q -VFCA: q -state Vector-valued Fuzzy Cellular Automata
Yuki Nishida, Sennosuke Watanabe, Akiko Fukuda, Yoshihide Watanabe
qq -VFCA: q -state Vector-valued Fuzzy CellularAutomata Y UKI N ISHIDA (cid:63) , S ENNOSUKE W ATANABE , A KIKO F UKUDA ,Y OSHIHIDE W ATANABE Department of Science of Environment and Mathematical Modeling, DoshishaUniversity, 1-3 Tatara Miyakodani, Kyotanabe-shi, Kyoto 610-0394, Japan Department of General Education, National Institute of Technology, OyamaCollege, 771 Nakakuki, Oyama-shi, Tochigi 323-0806, Japan Department of Mathematical Sciences, Shibaura Institute of Technology, 307Fukasaku, Minuma-ku, Saitama-shi, Saitama 337-8570, Japan Department of Mathematical Sciences, Doshisha University, 1-3 TataraMiyakodani, Kyotanabe-shi, Kyoto 610-0394, Japan
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Elementary fuzzy Cellular Automata (CA) are known as contin-uous counterpart of elementary CA, which are 2-state CA, viathe polynomial representation of local rules. In this paper, wefirst develop a new fuzzification methodology for q -state CA.It is based on the vector representation of q -state CA, that is,the q -states are assigned to the standard basis vectors of the q -dimensional real space and the local rule can be expressed bya tuple of q polynomials. Then, the q -state vector-valued fuzzyCA are defined by expanding the set of the states to the convexhull of the standard basis vectors in the q -dimensional real space.The vector representation of states enables us to enumerate thenumber-conserving rules of -state vector-valued fuzzy CA in asystematic way. Key words: cellular automata, fuzzy cellular automata, vector-valuedcellular automata, conservation law, number-conserving rule, periodicboundary condition (cid:63) email: [email protected] a r X i v : . [ n li n . C G ] F e b INTRODUCTION
One-dimensional Cellular Automata (CA) are linearly arranged arrays of cellsthat evolve simultaneously in accordance with the local update rule depend-ing only on their neighboring cells. Although rules of CA are very simple,they provide surprisingly rich applications and knowledges [1], e.g., trafficdynamics [2], evacuation process [3], cryptography [4], project scheduling[5], image processing [6], urban planning [7], and so on. Simple examplesof CA are 2-state 3-neighbor CA, called Elementary CA (ECA). Cattaneo etal. [8] apply the fuzzification to the boolean operators in the disjunctive nor-mal form of the local rule of ECA and obtain a kind of continuous CA whosecells have states in [0 , . Such CA are called elementary fuzzy CA and theirlocal rules are polynomials of the states of the neighbors. The asymptoticbehaviors of elementary fuzzy CA are studied for cases with a single seedin a zero background [9] and for periodic cases [10]. There are more de-tailed studies on dynamics for some specific rules, e.g., rule 90 [11, 12, 13],rule 110 [14], rule 184 [15]. The algebraic approach for elementary fuzzyCA using invariant theory is presented in [16]. However, to the best of ourknowledge, q -state fuzzy CA have not been successfully formulated yet. Forexample, if we try to fuzzify 3-state CA, we can simply come up with ex-pressing local rules by polynomials along a similar way to elementary fuzzyCA and expanding the states from { , , } to the continuous values in [0 , .Although local rules of elementary fuzzy CA are closed in [0 , , local rulesof 3-state fuzzy CA constructed above are not closed in [0 , . Moreover, thestate “1” is the middle value of “0” and “2”, which means that a state can beexpressed by other states.In this paper, we develop a new expression of q -state n -neighbor fuzzyCA. To treat the q states independently, we represent the q states by the stan-dard basis vectors of R q instead of { , , . . . , q − } . Then, the states of thecorresponding fuzzy CA are in the ( q − -simplex whose vertices are thestandard basis vectors of R q . In the case with q = 3 , states of the 3-statefuzzy CA belong to the interior or the boundary of the regular triangle withvertices (1 , , (cid:62) , (0 , , (cid:62) , (0 , , (cid:62) . We call such CA q -state Vector-valued Fuzzy CA ( q -VFCA). The vector representation of CA reminds us thequantum CA [17]. They are also continuous valued CA, but they do not havethe discrete counterparts. The q -VFCA in this paper are based on the conven-tional CA with q discrete states. Hence, we have one-to-one correspondencebetween the q -VFCA and the usual q -state CA. Other CA represented bythree-dimensional vectors are considered in the studies on the image process-2ng or the encryption of RGB color images [18, 19], but they do not comefrom 3-state CA. We also remark that fuzzy CA in this paper does not meanCA on fuzzy sets [20] or fuzzy choice of local rules [21].The existence of the conserved quantities is one of the fundamental prob-lem in the study of periodic CA. The additive conserved quantities of ECAand the elementary reversible CA are investigated in [22]. In the case of ECA,an example of the additive quantity is the sum of the states of the cells in aperiod, which is identical to the number of cells with state “1”. Fuk´s andSullivan [23] give a combinatorial characterization of the number-conservingCA rules and compute the number of such rules for q -state n -neighbor CA.This “number-conserving” means that the state “ k ” is equal to the numericalvalue k and the sum of these numerical values is conserved. On the otherhand, q -state vector-valued CA we focus on enable us to enumerate the rulesthat conserve the number of the cells with the specific state “ k ” by taking thesum of the k th entries of all vectors as the additive quantity. The concept ofnumber-conserving rules can be extended from q -state vector-valued CA to q -VFCA, as shown in [24] for elementary fuzzy CA.The present paper is organized as follows. In Section 2, we give defi-nitions of ECA and elementary fuzzy CA. In Section 3, q -state n -neighborvector-valued fuzzy CA are introduced. Although the method presented inthis paper can be applied to q -state n -neighbor vector-valued fuzzy CA, wedescribe -state -neighbor vector-valued fuzzy CA for the convenience ofthe notations. In Section 4, we enumerate all the number-conserving rulesof -state -neighbor vector-valued fuzzy CA, where similar computation isapplicable to any other additive conserved quantities. In this section, definitions and notations of ECA and elementary fuzzy CAare given.We denote the set of the states by Q . The neighboring cells of the cell i ∈ Z are given by the set N ( i ) = { i − n (cid:96) , i − n (cid:96) + 1 , . . . , i, . . . , i + n r } .CA are called q -state n -neighbor CA if | Q | = q and | N ( i ) | = n . Let x ti ∈ Q denote the state of the cell i at the time t ∈ Z ≥ . The local rule f : Q n → Q determines the evolutions of the cells by x t +1 i = f ( x ti − n (cid:96) , . . . , x ti , . . . , x ti + n r ) , i ∈ Z , t ∈ Z ≥ . We describe elementary fuzzy CA introduced in [8]. Let us consider theECA rule defined by Table 1. The local rule h : { , } → { , } is expressed3 yz
