arXiv:1407.2104v1 [math.OC] 14 Jun 2014 [email protected] h,21;L u,21;Fn,Yo u,2013; Cui, & Yao, Feng, 2012; Sun, & Li Yang, 2014; (Li, Chu, BCNs of have& kinds results different to these extended of further some be Moreover, Zhao, 2011). 2011; Cheng, Cheng, & 2013b; Li Valcher, Valcher, & & Fornasini Fornasini 2012; 2013a; 2013; Margaliot, Even, & & Laschov Margaliot, optimal 2009; Laschov, Qi, and & (Cheng decoupling control disturbance observabil- controllability, stabilization, as of ity, be such expression can BCNs theory algebraic to control and generalized in linear Qi, problems the Cheng, classical many by on BCNs, written Based book (2011). the the Li to Qi, of refer & framework we theory Cheng control BCNs, 2009; complete Qi, a & For collab- 2010a). his Cheng and 2009; Cheng (Cheng, by orators developed of was method BCNs product and semi-tensor BNs a Cheng, years, 2007; recent Ng, In 2009;). & Ching, sci- Hayashida, theory control (Akutsu, been and entists have biologists called by they attention usually and great are paid (BCNs), networks outputs BNs control and 2006). Boolean Thieffry, inputs & additional Chaouiya, to with Othmer, Naldi, & (1969) (Albert Faure, Kauffman regulation 2003; by cell analyze proposed and first func- describe are logical and They variables tions. logical by dynamical described discrete of systems kind a are (BNs) networks Boolean Introduction 1 pro semi-tensor the and BCNs r of with expression decomposition linear of the problem the investigates paper This Abstract neFudto NS)o hn ne rns11271194, 10701042. Grants and under 61074115 China 61273115, of (NNSF) Foundation ence ⋆ e words: Key of effectiveness computat the the validate reduce to to addressed co proposed are graphical is examples sufficient some method and effective necessary an a Third, Second, obtained. are rpitsbitdt Automatica to submitted Preprint mi addresses: Email Sci- Natural National by part in supported is research The ola oto ewrs eopsto ihrsett ou to respect with Decomposition networks, control Boolean colo ahmtclSine,NnigNra Universit Normal Nanjing Sciences, Mathematical of School [email protected] eopsto ihrsett outputs to respect with Decomposition Jadn Zhu). (Jiandong o ola oto networks. control Boolean for Yne Zou), (Yunlei uliZu inogZhu Jiandong Zou, Yunlei sett upt o ola oto ewrs(Cs.Firs (BCNs). networks control Boolean for outputs to espect h rpsdmethod. proposed the ut oeagbaceuvln odtosfrtedecompo the for conditions equivalent algebraic some duct, dto o h eopsto ihrsett upt sgiv is outputs to respect with decomposition the for ndition oa udni h elzto ftedcmoiin Final decomposition. the of realization the in burden ional norrcn ae o h 21) ehv proposed have we (2014), Zhu & the Zou paper recent our In decom- those achieving are for positions. model assumptions system the regularity on imposed Fur- additional analysis. some theoretical in thermore, advantage great displays it tt pc ehdi eyhayfrlarge for heavy very is method space state h adnlo h tt pc fan of 2 space is state the of the of cardinal to the space respect state with The functions 2010b). anal- Qi, state-space & an the (Cheng with method presented ysis are BCNs decomposition Li of the Kalman Cheng, form the form, by and normal form discussed controllable mat- normal the been observable a where has (2010), As topic Qi BCNs. this and fact, for of decomposition ter system exists there similar whether a ascertaining is topic system. interesting a of An components clear controllable makes tra- and observable which the the theory, in system control issue linear important ditional an is decomposition System 2012). Wang, & Li inadacntutv loih o h decomposition the for algorithm condi- constructive graphical a and a largest tion Moreover, the on subspace. condition be uncontrollable regularity can decomposition the & system without Zou the realized of that contribution in lies main (2014) A Zhu any spaces. use state not of did concepts we the But on assumption subspace. regularity uncontrollable the largest Cheng, under by (2010) Qi proposed and form Li normal controllable the just is n 2 eopsto ihrsett inputs to respect with decomposition n dmninlBNi enda h e fallogical all of set the as defined is BCN -dimensional eeal paig h opttoa udnof burden computational the speaking, generally , pt,Sm-esrpout Observability. product, Semi-tensor tputs, ,Nnig 103 RChina PR 210023, Nanjing, y, ⋆ n oia aibe.Since variables. logical n dmninlBCN -dimensional o Cs which BCNs, for 5Ags 2021 August 25 n although ,with t, sition en. ly, with respect to inputs are proposed. Then, a natural z1(t +1)= fˆ1(z1(t), ··· ,zs(t),u1(t), ··· ,um(t)), idea is reconsidering the observable normal form and the . Kalman decomposition form for BCNs. . zs(t +1)= fˆs(z1(t), ··· ,zs(t),u1(t), ··· ,um(t)),

