intriguing properties of global structure in some classes of finite cellular automata

Physica D 31 (1988) 318-338 North-Holland, Amsterdam

INTRIGUING PROPERTIES OF GLOBAL STRUCTURE IN SOME CLASSES OF FINITE CELLULAR AUTOMATA

Hiroyuki ITO* Department of Physics, Faculty of Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan

Received 17 October 1987 Revised manuscript received 2 March 1988 Communicated by M. Mimura

The simplest one-dimensional cellular automata introduced recently by Wolfram are analyzed. Particularly, the finite cellular automata with two special evolution rules are studied. Non-trivial correspondence in global structure is found between the state transition diagrams under the two rules: correspondence of cycle structure, basin volume of each cycle and structure of transient trees etc. Origin of the correspondence is explored on the basis of local properties of the evolution rules. It is shown that this correspondence becomes exact in the thermodynamic limit.

1. Introduction

Cellular automata introduced by Wolfram [1-5] present us some basic and naive questions about the behavior of complex systems. A cellular automaton consists of a huge number of identical elements. Each element takes a few discrete states and evolves according to local interaction rules. Many works have been done to see how the cellular automata evolves as a whole and what are the characteristic behaviors. These works are classified into two groups. In the first group, global structure of finite cellular automata, in which the system consists of a finite number of elements, has been analyzed [6-10]. Since a cellular automaton defines a transition diagram between states of the system, the analysis of qualitative and quantitative properties of transition diagrams, as the study of the network system, gives important information about the global structure of cellular automata. On the other hand, the works in another group are concerned with the analysis of local properties in

*Present address: Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan.

cellular automata [1, 2, 10, 11, 12]. Since local interaction rules lead to creation of local patterns in evolution of cellular automata, the characteriza- tion of these patterns gives other aspects of cellular automata. Some complementary approaches to global structure have been introduced in random cellular automata [13]. Because of random connections between distant elements, local properties become meaningless in this system. Nevertheless, the randomness leads to smooth statistical properties in global structure. While cellular automata with local interaction rules show smooth and stiff local properties, global structure tends to show complex behaviors, that is, number theoretic complexity depending on the system size [6, 8]. There- fore the analysis of global structure becomes quite difficult. An exceptional example was introduced by Martin et al. [6]. The extensive analysis of global structure was possible because of the additivity property of evolution. It is called a linear

rule. In the present paper we give first examples of

the extensive analysis on global structure in cellular automata without the additivity property. It is called a non-linear rule. The present study is, however, considerably different from that of

0167-2789/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

H. Ito / Global structure in finite cellular automata 319

Martin et al. They analyzed the dependence of global structure on the system size N under a special rule. On the other hand, the relation of global structure under two different rules is the main interest of the present study.

The structure of the continuous phase space in dynamical systems, such as time evolution of non-linear equations with continuous variables, has been extensively studied because of its differ- entiability [17]. On the other hand, a cellular automaton has a discrete phase space. Instead of the analysis of differentiable functions, we must use algebraic methods to investigate the phase space in cellular automata. Thus the structure of the phase space in cellular automata is drastically different from that of the continuous phase space. It is an important problem to investigate in what manner local interaction rules determine the global structure of the discrete phase space in cellular automata. The answer to this problem will give an instructive draft when we make models of physical systems, using cellular automata, and control the global property of the system by local interaction rules.

The present paper is formulated as follows. The simplest cellular automata and global structure in these systems are introduced in section 2. Non- trivial similarities in global structure between two different rules are shown by numerical data. In section 3, exact correspondence of cycle structure in transition diagrams under the two rules is proved. The transformation of elements between the two transition diagrams is also explained. We discuss the correspondence of node structure in the two transition diagrams in section 4. Owing to the locality of evolution rules, we can find local connections of configurations between different elements. This property makes the quantitative analysis of node structure possible. As a result of this analysis we obtain the correspondence of basin volume and structure of transient trees. The origin of irreversibility in evolution in these systems becomes clear through sections 3 and 4. In section 5, the behavior of above correspondence is discussed in the thermodynamic limit, that is, when the

system size N = oo. We find that the correspondence becomes exact in this limit. The behavior of cellular automata with N = ~ has been considered by many authors [7, 11, 14]. However, they have treated local properties of the system. Sec- tion 6 gives the summary of the present study and some discussions of the results which are compared with other works. Some detailed calculations are given in appendices A-C.