111 110 101 100 011 010 001 000 h ( x, y, z ) b b b b b b b b TABLE 1The local rule h of ECA. xyz
111 110 101 100 011 010 001 000 h ( x, y, z ) 1 0 1 1 1 0 0 0 TABLE 2ECA rule 184. by the polynomial obtained from the disjunctive normal form in boolean op-erations: h ( x, y, z ) = b xyz + b xy (1 − z ) + b x (1 − y ) z + b x (1 − y )(1 − z ) + b (1 − x ) yz + b (1 − x ) y (1 − z ) + b (1 − x )(1 − y ) z + b (1 − x )(1 − y )(1 − z ) . (1)For example, the polynomial expression of the local rule of ECA rule 184given in Table 2 is h ( x, y, z ) = xyz + x (1 − y ) z + x (1 − y )(1 − z ) + (1 − x ) yz = x − xy + yz. If we expand the domain of h to [0 , via the polynomial (1), then we caneasily check that the obtained function, denoted by f , satisfies f ([0 , ) ⊂ [0 , . The CA defined by the local rule f : [0 , → [0 , are called elemen-tary fuzzy CA. q -STATE CA AND q -STATE FUZZY CA We next consider the fuzzification of q -state CA. The states “0”,“1”, . . . , “ q − ” in the conventional CA are just symbols, not numerical values, that is, thestates , , , . . . can be replaced with A, B, C, . . . , for example. However,4f we set the state to the continuous value in [0 , q − in the fuzzificationprocess, then each state can not be regarded as an independent state. So weconsider the new formulation that the states , , . . . , q − are completelyindependent states.A simple way to express q independent states is assigning them to thestandard basis vectors e , e , . . . , e q of R q . We call the CA with Q = { e , e , . . . , e q } q -state Vector-valued CA ( q -VCA). In this section, we con-struct q -state Vector-valued Fuzzy CA ( q -VFCA). To avoid the use of com-plicated indices, we explain the case with q = 3 and n = 3 , although we caneasily construct q -VFCA with any q or n in the same way.As in the case of ECA, we use the polynomial expression of the localrule for the fuzzification of -VCA. The local rule h : { e , e , e } →{ e , e , e } of -VCA is expressed by the tuple of homogeneous polyno-mials of degree 3 as h ( x , y , z ) = (cid:88) j,k,(cid:96) =1 x j y k z (cid:96) h ( e j , e k , e (cid:96) )= (cid:88) h ( e j , e k , e (cid:96) )= e x j y k z (cid:96) (cid:88) h ( e j , e k , e (cid:96) )= e x j y k z (cid:96) (cid:88) h ( e j , e k , e (cid:96) )= e x j y k z (cid:96) , (2)where x = ( x , x , x ) (cid:62) , y = ( y , y , y ) (cid:62) , z = ( z , z , z ) (cid:62) . Indeed, eachmonomial x j y k z (cid:96) vanishes unless x = e j , y = e k and z = e (cid:96) . Conversely,each map h : { e , e , e } → { e , e , e } of the form h ( x , y , z ) = (cid:88) j,k,(cid:96) =1 a jk(cid:96) x j y k z (cid:96) (cid:88) j,k,(cid:96) =1 b jk(cid:96) x j y k z (cid:96) (cid:88) j,k,(cid:96) =1 c jk(cid:96) x j y k z (cid:96) , where ( a jk(cid:96) , b jk(cid:96) , c jk(cid:96) ) (cid:62) ∈ { e , e , e } , is the local rule of some -VCA.Let ∆ be the triangle in the three-dimensional space whose vertices are5 , e and e , i.e., ∆ = { ( x , x , x ) (cid:62) | x + x + x = 1 , x i ≥ , i = 1 , , } . Now we expand the domain of h to ∆ via (2). In the following proposition,we prove that the obtained function, denoted by f , satisfies f (∆ ) ⊂ ∆ . Proposition 1.
For any x , y , z ∈ ∆ , the summation of the entries of thevector f ( x , y , z ) is equal to 1, namely, (cid:88) h ( e j , e k , e (cid:96) )= e x j y k z (cid:96) + (cid:88) h ( e j , e k , e (cid:96) )= e x j y k z (cid:96) + (cid:88) h ( e j , e k , e (cid:96) )= e x j y k z (cid:96) = 1 . (3) Proof.