In this paper, we consider the decomposition with re- zs+1(t +1)= fˆs+1(z1(t), ··· ,zn(t),u1(t), ··· ,um(t)), spect to outputs in the framework of the linear algebraic . representation of BCNs. First, some necessary and suf- . (3) ficient algebraic conditions for the decomposition with z (t +1)= fˆ (z (t), ··· ,z (t),u (t), ··· ,u (t)), respect to outputs are proposed. Next, an equivalent n n 1 n 1 m ˆ graphical condition for the decomposition with respect y1(t)= h1(z1(t), ··· ,zs(t)), to outputs is derived. As the graphical condition is sat- . isfied, a constructive coordinate transformation to real- . ˆ ize the decomposition is obtained. Then, a method is yp(t)= hp(z1(t), ··· ,zs(t)). proposed to reduce the computational burden in realiz- ing the decomposition with respect to outputs. Finally, BCN (3) is called a decomposition with respect to outputs some examples are analyzed with our method. of order n − s. A decomposition with respect to outputs of the maximum order is called the maximum decompo- We organize this paper as follows. In section 2, some pre- sition with respect to outputs. We say that BCN (1) is liminaries and problem statement are provided. In sec- undecomposable with respect to outputs if the order of the tion 3, some algebraic conditions for the decomposition maximum decomposition with respect to outputs is 0. with respect to outputs are presented. In section 4, a graphical condition is obtained. In Section 5, a compu- Denote the real number field by R and the set of all the tation method is given. Finally, Section 6 gives a brief m × n real matrices by Rm×n. Let Col(A) be the set of conclusion. all the columns of matrix A and denote the ith column of i i A by Coli(A). Set ∆k = {δk|i =1, 2, ··· , k}, where δk = 2 Preliminaries and Problem Statement Coli(Ik) with Ik the k×k . For simplicity, 1 2 we denote ∆ := ∆2 = {δ2,δ2}. A matrix L ∈ Rm×n is called a logical matrix if Col(L) ⊂ △m. Obviously, the Let D = {True = 1, False = 0}. Consider a BCN de- T T scribed by the logical equations logical matrix L satisfies 1mL = 1n , where 1n denote the n−dimensional column vector whose entries are all equal x1(t +1)= f1(x1(t), ··· , xn(t),u1(t), ··· ,um(t)), to1.Thesetofallthe m×r logical matrices is denoted by . Lm×r. For simplicity, we denote the logical matrix L = . i1 i2 ir [δm,δm, ··· ,δm] by δm[i1,i2, ··· ,ir]. Set A ∈ Rm×n, xn(t +1)= fn(x1(t), ··· , xn(t),u1(t), ··· ,um(t)), B ∈ Rp×q, and α = lcm(n,p) be the least common y1(t)= h1(x1(t), ··· , xn(t)), (1) multiple of n and p. The left semi-tensor product of A . and B is defined as (Cheng, Qi, & Li, 2011): .

yp(t)= hp(x1(t), ··· , xn(t)), ⋉ α α A B = (A ⊗ I n )(B ⊗ I p ), where the state variables xi, the output variables yi and n+m where ⊗ is the Kronecker product. Obviously, the left the controls ui take values in D, fi : D → D and n semi-tensor product is a generalization of the traditional hi : D → D are logical functions. matrix product. So A ⋉ B can be written as AB. To ex- press the logical equations with linear algebraic method, Consider the logical mapping G : Dn → Dn defined by 1 elements in D are identified with vectors True ∼ δ2 and 2 False ∼ δ2 (Cheng, Qi, & Li, 2011). z1 = g1(x1, x2, ··· , xn), z2 = g2(x1, x2, ··· , xn), Proposition 1. (Cheng & Qi, 2010a) Let xi and ui ⋉n ⋉p . take values in ∆ and denote x = i=1xi, y = i=1yi, . (2) ⋉m u = i=1ui. Then the BCN (1) can be expressed in the zn = gn(x1, x2, ··· , xn). algebraic form x(t +1)= Lu(t)x(t), y(t)= Hx(t), (4) If G : Dn → Dn is a bijection, it is called a logical coordinate transformation. where L ∈ L2n×2n+m and H ∈ L2p×2n .

We say that BCN (1) is decomposable with respect to Let z = T x be the algebraic form of logical coordinate outputs of order n − s, if there exists a logical coordinate transformation (2), where T is a . [1] ⋉s [2] ⋉n transformation (2), such that (1) becomes Set z = i=1zi and z = i=s+1zi. Then the decom-