2. Global structure in cellular automata

We restrict our attention to the simplest one- dimensional cellular automata introduced by Wolfram [1]. In this model, the system is composed of N lattice cells in one dimension. Each cell has one variable 0i (i = 1 . . . . . N) which takes either of the Boolean variables B = (0,1). Thus, the volume of the configuration space is 2 8 . To define a dynamical system on this configuration space, we introduce a local evolution rule F on each site. In the simplest cellular automata, the present state of each site and those of the nearest- neighbor sites determine the state in the next time step, that is,

~b~ t+t)= F(~Ox, ~ ' ) , ~!~1), (2.1)

where t represents the discrete time step. Since the three arguments of the rule function F take 23 different combinations, there are 28 different rules corresponding to the decimal numbers 0 to 255 [1]. We will use these numbers to represent the corresponding rules.

As mentioned in the introduction, there are two main approaches to investigate the properties of this dynamical system. In this paper, we study the global structure in this system. Every configuration in this system {~'i} (i = 1 . . . . . N) is an element of the set B N, with the volume V{B N } = 2 N. Thus a rule function F defines a self-mapping in B N. Every element a ~ B N has its image R~i°)(a) in B N, where id. represents a decimal code number of the corresponding evolution rule. This image

3 2 0 H. Ito / Global structure infinite cellular automata

can be obtained by a one-step time evolution according to the corresponding rule F synchro- nously on every site. We assume the periodic boundary condition for site values. Here let us define the set S Oa') by

s ~ i d ' ) = {(a,b)la~BN, b=R(id)(a)~BN},

(2.2)

and name this set the graph o/the mapping R 0d'). Any information on the global structure is contained in the graph S~ d). Thus the graph S~ d) determines the transition diagram which represents the successive mappings of each element. The transition diagram consists of two parts: cycles and transients. Every cycle has the transient tree

parts which fall into one of the elements in the cycle. Since the number of elements in B N is finite, evolution eventually falls into some cycle, starting from any configuration. The set of elements which eventually fall into a cycle is called the basin o/the cycle. The elements of the cycle itself belong to its basin. Of main interest in the global structure are structure of the cycles, transient trees and basin of each cycle.

In the present study, we investigate the global structure of the system under two special rules. One is Rule 18 ( F(0, 0,1) = F(1, 0, 0) --1, and 0 otherwise) and the other is Rule 126 (F(0 ,0 ,0 )= F(1,1,1) = 0, and 1 otherwise). Both rules belong to the type III class in Wolfram's classifications [1, 2]. The system shows complex chaotic evolution of patterns as shown in figs. l(a) and l(b). Some quantitative properties of the global structure un-

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Rule 126

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Fig. 2. Global structure of state transition diagrams under Rule 18 (a) (upper graphs) and that under Rule 126 (b) (lower graphs). Each graph shows values plotted as a function of the system size N, varying from 3 to 18. (i) Mean cycle length: average length of cycles weighted by their basin volumes. (ii) Maximum cycle length: longest cycle in the transition diagram. (in') Number of cycles: total number of distinct cycles in the transition diagram (oven in logarithmic form). (iv) Fraction of cycle elements: fraction of whole configuration elements (2 N) which compose cycles (in logarithmic form). (v) Complexity of basin structure: this quantity is obtained by - ~ i p i l n pi, where the summation is taken over all distinct cycles in the transition diagram and p~ is a ratio of the basin volume of each cycle to the whole configuration elements 2 N [8]. This quantity is very sensitive to the structure of cycles and basin volumes. (vi) Mean transient length: average length of transient time steps until falling into a cycle. (vii) Maximum transient length: maximum length of transient time steps in the transition diagram.

322 H. Ito / Global structure in f in i te cellular automata

Table I Lists of cycle compositions under Rule 18 and under Rule 126 in a system size N (N= 14,15,16). Li: period of each cycle, BVi: basin volume of each cycle, MT/: average length of transient before falling into each cycle (averaged over basin elements of each cycle), and ML~: multiplicity of each cycle.

Rule 18 Rule 126

N L i B V i M T i M L i L i B V i MT~ M L i

14

15

16

1 11960 7.321 1 1 11932 8.305 1 2 5 0.600 14 2 6 0.667 14 2 4 0.500 21 2 4 0.500 21

14 582 1.718 7 14 588 2.296 7 14 98 0.857 2 14 84 1.167 2

1 32448 9.280 1 1 32448 10.271 1 2 10 0.800 5 2 10 0.800 5 2 6 0.667 45 2 6 0.677 45

1 31460 10.795 1 1: 31560 11.813 1 2 11 0.818 16 2 10 0.800 16 2 8 0.750 24 2 8 0.750 24 2 2 0.000 2 2 4 0.500 2 6 170 1.082 4 6 180 1.256 4

14 1 9 6 8 1.917 8 14 1968 2.214 8 14 908 4.663 8 14 904 5.029 8 14 626 6.032 16 14 620 6.126 16

de r Rules 18 and 126 are shown in fig. 2(a) and

fig. 2(b), respect ively, for system sizes N = 3-18 .