Since each monomial x j y k z (cid:96) appears exactly once in the summandsof the left-hand side of (3), we have (cid:88) h ( e j , e k , e (cid:96) )= e x j y k z (cid:96) + (cid:88) h ( e j , e k , e (cid:96) )= e x j y k z (cid:96) + (cid:88) h ( e j , e k , e (cid:96) )= e x j y k z (cid:96) = ( x + x + x )( y + y + y )( z + z + z )= 1 . The last equality follows from x , y , z ∈ ∆ . -state Vector-valued Fuzzy CA ( -VFCA) are continuous CA whose stateset is Q = ∆ and local rule is given by f .To visualize the evolution of -VFCA, we use the RGB color system. Thestates e , e and e are associated with red, green and blue, respectively.An inner point of ∆ is expressed by the mixture of the three colors, whichis illustrated in Figure 1. Space-time diagrams of -VCA and -VFCA areshown in Figure 2. They correspond to rule 6213370633633 for usual -state -neighbor CA, where the rule numbers are the decimal number convertedfrom 27 digits ternary number of rules. q -VFCA For q -VCA or q -VFCA, let x ti = ([ x ti ] , [ x ti ] , . . . , [ x ti ] q ) (cid:62) denote the stateof the cell i ∈ Z at the time t ∈ Z ≥ . In this section, we consider periodicCA with the positive integer period L , i.e., we assume x ti = x ti + L for i ∈ Z and t ∈ Z ≥ . 6 IGURE 1The color distribution on ∆ . For a function F : ( R q ) p → R , the sum of the form Φ t = L (cid:88) i =1 F ( x ti , x ti +1 , . . . , x ti + p − ) is called an additive quantity of q -VCA or q -VFCA. An additive quantity Φ t is called an additive conserved quantity if it satisfies Φ t +1 = Φ t for any t ∈ Z ≥ and any initial configuration ( x , x , . . . , x L ) . An example of additivequantities for q -VCA is the number of the cells in a period whose states are e k , which is counted as ν t ( k ) = L (cid:88) i =1 [ x ti ] k . These additive quantities can be extended for q -VFCA. A rule of q -VCA or q -VFCA is called a k -number-conserving rule if ν t ( k ) is the additive conservedquantity. In particular, a rule is called a complete number-conserving rule if ν t ( k ) is the additive conserved quantity for all k = 1 , , . . . , q .The vector representation of q -state CA or q -state fuzzy CA enables us toenumerate the rules that admit the various kinds of additive conserved quan-tities. We now demonstrate the enumeration of the number-conserving rulesof -VFCA with three neighboring cells. Note that the computation belowshows that a rule of -VFCA is k -number-conserving if and only if the corre-sponding -VCA is k -number-conserving.7 ti m e space t i t ti m e space t i FIGURE 2Space-time diagrams of -VCA (left) and -VFCA (right). Recall that the local rule of -VFCA is of the form f ( x , y , z ) = (cid:88) j,k,(cid:96) =1 a jk(cid:96) x j y k z (cid:96) (cid:88) j,k,(cid:96) =1 b jk(cid:96) x j y k z (cid:96) (cid:88) j,k,(cid:96) =1 c jk(cid:96) x j y k z (cid:96) , (4)where ( a jk(cid:96) , b jk(cid:96) , c jk(cid:96) ) (cid:62) ∈ { e , e , e } . If the rule of -VFCA defined bythe local rule f is 1-number-conserving, we have L (cid:88) i =1 [ x ti ] = L (cid:88) i =1 [ x t +1 i ] = L (cid:88) i =1 3 (cid:88) j,k,(cid:96) =1 a jk(cid:96) [ x ti − ] j [ x ti ] k [ x ti +1 ] (cid:96) . (5)Using the identities [ x ti ] + [ x ti ] + [ x ti ] = 1 , i = 1 , , . . . , L, L (cid:88) i =1 [ x ti − ] k = L (cid:88) i =1 [ x ti ] k = L (cid:88) i =1 [ x ti +1 ] k , k = 1 , , , L (cid:88) i =1 [ x ti − ] k [ x ti ] (cid:96) = L (cid:88) i =1 [ x ti ] k [ x ti +1 ] (cid:96) , k, (cid:96) = 1 , , , the most right-hand side of (5) is computed as a S (0 , , a + a + a − a ) S (0 , , a + a + a − a ) S (0 , , a − a + a − a − a + 2 a ) S (1 , , a − a − a + a − a − a + 2 a ) S (2 , , a − a + a − a − a − a + 2 a ) S (1 , , a − a + a − a − a + 2 a ) S (2 , , a − a − a + a ) S (1 , , a − a − a + a ) S (2 , , a − a − a + a ) S (1 , , a − a − a + a ) S (2 , , a − a − a + a − a + a + a − a ) S (1 , , a − a − a + a − a + a + a − a ) S (2 , , a − a − a + a − a + a + a − a ) S (1 , , a − a − a + a − a + a + a − a ) S (2 , , a − a − a + a − a + a + a − a ) S (1 , , a − a − a + a − a + a + a − a ) S (2 , , a − a − a + a − a + a + a − a ) S (1 , , a − a − a + a − a + a + a − a ) S (2 , , . Here, S ( j, k, (cid:96) ) = (cid:80) [ x ti − ] j [ x ti ] k [ x ti +1 ] (cid:96) for j, k, (cid:96) = 0 , , , [ x ti ] = 1 , andeach summation is taken over i = 1 , , . . . , L . Regarding [ x ti ] , [ x ti ] , i =1 , , . . . , L, as independent indeterminates and comparing the coefficientsin (5), we have 19 equations. Solving them under the condition a jk(cid:96) ∈{ , } , we obtain nine solutions for ( a , a , . . . , a ) , see Appendix A. If a jk(cid:96) = 1 , then we have b jk(cid:96) = c jk(cid:96) = 0 . On the other hand, if a jk(cid:96) = 0 , then9e have two possibilities: b jk(cid:96) = 1 , c jk(cid:96) = 0 and b jk(cid:96) = 0 , c jk(cid:96) = 1 . Thenumber of the -number-conserving -VFCA rules is computed as × .In the same way, we can obtain the conditions for b jk(cid:96) (resp. c jk(cid:96) ) for the -number conserving (resp. the -number-conserving) rules, which are shownin Appendix B.To find complete number-conserving rules, we seek local rules satisfyingboth the conditions for a jk(cid:96) and for b jk(cid:96) . Note that if ν t (1) and ν t (2) areadditive conserved quantities, so is ν t (3) . A pair of ( a , a , . . . , a ) and ( b , b , . . . , b ) gives the local rule of a -VFCA if and only if a jk(cid:96) + b jk(cid:96) ≤ for all ≤ j, k, (cid:96) ≤ . Hence, checking × pairsof ( a , a , . . . , a ) and ( b , b , . . . , b ) , we obtain 15 completenumber-conserving rules, shown in Appendix C. In this paper, we first introduce the vector representation of q -state n -neighborCA called q -VCA. The q states are assigned to the standard basis vectors e , e , . . . , e q of R q and the local rule h can be expressed by a tuple of q polynomials that are homogeneous of degree n . Then, we consider CA withthe states in the convex hull of e , e , . . . , e q , where we naturally expand thevector-valued map h via the polynomial expression. We call them q -VFCA.If q = 2 , the local rule for -VFCA is equivalent to that for elementary fuzzyCA obtained from the disjunctive normal form in boolean operations. We alsoexplain how to enumerate the number-conserving rules of these vector-valuedCA. If we focus on the additive quantity F ( x ti ) = (cid:80) qk =1 ( k − x ti ] k , we candiscuss the usual number-conserving rules [23], which conserve the numberof particles, that is, the sum of the values. It is a future work to investigate theproperties of q -VFCA, such as the asymptotic behavior. REFERENCES [1] Wolfram, S. (2002). A New Kind of Science. 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Journal of Cellular Automata, 2, 141–148.[24] Betel, H., Flocchini, P. (2011). On the relationship between fuzzy and Boolean cellularautomata. Theoretical Computer Science, 412, 703–713. SOLVING EQUATIONS FOR -NUMBER-CONSERVING RULES Comparing the coefficients of (5), we have the following 19 equations. a = 0 , (6) a + a + a − a = 1 , (7) a + a + a − a = 0 , (8) a − a + a − a − a + 2 a = 0 , (9) a − a − a + a − a − a + 2 a = 0 , (10) a − a + a − a − a − a + 2 a = 0 , (11) a − a + a − a − a + 2 a = 0 , (12) a − a − a + a = 0 , (13) a − a − a + a = 0 , (14) a − a − a + a = 0 , (15) a − a − a + a = 0 , (16) a − a − a + a − a + a + a − a = 0 , (17) a − a − a + a − a + a + a − a = 0 , (18) a − a − a + a − a + a + a − a = 0 , (19) a − a − a + a − a + a + a − a = 0 , (20) a − a − a + a − a + a + a − a = 0 , (21) a − a − a + a − a + a + a − a = 0 , (22) a − a − a + a − a + a + a − a = 0 , (23) a − a − a + a − a + a + a − a = 0 . (24)From (6), (8), (12), (16) and (24), we have a = a = a = a = a = a = a = a = 0 . We consider three cases for (7).1. a = 1 and a = a = 0 .We have a = a = a = 1 , a = a = a = a = 0 from (10), (13), (14), (15), (17) and (20). Now, equations (9) and (11)turn to a + a = 1 , a + a = 1 . ( a , a , a , a ) . If a =1 , a = 0 , a = 0 and a = 1 , equation (21) implies a = 2 ,leading to a contradiction. Thus, we obtain the following three cases,where a , a , a , a and a are uniquely determined by (18),(19), (21), (22) and (23). ( a , a , a , a , a , a , a , a , a )= (1 , , , , , , , , , (0 , , , , , , , , , (0 , , , , , , , , . a = 1 and a = a = 0 .We have a = a = a = 1 , a = a = a = 0 from (9), (13), (14), (15) and (17). Equations (10) and (11) turn to a + a = 1 , a + a = 1 . Again, we have four possibilities for ( a , a , a , a ) . If a =0 , a = 1 , a = 1 and a = 0 , equation (23) implies a = − ,leading to a contradiction. Thus, we obtain the following three cases. ( a , a , a , a , a , a , a , a , a , a )= (1 , , , , , , , , , , (1 , , , , , , , , , , (0 , , , , , , , , , . a = 1 and a = a = 0 .In the same way, we have a = a = a = 1 , a = a = a = a = 0 from (11), (13), (14), (15), (17) and (23), and ( a , a , a , a , a , a , a , a , a )= (1 , , , , , , , , , (1 , , , , , , , , , (0 , , , , , , , , . from other equations. 13ence, we have the following nine solutions. ( a , a , . . . , a ) = (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 , , , , , , , , , , , , , , , , , , , , , , , , , , . (25) B SOLUTIONS FOR OTHER NUMBER-CONSERVING RULES
A rule of -VFCA is -number-conserving if and only if the coefficients ofthe local rule (4) satisfies ( b , b , . . . , b ) = (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , . (26)14 rule of -VFCA is -number-conserving if and only if the coefficients ofthe local rule (4) satisfies ( c , c , . . . , c ) = (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 , , , , , , , , , , , , , , , , , , , , , , , , , , . C COMPLETE NUMBER-CONSERVING RULES
Table 3 shows the rule numbers and the local rules of complete number-conserving -VFCA. The rule numbers of -VCA or -VFCA are determinedas follows. Let h : { e , e , e } → { e , e , e } be the local rule of -VCA.Then, we consider the sequence h ( e , e , e ) h ( e , e , e ) · · · h ( e , e , e ) . Replacing e with , e with , and e with , we have 27 digits ternarynumber. We can compute the rule number in the decimal number by convert-ing this ternary number. The rule numbers of -VFCA are the same as thoseof the corresponding -VCA. We also express the rule numbers in the -adic numbers, which are expressed by , , . . . , , a , b , . . . , q , since they seemto capture the combinatorial characteristics of complete number-conservingrules. The pair ( µ, ν ) in the table means that the rule is determined by the µ thsolution in (25) and the ν th solution in (26).15ecimal rule number -adic rule number pairlocal rule , y , y , y ) (cid:62) (cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96) (9 , z , z , z ) (cid:62) , x , x , x ) (cid:62) , x y + y z + y z , x y + x y + y z , y ) (cid:62) , x y + y z + y z , y , x y + x y + y z ) (cid:62) , y , x y + y z + y z , x y + x y + y z ) (cid:62) , y , x y + x y + y z , x y + y z + y z ) (cid:62) , x y + x y + y z , y , x y + y z + y z ) (cid:62) , x y + x y + y z , x y + y z + y z , y ) (cid:62) (cid:96) qq (cid:96) d0 (cid:96) d0 (8 , x y + x y + y z , x y + x y + y z , x y + x y + y z ) (cid:62) , x y + y z + y z , x y + y z + y z , x y + y z + y z ) (cid:62) , x y + y z + y z , x y + y z + y z , x y + y z + y z ) (cid:62) (cid:96) qd (cid:96) (cid:96) (4 , x y + x y + y z , x y + x y + y z , x y + x y + y z ) (cid:62) (cid:96) (cid:96) dq (cid:96) , x y + x y + y z , x y + x y + y z , x y + x y + y z ) (cid:62) , x y + y z + y z , x y + y z + y z , x y + y z + y z ) (cid:62) TABLE 3Complete number-conserving rules of -VFCA.-VFCA.