2 position form (3) can be rewritten in the algebraic form n−s ≥ n−s˜, i.e. s ≤ s˜. On the other hand, from the defi- nition of largest unobservable subspace, it follows that [1] [1] z (t +1) = G1u(t)z (t), zs+1, ··· ,zn ∈ Oc = F(˜zs˜+1, z˜s˜+2, ··· , z˜n). (7) [2] z (t +1) = G2u(t)z(t), (5) Thus, (7) implies that n − s ≤ n − s˜, namely s ≥ s˜. y(t) = Mz[1](t), Therefore, we have proved that s =s ˜, which implies that (6) is a maximum decomposition with respect to outputs of system (4). Conversely, bys ˜ = s, (7) and where G ∈L s× s+m , G ∈L n−s× n+m and M ∈L p× s . 1 2 2 2 2 2 2 2 Theorem 13 of Cheng, Li, & Qi (2010), we obtain that {zs+1, ··· ,zn} is a regular basis of Oc, that is, (5) is an For the BCN (1) with algebraic form (4), the problem observable normal form. ✷ of decomposition with respect to outputs is to find a co- ordinate T such that (4) has the Remark 1. Proposition 3 implies that the decomposition form (5). The problem of maximum decomposition with with respect to outputs is a generalization of the observ- respect to outputs is to find a coordinate transformation able norm form. From the proof of Proposition 3, we see matrix T such that the decomposition with respect to that, if the largest unobservable subspace Oc is regular, outputs has the maximum order n − s. then z[2] and z˜[2] can be logically expressed in terms of each other, i.e. there exists a logical matrix R such that In Cheng, Li and Qi (2010) , the observable normal form T T (1 s ⊗ I n−s )T = RT˜ (1 s ⊗ I n−s ), i.e. of a BCN is proposed. Here, we rewrite the result in the 2 2 2 2 algebraic form as follows. 1 T T R = (1 s ⊗ I n−s )T T˜ (1 s ⊗ I n−s ). (8) 2s 2 2 2 2 Proposition 2. (Cheng, Li & Qi, 2010) Consider BCN (1) with the algebraic form (4). Assume the largest If there exist two maximum decompositions with respect unobservable subspace Oc is a regular subspace with to outputs described by (5) and (5) respectively such that { z˜s˜+1, z˜s˜+2, ··· , z˜n } as its basis. Then, under the coor- the right side of (8) is not a logical matrix, then the largest dinate transformation z˜ = T˜ x, the BCN (4) becomes uncontrollable subspace is not regular. [1] [1] z˜ (t +1) = G˜1u(t)˜z (t), 3 Algebraic conditions for the decomposability [2] ˜ (6) z˜ (t +1) = G2u(t)˜z(t), with respect to outputs y(t) = M˜ z˜[1](t),

[1] s˜ [2] n In this section, based on the definition of the decompos- where z˜ = ⋉ z˜ , z˜ = ⋉ z˜ , G˜ ∈ L s˜× s˜+m , i=1 i i=˜s+1 i 1 2 2 ability with respect to outputs, we derive some equiva- ˜ ˜ G2 ∈ L2n−s˜×2n+m and M ∈ L2p×2s˜ . The decomposition lent algebraic conditions. form (6) is called the observable normal form of (4).

Lemma 1.Assume that M1,M2, ··· ,Ml ∈ Rm×n are T T A natural question is whether the defined maximum de- non-negative matrices satisfying 1mMk = 1n for every composition with respect to outputs and the observable k =1, 2, ··· ,l. If M1 +M2 +···+Ml = lG, with G being normal form proposed by Cheng, Li and Qi (2010) are a logical matrix, then M1 = M2 = ··· = Ml = G. the same one. Actually, with the regularity assumption, they are the same one. Swap matrix W[m,n] is an mn × mn logical matrix, de- 1 2 m fined as W[m,n] = [In ⊗ δm, In ⊗ δm, ··· , In ⊗ δm]. For the basic concepts of the state-space method men- tioned above, please refer to Cheng & Qi (2010) and Lemma 2. (Cheng, Qi, & Li, 2011) Let W[m,n] ∈ Cheng, Li, & Qi (2010b). Comparing (5) with (6), we T −1 R × find that the maximum decomposition with respect to mn mn be a swap matrix. Then W[m,n] = W[m,n] = outputs and the observable normal form have the same W[n,m] and W[m,1] = W[1,m] = Im, where Im is an structure. identity matrix.

Proposition 3. Assume that the largest unobservable Lemma 3. (Cheng, Qi, & Li, 2011) Let A ∈ Rm×n, subspace Oc is regular, then (5) is a maximum decompo- B ∈ Rp×q. Then W[m,p](A ⊗ B)W[q,n] = (B ⊗ A). sition with respect to outputs if and only if it is an ob- servable normal form described by Proposition 2. Theorem 1. Consider BCN (1) with the algebraic form Proof. Assume that (5) is a maximum decomposition (4). Let L = [L1,L2, ··· ,L2m ], where Li ∈ L2n×2n . with respect to outputs of order n−s. We first prove that Then the following statements are equivalent: s =s ˜. On the one hand, by the definition of the max- (i) the system (1) is decomposable with respect to outputs imum decomposition with respect to outputs, we have with order n − s;