In these figures, the mean cycle length is the

average of the lengths of all cycles weighted by

the i r bas in volumes. The mean t ransient length is

the average of the t ransient t ime steps before

fa l l ing in to a cycle. The elements of cycle itself

have a t rans ien t length zero. W e can see s t r iking

s imi lar i t ies be tween the two rules for the quant i -

t ies in fig. 2*. In fact, the m a x i m u m cycle length,

the n u m b e r of cycles and the f ract ion of cycle

e l emen t s a re exact ly the same for the two rules. To

see the s imilar i t ies more precisely, the deta i led

l ists of all cycles are presented in table I for

N = 14-16 . W e find that the compos i t ions of

cycles, pe r iods and mult ipl ic i t ies are exact ly the

same, and the bas in volumes of cor responding

cycles are qu i te similar. Concerning the t rans ient

par t s , we also find qui te a good cor respondence

*Some parts of the data in fig. 2 have been presented in table 14 of ref. [5]. The similarity between the two rules can be seen also there.

Table II The distribution of the transient length before the zero-state is reached in evolution from its basin elements in a system size N=16.

Rule 18 Rule 126

Transient Transient length Distribution length Distribution

0 1 0 1 1 2207 1 1 2 20 2 2206 3 32 3 8 4 208 4 32 5 272 5 224 6 720 6 256 7 2112 7 704 8 5488 8 2112 9 2944 9 5472

10 1344 10 2912 11 4048 11 1248 12 704 12 4096 13 656 13 800 14 3296 14 704 15 1664 15 3328 16 1440 16 1728 17 2544 17 1568 18 704 18 2528 19 544 19 608 20 512 20 512

21 512

Basin volume 31460 31560 Mean transient 10.795 11.813

length

be tween the two rules. Since under bo th Rules 18

a n d 126 the conf igurat ion of all zero sites (zero-

s ta te) { ~ i = 0 } is m a p p e d into itself, this s ta te

fo rms a cycle wi th the per iod 1 ( l imit point) . The

t ree s t ruc ture of this cycle is shown in table I I for

N = 16. The d is t r ibu t ions of the t ransient length

be fo re fa l l ing in to the zero-s ta te is shown for all

b a s i n e lements . In add i t ion to the s imilar i ty in the

b a s i n volumes, there is a non- t r iv ia l correspon-

dence in the d is t r ibu t ions of the t rans ient length.

W e can o b t a i n qui te a good cor respondence no t

on ly qua l i ta t ive ly bu t also quant i ta t ive ly if the

t r ans ien t l ength is increased by one unit for Rule

18. N o w there arise two quest ions: " W h y do these

p rope r t i e s show such a good cor respondence?"


Table III Global structure under Rules 18 and 126 in a system size N = 30. Uniform Monte Carlo samplings of 106 configurations for each rule. L i represents the period of the destination cycles. Pi is the fraction of samplings which fall into those cycles. The mean transient length to the zero-state is the average over its basin elements.

Rule 18 Rule 126

1 0.783 1 2 3.30 × 10 -5 2 4 9.00 × 10-5 4 6 1.35 × 10 - 4 6

10 7.04 × 10 -4 10 28 2.83 × 10 -3 28 3O 0.214 30

0.783 3.40 × 10-5 8.90 × 10 -5 1.25 X 10 -4 7.42 × 10 -4 2.88 × 10 -3 0.213

Mean cycle 7.281 7.261 length

Mean transient 17.642 18.591 length

Mean transient 20.544 21.536 length to zero-state

and "how does this correspondence evolve in the thermodynamic limit ( N = o0)?". To answer the latter, question, uniform Monte Carlo samplings are performed. Selecting a configuration element from the whole configuration set B N in a uni- formly random way, we evolve it until it falls into a cycle. The transient length and the period of the destination cycle are measured. The statistical data from 106 samples in the system size N = 30 are shown in table III. Since the volume of configuration space is 23°- 10 9, the fraction of these samplings is only 0.1% of the whole configuration space. However the data dearly shows the similarities between the two rules as in the cases of small size systems. It is probable that the non-trivial correspondence in the global structure under the two different rules does hold even in the thermodynamic limit.

3. Correspondence of cycle structure

In this section, we show the first key to reveal the non-trivial similarity in the global structure

between the two different rules. It is the correspondence of configuration elements between the two rules, which is derived from the following property:

For every element ( a, b) in the graph S~ 8), there always exists an uniquely corresponding element (a', b') in the graph S~ 26), where a' = T(a) and b '= T(b).

