomes {δ51 , δ54 }. Second consider δ51 and δ54 . From A(δ51 ,δ54 ) of Fig. 13, choose U2 = δ21 ∈ / L(A(δ51 ,δ54 ) ) to ensure that (HL)1δ1 (U2 ) 6= 5 1 (HL)δ4 (U2 ). One has that (HL)1δ1 (U2 ) = δ21 , (HL)1δ4 (U2 ) = 5 5 5 δ22 , (HL)1x0 (U2 ) = δ22 , so x0 6= δ51 . Remove δ51 from Sx0 , then Sx0 becomes {δ54 }. Hence x0 = δ54 . For Definition 9: Denote Vn = {(δ51 , δ52 ), (δ52 , δ51 ), (δ51 , δ54 ), (δ54 , δ51 ), (δ52 , δ54 ), 4 2 (δ5 , δ5 ), (δ53 , δ55 ), (δ55 , δ53 )}. Then the transition graph of DFA AVn generated by Algorithm 5.10 is as shown in Fig. 14. / L(AVn ) to ensure From Fig. 14, choose U = δ21 δ21 ∈ that (HL)2δ1 (U ), (HL)2δ2 (U ) and (HL)2δ4 (U ) are pair-wise 5 5 5 distinct, and (HL)2δ3 (U ) and (HL)2δ5 (U ) are also distinct. 5 5 One has that (HL)2δ1 (U ) = δ21 δ21 , (HL)2δ2 (U ) = δ21 δ22 , 5 5 (HL)2δ4 (U ) = δ22 δ21 and (HL)2x0 (U ) = δ22 δ21 . Then x0 = δ54 . 5 Note that Nnd = (3∗2+2∗1)/2 = 4. Hence (Rδ54 −1)Nnd = (3 − 1) ∗ 4 = 8. So using our approach, to determine δ54 , the

upper bound on the smallest length of input sequences is 8. However in fact, using our approach, one directly get that 2 is the length of the shortest input sequences by using which one can determine the initial states. On the other hand, according to [20, Theorem 2], one has that s∗ = 5, so according to their approach, the upper bound on the smallest length of onput sequences is s∗ Nnd = 5 ∗ 4 = 20. Besides, their approach cannot give the length of the shortest input sequences by using which one can determine the initial states. For Definition 10: Since Nnd = 4, we choose an input sequence U = δ21 δ21 δ21 δ21 of length 4. One has that (HL)4δ1 (U ) = δ21 δ21 δ21 δ21 , (HL)4δ2 (U ) = 5 5 1 2 1 2 δ2 δ2 δ2 δ2 , (HL)4δ4 (U ) = δ22 δ21 δ22 δ21 and (HL)4x0 (U ) = 5 δ22 δ21 δ22 δ21 . Then x0 = δ54 . According to [22, Remark 6], the upper bound on the smallest length of input sequences to determine the initial state is (N + 1)(N − 2)/2 = (5 + 1) ∗ (5 − 2)/2 = 9. VII. C ONCLUDING

REMARKS

In this paper, how to determine all the observability of BCNs that are presented in the existing literatures have been solved by using a unified approach proposed in the present paper based on the theories of finite automata and formal languages. Based on the unified approach, for each observability, an equivalent test criterion has been given, an algorithm has been designed to determine the observability, an upper bound on the smallest length of input/output pair sequences that determine the initial state and a method to determine the initial state have also been given. In this paper, we proved that there are algorithms to determine the observability of BCNs. However, from Remark 3.1, one sees that determining each observability is NPhard. And the algorithms to determine the observability of BCNs designed in this paper are all in exponential times. Then whether one can reduce their computational complexity effectively is a challenging and urgent problem to be solved. The challenging problem can be stated as “Is determining the observability of BCNs an NP problem?” As for the computational complexity, a related in-depth work was proved in [25]: The determination of the existence of fixed points for a Boolean network is NP-complete, that is, both NP and NP-hard. A PPENDIX A D ECIDABILITY THEORY In this section, we briefly introduce decidability theory, that is, the theory on whether a decision problem can be solved by algorithms. In the field of computer science, a decision problem is formulated as “Does an instance have the property P ?”. For example, “Is a time-variant BCN controllable?” studied in [19] is a decision problem. Given a decision problem, the set of all instances that have the property P is denoted by A, then a deterministic mechanical procedure A is called a semialgorithm, if it receives an instance x and after a finite number

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of processing steps, it returns “Yes”, if x ∈ A. Note that if x∈ / A, it may run forever and never halts. A decision problem is called semi-decidable, if it possesses a semi-algorithm. In [19], we gave an equivalent condition for the controllability of time-variant BCNs, which is stated as “a time-variant BCN is controllable iff an assumption holds for some positive integer”. Then “a time-variant BCN is not controllable iff for any positive integer this assumption does not hold”. Then using the equivalent condition for the controllability of time-variant BCNs, one can design a semi-algorithm. If the semi-algorithm receives a time-variant BCN, then it returns “Yes” after a finite number of processing steps, if the BCN is controllable; and never halts if the BCN is not controllable. Given a semidecidable decision problem, then it possesses a semi-algorithm A. The semi-algorithm A is called an algorithm if it receives an instance x and after a finite number of processing steps, it returns “No”, if x ∈ / A. A decision problem is called decidable, if it possesses an algorithm; and called undecidable otherwise. For example, [13] proved that the decision problem “Is a BCN controllable?” is decidable. However, it is still unknown whether the decision problem “Is a time-variant BCN controllable?” is decidable or not. Since the former semialgorithm for the controllability of time-variant BCNs is not an algorithm. There are at most countably infinitely many decidable decision problems, while there are uncountably infinitely many undecidable decision problems [1, Section 3.1]. For example, the decision problems “Turing machine halting problem: Does a given Turing machine eventually halt when it is started on an empty tape?” [1], “Is a Turing machine with outputs observable?” [23] and “Is a ≥ 2-dimensional cellular automaton reversible or surjective?” [24] are all undecidable. In this paper, we essentially prove that the decision problem “Is a BCN is observable?” is decidable with respect to any observability. ACKNOWLEDGMENT The authors thank Prof. Jarkko Kari, Dr.s Charalampos Zinoviadis, Ville Salo and Ilkka T¨orm¨a (University of Turku, Finland) for fruitful discussions and thank the anonymous reviewers and the associate editor for their valuable and constructive comments to improve the readability and the quality of this manuscript. R EFERENCES [1] J. Kari. A Lecture Note on Cellular Automata. http://users.utu.fi/jkari/ca/, 2013. [2] J. Kari. A Lecture Note on Automata and Formal Languages. http://users.utu.fi/jkari/automata/, 2013. [3] P. Linz. An Introduction to Formal Languages and Automata. Sudbury, Mass. : Jones and Bartlett Publishers, 2013. [4] T. Ideker, T. Galitski, L. Hood (2001). A new approach to decoding life: systems biology, Annu. Rev. Genomics Hum. Genet., 2, 343–372. [5] H. Kitano (2002). Systems Biology: A brief overview, Sceince, 259, 1662–1664. [6] S. A. Kauffman (1969). Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theoretical Biology, 22, 437–467. [7] T. Akutsu, S. Miyano, S. Kuhara (2000). Inferring qualitative relations in genetic networks and metabolic pathways, Bioinformatics, 16, 727–734. [8] R. Albert, A-L Barabasi (2000). Dynamics of complex systems: scaling laws or the period of Boolean networks, Phys. Rev. Lett., 84, 5660–5663.

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