3 (ii) there exist a permutation matrix T ∈ L2n×2n , logical where Pi ∈ L2s×2s are non-negative matrices. By (19), matrices G1 ∈ L2s×2m+s and M ∈ L2p×2s such that Lemmas 2 and 3, we have T T T (I2s ⊗ 1 n−s )TL(I2m ⊗ T )= G1(I2m+s ⊗ 1 n−s ), (9) 2 2 T n−s T T G1i = QLiQ /2 HT = M(I2s ⊗ 12n−s ). (10) T n−s = [P1,P2,··· ,P2n−s ]W[2s,2n−s]TQ /2 (iii) there exist a permutation matrix T ∈ L2n×2n , G1 = n−s = [P1, P2, ··· , P2n−s ](12n−s ⊗ I2s )/2 , [G11, G12, ··· , G12m ] ∈ L2s×2m+s and M ∈ L2p×2s such that that is, QLi = G1iQ and H = MQ (11) − 2n s m T − hold for each i =1, 2, ··· , 2 , where Q = (I s ⊗1 n−s )T n s 2 2 Pk =2 G1i. (20) s× s and G1i ∈ L2 2 ; k=1 n n X (iv) there exists a permutation matrix T ∈ L2 ×2 such T that Multiplying (19) on the left by 12s yields T n−s T n−s QLiQ /2 and HQ /2 (12) T T T 12s [P1, P2, ··· , P2n−s ]= 12n W[2n−s,2s] = 12n , (21) s T are logical matrices, where Q = (I2 ⊗ 12n−s )T . Proof. (i) ⇔ (ii) Assume that BCN (1) is decomposable T T n−s with respect to outputs of order n − s. Then (3) can be namely, 12s Pk = 12s for every k =1, 2, ··· , 2 . Thus, converted in (5). By (3), we have from Lemma 1 and (20), it follows that Pk = G1i. Con- sidering (19), we have [1] T z (t +1)=(I2s ⊗ 12n−s )z(t + 1) T T n−s s − s T T QLiT W[2 ,2 ] = G1i(12n s ⊗ I2 ). (22) = (I2s ⊗ 12n−s )TL(I2m ⊗ T )u(t)z(t), (13) y(t)= HT Tz[1](t)z[2](t). (14) Thus T T QLiT = G1i(I2s ⊗ 12n−s ), (23) By (5), we have which implies QLi = G1iQ. With a same procedure above, we obtain H = MQ. [1] T ✷ z (t +1)= G1(I2m+s ⊗ 12n−s )u(t)z(t), (15) (iii) ⇒ (ii) The proof is trivial. T [1] [2] y(t)= M(I2s ⊗ 12n−s )z (t)z (t). (16) 4 A graphical condition for the decomposability From (13) and (15), we get (9). From (14) and (16), we with respect to outputs get (10). Conversely, with the similar procedure, it fol- lows from (9) and (10) that (5) holds. Thus (i) is proved. Consider BCN (4) with L=[L1,L2, ··· ,L2m]∈L2n×2n+m . (ii) ⇒ (iii) Multiplying (9) on the right by I m ⊗T yields 2 Each logical matrix Lj can be regarded as an of a Gj . The vertex set of Gj is T n QL = G1(I2m ⊗ ((I2s ⊗ 12n−s )T )) A = {1, 2, 3, ··· , 2 }, and Gj has a directed edge (q, k) if = [G11, G12, ··· , G12m ](I2m ⊗ Q) and only if (Lj )kq 6= 0. We say that k is an out-neighbor of q with respect to G if (L ) 6= 0. We denote the out- = [G Q, G Q, ··· , G m Q], (17) j j kq 11 12 12 j neighborhood of set S in graph Gj by N (S). which implies QLi = G1iQ. Multiplying (10) on the Definition 1. Let A be the vertex set of a graph G, and right by T gives H = MQ. µ Φl,l = 1, 2, ··· ,µ be subsets of A. {Φl} is called a (iii) ⇒ (iv) A straightforward calculation shows that µ l=1 vertex partition of A, if ∪ Φl = A and Φi ∩ Φj = ∅ l=1 µ T T T n−s for any i 6= j. A vertex partition {Sl}l=1 of A is called QQ = (I2s ⊗ 1 n−s )TT (I2s ⊗ 12n−s )=2 I2s . 2 an equal vertex partition if |Sl| = |A|/µ for every l = 1, 2, ··· ,µ. Thus, it follows from (11) that In Zou & Zhu (2014), the concept of perfect equal vertex T n−s T n−s G1i = QLiQ /2 and M = HQ /2 . (18) partition (PEVP) is proposed for the decomposition with respect to inputs. Here, we further extend this concept Thus the matrices in (12) are logical matrices. to deal with the decomposition with respect to outputs.