The function T(a) is a transformation operator of the configuration element a E B N under the following rules. Every local sequence of the successive 1 sites sandwiched in between the two 0 sites (such as 010 or 011.. . 110) in the configuration a should be transformed into a new sequence (such as 011 or 011.. . 111, respectively) by changing the site value 0 on the right-side into 1". It gives the configuration a ' . An example is given in fig. 6. The existence of this property can be examined through direct evolution of configurations according to the rules. Here it must be noted that the transformation T is neither a surjection (onto mapping) nor an injection (one-to-one mapping), so that the transformation T induces the contraction of state elements (e.g. seven different states with the length four: 1111, 1110, 1101, 1011, 0111, 1010 and 0101, are transformed into a unique state 1111 by the mapping T). Let C N be the range of the set BN by the transformation T, that is

CN= ( T( a) ~ BNla ~ B N }. (3.1)

The volume (number of elements) of the set C N will turn out to be V(CN}--(1/s)N---- (1.7549) N, (cf. V{B N } = 2N), where K is the single real root of the algebraic equation x 3 - x 2 + 2x - 1 = 0.

*Owing to the spatial symmetry in the evolution under the present rules, the whole discussion holds also when the new sequence is given by 110 or 111... 110, respectively, which is obtained by changing the left-side site value from 0 to 1.

324 H. Ito / Global structure infinite cellular automata

Furthermore we define a subset of S(N 126) by

S} O26) = { ( a ' , b ' ) • S~26)1a' = r ( a ) , b' = r ( b ) ,

~O8)} (3.2a) ( a , b) • ~'N ,

or equivalently

S / v ( I 2 0 = { ( a ' , b ' ) E s ~ Z 6 ) I a ' ~ C N } . (3.2b)

Because of the contraction of state elements by the mapping T, the number of elements contained in the graph S~ O26) is quite small compared with that of the whole elements in the graph S~ 26) (i.e., V{ C N } a: V{ B N } where N >> 1). Since no element (a ' , b ' ) with b' ~ CN is contained in S~ O26), where the set C N is the complementary set of C N (i.e., CN -- B N -- CN), the elements in S~ 020 con- struct the self-contained state transition diagram within the whole elements in the set C N. Let us name this state transition diagram, constructed by the elements in S~ (120, the skeleton diagram in the global structure under Rule 126. The transformation T shows that C N is the set of configurations which contain isolated 1 sites such as 010 in their sequences. Besides, it can be shown that the time evolution according to Rule 126 cannot create configurations which contain the isolated 1 sites; that is, these configurations are unreachable by the time evolution under Rule 126. Thus the remaining elements which are not contained in the skeleton diagram evolve to an element in the skeleton diagram by only a one-step time evolution. These elements are the leaves attached to the skeleton diagram. Therefore any cycle in the transition diagram under Rule 126 should be contained in the

skeleton diagram. Every cycle in the transition diagram under Rule 18 is composed of the set of elements in S °s) such as ( ( a , b ) , ( b , c), ( c , d ) . . . . . (*, a)} and by the mapping T it is transformed into the set of elements in S~ °26), { ( a ' , b ' ) , ( b ' , c ' ) , ( c ' , d ' ) . . . . . (* ', a ')}, which also constructs the corresponding cycle in the skeleton diagram under Rule 126. Here the one-to-one correspondence of cycles in the global structure

between the two rules can be derived by this relation. However one would wonder that the unfavorable breakdown might happen for the one- to-one correspondence because of the contraction of elements by the mapping T. For example, two distinct cycles in the transition diagram under Rule 18 might be transformed into the same cycle. The transformed cycle might split into small cycles because two distinct elements in the cycle might be transformed into the same element, or new cycles might be created because of the appearance of the same elements in the distinct parts by the transformation T. Nevertheless, it will be shown that these unfavorable modifications never happen, so that the one-to-one correspondence of the cycle compositions between Rule 18 and Rule 126 can be proved exactly.

Now, we concentrate our attention on the problem: in what manner does the contraction of state elements occur by the mapping T? First, let us define another subset of BN:

D N = { a • B Nlthe configuration of the element a

contains no sequence of isolated 0

such as 101}. (3.3)

The volume V{ D N } is equal to 1I{ C N }, since the set C N consists of all the elements which contain no sequence of isolated 1. Configuration of every element a • D N is composed of blocks. Each block is the sequence of 1 sites which is sandwiched in between the two 0 sites (such as 010 or 011.. . 110). There may be successive 0 sites between them*. An example is given by the element a in fig. 6. For every element a • D N, the variations of a are obtained by the following procedures: in each block in the configuration a, the values on some sites may be changed into 0, if it does not give rise to two or more successive 0 sites. For example, a block 0111110 has an allowed variation 0101010 but 0100110 and 0111100 are not allowed. For

*Note that both the configuration of all 0 rites (zero-state) and that of all 1 sites are contained in D N.


every dement a ~ DN, the set of dements G[a] is defined as

G[ a] = { a ~ BNla is one of the variations of a },

a ~ D N. (3.4)

Since every element a E B N is contained in the class G[a], where the dement a ~ D N is obtained by changing all isolated 0 sites in the configuration a into 1, the following decomposition is derived:

G[a,]

2 = R(18)( o(, ) R(126)(#,.') = T( ~ )

Fig. 3. Contraction of state elements by the mapping T. Whole elements in G[a] (a ~ DN, the element a is indicated by an open circle) evolve to t~ by a one-step time evolution under Rule 18; furthermore, by the mapping T, these elements are transformed into a unique element a ' in the skeleton diagram under Rule 126.

B E (3.5) aED N

The dements in the class G[a] have the following two special properties. First, all elements in G[a] are transformed into the dement a ' by the mapping T, where or'-- T(a). It should be noted that the restriction of the mapping T within the set D N leads to a bijection mapping between D N and C N. And secondly, all dements in G[a] are transformed into the dement a by a one-step time evolution according to Rule 18, where t~= R(XS)(a). This is because any sequence of symbols 0 and 1 not containing two or more successive 0 sites evolves to the sequence of all 0 sites by a one-step time evolution under Rule 18".

In order to clarify the nature of state contractions by the mapping T more definitely, schematic drawings are depicted in fig. 3. Finally, since the contractions of dements by the mapping T occur in this manner, the unfavorable modifications of cycle compositions discussed previously never happen. Therefore, the one-to-one correspondence of cycle structure between Rule 18 and Rule 126 has been proved exactly in any system size N.

*These appearances of successive 0 sites correspond to the appearance of many triangle patterns in the time evolution of the configuration pattern as shown in fig. l(a). Further this contraction of state elements in the time evolutions causes the irreversible nature of the evolution in cellular automata.

4. Correspondence of node structure

In this section, we investigate why both the transient tree structure and the basin volume of each cycle show such a good correspondence between Rules 18 and 126.

In the transition diagram, every reachable dement has some incoming states and one outgoing state. Thus such dements form nodes in the transition diagram. We first show the correspondence of node structure in the transition diagrams between the two rules. First, let lOS)(&) be the set of pre-images of a state & ~ B N under Rule 18. It is composed of such elements which evolve to the state 5 after a one-step time evolution. In order to investigate the quantitative properties of the basin volume or the transient tree structure, we need to know how many pre-images every element has. In other words, how many states come into every node in the transition diagram?

First, Rule 18 is considered. Some authors [6, 7] have shown that configurations must satisfy some conditions in order that they are reachable, having pre-images. A reachable configuration must not contain the sequence of site values 111, because the time evolution under Rule 18 never creates such a local sequence of site values. Furthermore, any reachable configuration ~ ~ B N belong to either of the following two cases because of some parity restrictions.

i) Case 1: In the configuration t~, the sequence 11 does not appear, the 1 sites being isolated, and

326 H. Ito / Global structure in finite cellular automata

there are even number of 1 sites. Noting that the 1 site is shifted by a one-step time evolution, we can easily find two different pre-images of 5 within the set D N. We denote them by a t and a2, an example being given in fig. 4(a). In fig. 4(a), lines

Case 1

/ \ 3/ \ / / \ 1 o~1-" 0 , • , 0 0 1 1 , 1 1 0 0 0 1 1 0 0 , , , 0 0 . 1 0 0 - • , 0 0 1 1 -1 0 0 0 I \ / ~ i x / \

, . .

tioned in the previous section, every configuration element a ~ D N is composed of some block sequences 1411, W2,..., W k (k > 0), where each W/ is the local sequence of I~ successive 1 sites sandwiched in between the two 0 sites (4 > 1), and there may be successive 0 sites between them. From the definition of the variations of a, it can be easily found that V(G[a]} is obtained by

V{ G[a]} = I-I $2t,, (4.1) i = 1

Fig. 4(a).

are drawn in order to clarify the fact that the shift-movements of site values 1, which locate at the left-side and fight-side boundaries of each block sequence of 1 sites in the pre-images, completely determine two distinct pre-images of the state ~ which are the elements of DN*. Further- more, as shown in the previous section, each variation of a t and a 2 also evolves to the same state &. The set I(lS)(~) consists of two classes G[at] and G[a2].