2m m (iv) ⇒ (iii) Denote the logical matrices in (12) by G1i Definition 2. Consider the set {Gj }j=1 composed of 2 and M respectively. Let digraphs with the common vertex set A. Assume that dif- ferent colors are assigned to all the vertices of A. An equal µ T − − QLiT W[2n s,2s] = [P1, P2, ··· , P2n s ], (19) vertex partition S = {Sl}l=1 of A is called a common

4 [1] [1] j S3 1 8 S2 z (t +1) = G1j z (t) for any fixed u(t) = δ2m , i.e. the state z(t) transmits from one Sz1···zs to another for 2 3 any given graph Gj . This is just why we propose the S1 concept of CC-PEVP and try to reveal the relationship 7 6 between CC-PEVP and the decomposition with respect to outputs. 4 5 S4 Proof of Theorem 2. (Necessity) By Theorem 1, there Fig.1. CC-PEVP of two graphs G1 and G2. exist a permutation matrix T ∈ L2n×2n and logical ma- m trices G1j ∈ L2s×2s (j =1, 2, ··· , 2 ), M ∈ L2p×2s such concolorous perfect equal vertex partition (CC-PEVP) of m 2m that the equalities in (11) hold for each j =1, 2, ··· , 2 . {Gj }j=1 if m Set (i) S is perfect for each j = 1, 2, ··· , 2 , i.e. for any T s 1 n−s s n j j Q = (I2 ⊗ 2 )T = δ2 [i1, ··· ,i2 ]. (25) given l and j, there exists an αl such that N (Sl) ⊂ Sαj ; l Thus, (11) can be rewritten as (ii) for any given l = 1, 2, ··· ,µ, all the vertices in Sl have the same color. δ2s [i1, ··· ,i2n ]Lj = G1j δ2s [i1, ··· ,i2n ], (26) H = Mδ s [i , ··· ,i n ]. (27) We give a simple example shown in Fig. 1 to explain 2 1 2

Definition 2 intuitively. In Fig. 1, each vertex has red s 2 or green color. The digraph with blue edges is denoted Let Sl = {q|iq = l}. Then {Sl}l=1 is an equal vertex partition of A with |S | =2n−s. For any l, we have that by G1 and the other one with black edges is denoted by l G2. Obviously, {S1,S2,S3,S4} shown in Fig.1 forms an αj equal vertex partition and each Sl contains vertices of j l l l βl ∃ α , βl, s.t. G1j δ s = δ s , Mδ s = δ p . (28) the same color. Moreover, from Fig.1, we have l 2 2 2 2 j N 1(S )=N 1(S )={3}⊂S , N 1(S )=N 1(S )={1}⊂S , For any k ∈ N (Sl), there exists q ∈ Sl, i.e. iq = l, such 1 2 1 3 4 2 k 2 2 2 2 that Colq(Lj )= δ2n . Then, by (28), we have N (S1)=N (S2)=N (S3)=N (S4)={4, 5}⊂S4. j ik iq l αl δ s =δ2s[i1,··· ,i2n]Colq(Lj)=G1j δ s =G1j δ s=δ s . (29) Therefore, by Definition 2, {S1,S2,S3,S4} is a CC- 2 2 2 2 PEVP of {G , G }. 1 2 j Thus ik = αl , which implies k ∈ S j . Therefore, we have αl Based on Definition 2, we propose an equivalent graph- j N (Sl) ⊂ Sαj . Moreover, for every k ∈ Sl (ik = l), ical condition for the decomposability with respect to l outputs in the following theorem. k ik l βl Hδ2n = Mδ2s = Mδ2s = δ2p , Theorem 2. Consider the BCN (1) with algebraic form (4). Let L = [L1,L2, ··· ,L2m ] with each Lj ∈ L2n×2n . which implies that all the vertices in Sl have the same 2m color. Denote by A the common vertex set of {Gj }j=1, where each Gj is induced by Lj . Color all the vertices in A in 2s such a way that any two vertices µ and λ are of the same (Sufficiency) Since {Sl}l=1 is a CC-PEVP, by Definition µ λ 1 and the definition of the coloring, we have that color if and only if Hδ2n = Hδ2n . Then BCN (1) is decomposable with respect to outputs with order n − s if 2m 2s s m j j and only if {Gj }j=1 has a CC-PEVP {Sl}l=1 with |Sl| = ∀ 1 ≤ l ≤ 2 , 1 ≤ j ≤ 2 , ∃ α , s.t. N (Sl) ⊂ S j (30) l αl 2n−s. and Before the proof of Theorem 2, we give an intuitive ex- planation on the motivation. From the decomposition s q βl ∀ 1 ≤ l ≤ 2 , ∃ βl, s.t. ∀ q ∈ Sl, Hδ2n = δ2p . (31) form (3), it follows that, as z1, ··· ,zs are fixed, the set s For any q ∈ Sl (l = 1, 2, ··· , 2 ), let iq = l. We denote S ··· ={(z ,··· ,z ,z ,··· ,z ) | z ∈{0,1},s+1≤j ≤n} n−s z1 zs 1 s s+1 n j Q = δ2s [i1, ··· ,i2n ]. Since |Sl| = 2 , there exists a T s − n−s permutation matrix T such that (I2 ⊗ 12n s )T = Q. has 2 states and these states have the same output. For any l ∈{1, 2, ··· , 2s}, we have Then the family T T q Coll(QLjQ )= QLjColl(Q )=QLj δ2n {Sz1···zs | zj ∈{0, 1}, j =1, 2, ··· ,s}, (24) q∈S Xl n forms a concolorous equal partition of all the 2 states. = QColq(Lj ). (32) From the algebraic representation (5), it follows that q∈S Xl