ii) Case 2: In the configuration ~, the sequence 11 may appear, but between any two neighboring 11 sequences there should be even number of 1 sites. Since the local sequence 0110 in the configuration of t~ is always created from the sequence 1001 in the pre-images, the configuration fi has only one pre-image within the set DN. Let it be denoted by a. An example is given in fig. 4(b). Therefore the set of pre-images of the configuration & is given by the single class G[a].

where the quantity $2L (L > 1) is the number of strings made of L symbols of 0 or 1, without 00 sequence, in which both the first and the Lth symbols are 1. In order to calculate $2L, we introduce a subsidiary quantity ilL, which is the number of such strings with the length L that the first symbol is 1, the 00 sequence is not found in the first L - 2 symbols and the last three should be 001. Then the following relations are obtained:

$2L+1 + flL+X = 252t., (4.2)

/~L+a +/~L+2 = $2L (L >_ 3),

with $21 = $22 ~- 1, $23 = 2, $24 = 3, 13 3 = 0, and f14 - ~ 1 .

Introducing the generating functions [6] by

oo

A ( , ) = E $2m+: m, r a m 0

oo

B ( Z ) = E tim+3 Zm" m--O

(4.3)

Case 2

: o. . .oo , / , . , ,oo:( . . . ,,oooo, ,o : o . . . o,{oo, oo%1 oo. oo 1,o,~ oo. oo I,t oo. oo 1,oo: ool,

/ A X /k v \

Fig. 400).

Next, let us investigate the volume V{ G[a]}, the number of elements in the class G[a]. As men-

We rewrite eq. (4.2) to obtain

2 A ( z ) - ½A(z ) + 1 B ( z ) 2 - z '

(1 5) 1

which lead to

(4.4)

*The configuration of all 0 site values (zero-state) has two pre-images within DN, that is, the configuration of all 1 site values and itself.

= - ( z + 2) z 2 + z - - 1 "

(4.5)


Thus we get

1 1 L ], (4.6)

where 7+ = ( - 1 + v~)/2 and 7_ = ( - 1 - ~/3-)/2 are the roots of the equation x 2 + x - 1 = 0 and the first ten values of 12 z (L > 1) are 1, 1, 2, 3, 5, 8, 13, 21, 34 and 55. It is the Fibonacci sequence. Now V(G[a]} can be obtained explicitly, using eqs. (4.1) and (4.6).

At every node element & in the transition diagram under Rule 18, ios)(&) is composed either of a class G[a] or of two classes G[al] and G[a2], depending on if there are the site values 11 or not in the configuration a. By the mapping T all elements of each class G[a] are transformed into the unique element T(a) in the skeleton diagram under Rule 126. We will next ask the question: " In the complete transition diagram under Rule 126, how many pre-images does the state T(a) have?". The relation of node structure between the two transition diagrams are schematically depicted in fig. 5, where I020[T(a)] means the set of pre- images of the state T(a) under Rule 126. It must be noted that in fig. 5 only the local parts at the node a in the transition diagram are depicted,

(a) >

T(;Z)

(b)

I(126}[I(oc)]

I(12S)CT(~1)] I(126~T(~.2)] G[~I] G[o~ 2]

~ ~ T(w"I) T(ot,2) > ' ~ 2 T

Fig. 5. Correspondence of node structure between the state transition diagram under Rule 18 (left) and that under Rule 126 (right). Structure of the skeleton diagram under Rule 126 is also shown (middle). The set of pre-images of ,~ under Rule 18 is composed of either one class (a) or two classes (b) depending on the appearance of the site values 11 in the configuration &

thus the elements in the class G[a] are not neces- sarily at the periphery of the transition diagram. Every node in the transition diagram has the property shown in fig. 5.

We now show how the number of pre-images of T(a), V( I(t20[T(a)]}, can be calculated. As mentioned above, every element in the class G[a], a e D~, is transformed into the unique configuration T(a) through the mapping 7'. According to the transformation rule of T, each block sequence W~ in the configuration a is transformed into a new local sequence in the configuration T(a) by changing the 0 site in the right-side in W~ into 1. The new local sequence contains l i + 1 successive 1 sites as shown in fig. 6.

d, :

T(~)"

w~ w 2 w 3 . . . w,_, w,

7 r 11"'7 1 I I II 0 0 . . - 0 0 1 . . . 1 0 0 1 0 0 0 1 . o . 100 . . . . 0 0 1 1 0 0 1 o . . 1

] I If'] I I I II IO...ooi.-. 11o110oi,.. 11o .... ooi 11oi... I

.04"1 9.2+I ~.3~-1 • • . .~k_1+1 ~-k+t

Fig. 6.