5 kq Since Lj is a logical matrix, we can let Colq(Lj) = δ2n , 1, 2, 3, 4) given in Fig.1 is a CC-PEVP of {G1, G2}. Thus j i.e. kq ∈ N ({q}). Thus it follows from (32) and (30) BCN (35) is decomposable with respect to outputs of that order 1. Let

T kq ikq T Coll(QLjQ )= Qδ2n = δ2s (I4 ⊗ 12 )T = Q = δ4[2 3 1 4 4 1 3 2]. q∈Sl q∈Sl X j X αl j j It follows that T = δ8[3 6 1 8 7 2 5 4]. The coordinate = δ2s (kq ∈N ({q})⊂N (Sl)⊂S j ) αl transformation matrix T is the same as that given in q∈S Xl Example 10.3 of Cheng, Qi, & Li (2011). Therefore, the αj n−s l decomposition with respect to outputs is obtained as =2 δ2s . (33)

From (31), it follows that z1(t +1)= u(t),

T T q z2(t +1)= z1(t) ∧ u(t), Coll(HQ )= HColl(Q )= Hδ2n  q∈S z3(t +1)= z3(t) → u(t). Xl βl n−s βl y(t)= z (t) → z (t). = δ2s =2 δ2s . (34)  1 2 q∈S Xl T n−s T n−s By (33) and (34), both QLjQ /2 and HQ /2 5 Searching a CC-PEVP are logical matrices. Therefore, by (iv) of Theorem 1, the sufficiency is proved. ✷ To achieve the decomposition with respect to outputs of the maximum order n − s, we need to find a CC- From the sufficiency proof of Theorem 2, we see that, if s s 2 2 PEVP {Sl}l=1 of the minimum s. Since the number of a CC-PEVP {Sl}l=1 is given, a constructive procedure to calculate the transformation matrix T is obtained as all the equal vertex partitions is finite, a straightforward follow: method is to check whether each one is a CC-PEVP (i) let i = l for any l =1, 2, ··· , 2s and q ∈ S ; or not, but the computational burden of this method q l is very heavy. To reduce the computational burden, we (ii) let Q = δ2s [i1, ··· ,i2n ]; (iii) compute permutation matrix T from investigate some necessary conditions for the existence of a CC-PEVP. T (I s ⊗ 1 n−s )T = Q. 2 2 For BCN (1) with the algebraic form (4), let To display the effectiveness of Theorem 2, we recon- j1j2···jr k sider Example 10.3 of Cheng, Qi, & Li (2011) and con- Rk = {q| Colq(HLj1 Lj2 ··· Ljr )= δ2p } (36) struct the coordinate transformation using the graphical p m method. for any k =1, 2, ··· , 2 , r ≥0 and 1≤j1, j2, ··· , jr ≤2 . Here, r = 0 means that all the ji vanish, i.e. Example 1. Consider the following system k x1(t+1)=x3(t) ∨ u(t), Rk = {q| Colq(H)= δ2p }.

x2(t+1)=(x1(t)∧¬x3(t))∨(¬x1(t)∧(x3(t)↔u(t))), (35) From (36), we get the following result. x3(t+1)=x3(t) → u(t), Lemma 4. For any given j1, j2, ··· jr, the fam- y(t) = (x1(t) ↔ x3(t)) → (x2(t)∨¯x3(t)). ··· p j1 j2···jr j1j2 jr 2 ily R = {Rk }k=1 is a partition of A = Let x(t)= x (t)x (t)x (t). Then we have n 1 2 3 {1, 2, ··· , 2 }. For any q1, q2 ∈ A, they are in the same ··· Rj1j2 jr if and only if x(t +1)= Lu(t)x(t), y(t)= Hx(t), k

Colq (HLj ··· Lj )= Colq (HLj ··· Lj ). where L = [L1,L2] ∈ L8×16, with 1 1 r 2 1 r

L1 = δ8[3 1 3 1 1 3 1 3],L2 = δ8[4 5 4 5 4 5 4 5] and Given a partition R of set A, for any q1, q2 ∈ A, we H = δ [2 1 1 1 1 1 1 2]. R 2 say that q1 is equivalent to q2, denoted by q1 ∼ q2, if The digraphs G1 and G2 corresponding to BCN (35) and only if there exists a R ∈ R such that q1, q2 ∈ R. is just shown in Fig.1. The vertex partition Si (i = Conversely, each equivalence on A induces an

6 m j1j2···jr associated partition R of set A. Therefore, by Lemma 4, for any 1 ≤ j1, ··· , jr ≤ 2 . Choose βl such that we have j1j2···jr βl l j1···jr R δ2p = MG1j1 ··· G1jr δ2s . (40) q1 ∼ q2 ⇔ Colq1(HLj1···Ljr)=Colq2(HLj1···Ljr). (37)

j1j2···jr Now we are in a position to prove Sl ⊂ R j1j2···jr . In βl fact, for every k ∈ Sl (ik = l), it follows from (39) that Let us briefly recall some basic and well-known terms, notions and facts, without proofs, concerning partitions k Colk(HLj1 ··· Ljr )=MG1j1 ··· G1jrQδ2n and equivalence relations. For details, please refer to ik =MG ··· G δ s Bor˚uvka (1974) and Pot˚uˇcek (2014). 1j1 1jr 2 l =MG1j1 ··· G1jrδ2s Definition 3 j1j2···jr . Let U and V be partitions of a set A. As- βl sume that, for every U ∈U, there exists V ∈V such that = δ2p . (41) U ⊂ V . Then partition U is said to be a refinement of V, ⊏ j1j2···jr ✷ which is denoted by U V. Thus it follows from (36) that k ∈ R j1j2···jr . βl Definition 4 . A meet of two partitions U and V of a set From Proposition 4, it seems that an infinite number A, denoted by U⊓V, is a set of all intersections U ∩ V , of partitions need to be considered. But actually only where U ∈U and V ∈V. a finite number of partitions are necessary due to the following result. Lemma 5. Let V1 and V2 be two partitions of a set A and let V = (V ⊓V ) \ {∅}. Then V is a partition of 1 2 Lemma 7 (Cheng, Qi, & Zhao, 2011). Define a set of A satisfying V ⊏ V and V ⊏ V . Moreover, if U is a 1 2 logical matrices as partition of A such that U ⊏ V1 and U ⊏ V2, then U ⊏ V.

H0 ={H}, Definition 5. The partition V given in Lemma 5 is called m Hr ={HLj Lj , ··· ,Lj | j , j , ··· , jr ∈{1, 2, ··· , 2 }}, the greatest common refinement of the partitions V1 and 1 2 r 1 2 gcr V2 denoted by (V1, V2). ∗ where r = 1, 2, ···∗. Then there exists the minimum r r ∗ Definition 6. The greatest common refinement of the such that Hr ⊂ ∪r=0Hr for any r > r . partitions V1, V2, ··· , Vµ of A is inductively defined as gcr(V1, V2, ··· , Vµ)=gcr(gcr(V1, V2, ··· , Vµ−1), Vµ). Denote H = H0 ∪ H1 ∪···∪Hr∗ (42) Lemma 6. Let W = gcr(V1, V2, ··· , Vµ). Then, for any and let |H| = τ. For convenience, we denote the τ ma- q1, q2 ∈ A, we have trices in H by Hi = δ2p [hi1,hi2, ··· ,hi2n ] and the corre- sponding partitions by Ri (i = 1, 2, ··· , τ). By Propo- W Vi sition 4 and Lemma 7, we see that it is only needed q1 ∼ q2 ⇔ (q1 ∼q2, i =1, 2, ··· ,µ). (38) 2s to search a CC-PEVP {Sl}l=1 from the partitions Ri Now, let us come to our main result: (i =1, 2, ··· , τ). Let

Proposition 4. Assume that BCN (4) with L = h11 h12 ··· h12n [L1, ··· ,L2m] is decomposable with respect to outputs  h21 h22 ··· h22n  with order n − s, where each Li ∈ L2n×2n . Denote by O = . . . . , (43) Gj the directed graph with the adjacency matrix Lj. Let  . . .. .  2s n−s 2m   S={Sl}l=1 (|Sl|=2 ) be a CC-PEVP of {Gj}j=1. Then    h h ··· h n   τ1 τ2 τ2  j1···jr m   S ⊏ R , ∀ r ≥ 0, ∀ j1, ··· , jr ∈ {1, 2, ··· , 2 }. which is just the observability matrix proposed in Cheng & Qi (2009). s Proof. For any l = 1, 2, ··· , 2 and q ∈ Sl, let iq = l. Corollary 1. Under the conditions of Proposition 4, we Set Q = δ2s [i1, ··· ,i2n ]. Then by the sufficiency proof of Theorem 2, we obtain (iv) of Theorem 1, which is have ⊏ equivalent to (iii) of Theorem 1. From (11), we have S gcr(R1, R2, ··· , Rτ ) =: C. (44) H =MQ and By Corollary 1, we only need to search the CC-PEVP 2s HLj1 ··· Ljr=MQLj1 ··· Ljr=MG1j1 ··· G1jrQ (39) S = {Sl}l=1 of the minimum s from partition C.