Here, we make some preliminary calculations for a later convenience. First, let us estimate the number of pre-images which evolve to the strings with length L + 2 by a one-step time evolution under Rule 126. The strings are composed of L successive 1 sites sandwiched in between two 0 sites (L >__ 2). Periodic boundary condition is not assumed, the pre-images being the strings with length L + 4. As shown in fig. 7, we denote the first and last three site sequences of the pre-images by C 1 and (72, respectively. The Rule 126 is given by F(1,1,1) = F(O, O, O) = O, and 1 otherwise. The set of C1 and C2 must be one of the following four

C~ C2 Fn7F- - - ln -n 011..,110

L Fig. 7.

328 H. Ito / Global structure infinite cellular automata

cases. Case I: (C1, C z ) = ( l l l , l l l ) , case II: (111,000), case III: (000,111), and case IV: (000,000). Let F~ I) and /,~n) be the numbers of pre-images with length L + 4 (L > 2) in the cases I and II, respectively. Similar definitions may also be introduced /-~in) and F~ Iv), however, we need not consider the cases III and IV, because of a special property in the evolution under Rule 126. It is called conjugation property: if ( a, b) ~ S~ 26), then also (C(a), b)~ S~ 26), where C(a) is the conjugation of a, which is obtained by changing the value on every site in the configuration a into its conjugation value (i.e., 1 ~, 0). Thus it is shown that

relations are obtained:

F ( I ) _ /-,(11) = _ _ _ 2 sin [ 32~(L - 1) ]

- 1 , f o r L - 3 m + 2 , 0, f o r L - - - 3 m + l ,

+1, f o r L = 3m, m = 0 , 1 , . . . ,

and

(4.10)

r (u ) (4.11)

- ( i n ) = F ( I I ) ,

=

(4.7)

After some straightforward calculations, which are given in appendix A, the quantity/,~i) turns out to be

1 V~- s i n [ Z ~ ( L - 1)] ( L ~ 2 ) . (4.8)

The first ten values for L > 2 are 0, 1, 1, 1, 3, 4, 6, 11, 17 and 27. The other quantity F~ n) is given by

Now, let us return to the original problem of the number of pre-images of T(a) under Rule 126, v(ItI26)[T(a)]}. As shown in fig. 6, the configuration T(a) is composed of k block sequences. Each block consists of li + 1 successive 1 sites (4 > 1, i = 1 . . . . . k). There are one or more 0 sites between them. Now, we can consider new blocks, each block consisting of one or more 0 sites sandwiched in between two 1 sites. In the configurations of the pre-images of T(a), the local block sequences corresponding to these new blocks in T(a) are indicated by boxes in fig. 8. They must

C~ C~ C 3 C~...Ck_~ C k i I I - - I t I I ~ - " -CE3 I ~

....

~1+1 J[2÷I J[3+1 • • • ~.k_~+l ~k+l

Fig. 8.

1 sin (L>_2). (4.9)

The first ten values for L > 2 are 1, 0, 1, 2, 2, 4, 7, 10, 17 and 28, where -f+ and ~,_ are the roots of the equation x2 + x - 1 = 0. From the above ex- pressions for Fz O) and/-~ii), the following valuable

be either all 0 sites or all 1 sites. We label these block sequences as Ci (i = 1 . . . . . k) as shown in fig. 8. First we consider the case in which the block sequence C1 consists of all 1 sites. If the block sequence C2 also consists of all 1 sites, the total number of local sequences between C 1 and C 2 i s F~I)+I (case I). If C2 consists of all 0 sites, it is given by Fff+I)x (case II). The periodic boundary condition for site values in the configuration excludes the possibility that the total number of local sequences with F OI) and F era) is

H. Ito / Global structure infinite cellular automata 329

odd. Using the relation (4.7), we finally obtain

V(I<126) [T(a)] } k

= r I : r : ' ) + r,f'+'l) t 4 +1 i ~ l , z

k + H (/~(I) __ r ,(II)] (4.12)

i _ l k li+1 ~li+i ]"

Here, it is taken into account that C 1 can consist of all 0 sites. Eq. (4.11) shows that the first term is equal to the volume V(G[a]} which is given by eq. (4.1). Furthermore, eq. (4.10) shows that the difference/'1~1 - F~I+~l takes one of the three values - 1 , 0 , +1. The second term in eq. (4.12) also takes only these values*.

Now, we summarize the discussion in this section. At every node (state) ti ~ B N in the transition diagram under Rule 18, the set of incoming states (pre-images) to ti consists of one class G[a] or two classes G[al] and G[a2] ( a, al, a 2 ~ D ,v ) as shown in fig. 5. The mapping T transforms all elements in each class G[a] into an unique element T(a) in the skeleton diagram under Rule 126. The number of pre-images to the element T(a) in the transition diagram under Rule 126, V{I(126)[T(et)]}, is almost equal to the volume V{G[a]}. The difference is at most 1.