7 T Proposition 5. For any q1, q2 ∈ A, we have where L = [L1,L2]= 12 W[2,2n] with n n n C L1 = δ2 [1, 3, 5, ··· , 2 −1, 1, 3, 5, ··· , 2 −1], n n q1 ∼ q2 ⇔ Colq1 (O)= Colq2 (O), (45) L2 = δ2n [2, 4, 6, ··· , 2 , 2, 4, 6, ··· , 2 ]

−1 −1 where partition C of the vertex set A is shown in (44). and 2n 2n Proof. By (37), Lemma 6 and (43), the proposition is H = δ [1, 1, ··· , 1, 2, 2, ··· , 2]. proved. 2 A straightforward computation shows that C = 2n z }| { z }| { Corollary 2. If there exists a C ∈ C with odd cardinal, {{i}}i=1, i.e. |C| = 1, ∀ C ∈ C. Thus the shift regis- then BCN (4) is undecomposable with respect to outputs. ter is observable and undecomposable with respect to outputs. To give an intuitive explanation, we list the Proof. By Corollary 1, C is a union of some Sl. Since n−s n−s procedure for the case of n = 3 as follows . |Sl| = 2 for each l, we see that 2 is a factor of |C|. Considering that |C| is odd, we obtain the order H = δ [1, 1, 1, 1, 2, 2, 2, 2], n−s = 0. Thus BCN (4) is undecomposable with respect 2 to outputs. HLi = δ2[1, 1, 2, 2, 1, 1, 2, 2] (i =1, 2), HL L = δ [1, 2, 1, 2, 1, 2, 1, 2] (i, j =1, 2), Lemma 8 (Cheng & Qi, 2009). Assume that BCN (4) i j 2 is globally controllable. Then it is observable if and only δ2[1, 1, 1, 1, 1, 1, 1, 1], k =1, if all the columns of O are distinct. HLiLj Lk = (δ2[2, 2, 2, 2, 2, 2, 2, 2], k =2. Corollary 3. Assume that BCN (4) is globally control- lable. If (4) is observable, then it is undecomposable with It is easy to check that all the columns of O are distinct. respect to outputs. Proof. From Lemma 8, it follows that |C| = 1 for each In order to display the procedure of finding a CC-PEVP, C ∈C. Thus the corollary is proved by Corollary 2. we reconsider the BCN given in Example 1.

Remark 2. The inverse of Corollary 3 is not correct. For Example 3. For BCN (35), we have example, consider a Boolean network (4) without control H=δ2[2, 1, 1, 1, 1, 1, 1, 2], (m = 0). Let L = δ4[1, 2, 3, 1] and H = δ2[1, 1, 1, 2]. HL1=δ2[1, 2, 1, 2, 2, 1, 2, 1], i Since HL = δ2[1, 1, 1, 1] for any i> 0, we have that 2 2 HL2 =HL2L1 =HL1 =HL2=δ2[1, 1, 1, 1, 1, 1, 1, 1], HL1L2=δ2[2, 2, 2, 2, 2, 2, 2, 2]. 1112 O = , " 1111 # From the observability matrix, we have C = {{1, 8}, {2, 4, 5, 7}, {3, 6}}. (47) which implies that C = {{1, 2, 3}, {4}}. Thus the BCN is unobservable and undecomposable. However, for the By (47), we first let S1 = {1, 8}, S2 = {3, 6}. Since traditional linear control systems, the observability and 2 N (S1)= {4, 5}, we let S3 = {4, 5} and S4 = {2, 7}. It the decomposability are equivalent. This is a property of 4 is easy to check that {Sl}l=1(|Sl| = 2) is the unique CC- BCNs different from that of the traditional linear con- PEVP. Thus, the system is decomposable with respect trol systems. Hence, we prefer to call (3) the decompo- to outputs. In fact, the partition obtained here is the sition with respect to outputs instead of the observability same one as shown in Fig.1. decomposition. A contribution of this paper lies in that the regularity To illustrate our method, we consider a practical exam- assumption on the unobservable subspace proposed by ple. Cheng, Li, & Qi (2010) is removed. To illustrate this point, we consider an example as follows. Example 2. A shift register is a cascade of flip-flops:

Example 3. Consider BCN (4) with L=[L1,L2]∈L8×16, xi(k +1)= xi+1(k), (i =1, 2, ··· ,n − 1), x (k +1)= u(k), n L1 = δ8[6, 8, 1, 8, 7, 8, 6, 8],L2 = δ8[6, 8, 7, 8, 1, 8, 6, 8] y(k)= x1(k), (46) and H = δ2[1, 1, 2, 1, 2, 1, 1, 1]. A straightforward com- where xi is the binary state of the ith flip-flop, u the putation shows that input and y the output. Multiplying all the equations yields 11212111 O = . (48) x(k +1)= Lx(k), y(k)= Hx(k), " 11111111 #

8 Thus References C = {{1, 2, 4, 6, 7, 8}, {3, 5}}. Albert, R., & Othmer, H. G.(2003). The topology We try to search a CC-PEVP S from C. Let S1 = {3, 5}. of the regulatory interactions predicts the expres- 1 2 It is easy to check that N (S1) = N (C2) = {1, 7}. So sion pattern of the segment polarity genes in we let S2 = {1, 7}. Assume S3 = {2, 6} and S4 = {4, 8}. drosophila melanogaster. Journal of theoretical bi- Then it is easily seen that ology, 223(1),1–18. Akutsu, T., Hayashida, M., Ching, W., & Ng, M. K. 1 2 N (S1)= N (S1)= {1, 7} = S2, (2007). Control of Boolean networks: hardness re- sults and algorithms for tree structured networks.

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