In figs. 9(a) and 9(c), two transition diagrams containing the zero-state (0 ~¢) under Rule 18 and under Rule 126 are depicted, respectively. The skeleton diagram under Rule 126 is shown in fig. 9(b) to clarify the correspondence between two

*It must be noted that the two configurations of all 0 site values (0 N) (zero-state) and all 1 site values (1 N) need special considerations. In the transition diagram under Rule 18, the configuration of all 0 sites has two sets of pre-images. One is the class G[(1N)], which is the set of variations of (1N). The other is the element (0 N) itself. Taking account of the periodic boundary condition, we can show that V{G[(1N)]} is obtained by 12 N + 212N_ 1 z ( l / y+ ) N + (1/.t_) N. By the mapping T, all elements in G[(1N)] are transformed into the state (1 N) in the skeleton diagram under Rule 126, and the element (0 N) is transformed into the state (0N). It can be also shown that in the transition diagram under Rule 126, V{I(126)(1N)} is given by 2(/~I)+3 - F(NI)I) z (1/'y+) N + ( l / 'y_) N + 2 cos(2~rN/3). Therefore we find that the difference between the two quantifies V{G[(1N)]} and V{I(126)[(1N)]} is at most 1 in this case, too.

transition diagrams. The quantitative properties of these transition diagrams are listed in table IV. As a result of the investigations in this section, we can understand the correspondence of the global structure between the two transition diagrams:

i) The numbers of basin elements falling into the zero-state (O N) show quite a good correspondence for the two transition diagrams. The difference (136- 132--4) is less than the number of distinct classes G[a] (37 in this case) contained in this transition diagram.

ii) From fig. 9 and table IV, it is known that the two distributions of the transient lengths of basin elements falling into the zero-state (0 ~') show quite a similar structure under the two rules not only qualitatively but also quantitatively, if the length is shifted by one unit for Rule 18. Thus the mean transient length under Rule 126 is about one unit larger than that under Rule 18.

Such similarity between the basin volumes always holds for other cycles with a longer period in the transition diagram. The transient length is defined as the length of time-steps from the basin element to an element in a cycle. For these cycles, the correspondence of node structure between the two rules cannot reflect the similarities of the distribution of transient lengths or the mean transient length so clearly. As shown in fig. 10, the mean transient length to the element T(t~) in a cycle under Rule 126 is the average over the elements which reach T(~) in their time-evolution. It is always about one unit length larger than that to the element ti under Rule 18. However, since T(a), which is the pre-image of T(ti), is also contained in the same cycle, the mean transient length to the cycle does not increase. Thus such quantities as the distribution of the transient length and the mean transient length, which are statistically averaged over the whole cycle compositions in the transition diagram, have dominantly clear correspondence between the two rules in the system with size N, in which the cycle of the zero-state (0 N) has a large basin volume which effectively determines the statistical quantities. It can be seen in table III for N = 30.


(a) Rule 18 (b)

(c) Rute 12(3

Fig. 9. State transition diagram containing the zero-state limit point under Rule 18 (a) and that under Rule 126 (c) in the system of size N = 8. The corresponding skeleton diagram under Rule 126 is also shown (b) to clarify the correspondence of the two transition diagrams. In the diagrams, circles and squares are used to distinguish two classes of pre-images at the nodes, and open symbols in the diagram (a) represent the element in the set DN. Every element evolves toward the center except for the transitions indicated by arrows. The quantitative properties of these diagrams are hsted in table IV.

5. Thermodynamic limit

In this section, we study the effect of the system size N on the correspondence of global structure

between Rule 18 and Rule 126. I t is impor tant to examine the thermodynamic limit N ---} oo in order

to discuss the statistical properties of cellular au-

tomata .

First we estimate the N-dependence of some quantifies. As discussed in section 4, the reachable configurations t~ e B N, by the time evolution under Rule 18, must satisfy some conditions. The total number of these reachable configurations is denoted by PN- For every reachable configuration

4, the set of pre-images is composed of only one

class G[a], if the configuration t~ contains the local


Table IV The distributions of the transient lengths before the zero-state is reached, in evolution from the basin elements in a system with size N ffi 8. The corresponding transition diagrams are shown in fig. 9.

Rule 18 Rule 126

Transient Transient length Distribution length Distribution

to the number of elements contained in the skeleton diagram under Rule 126, i.e., V{CN}. Since every pre-image fl ~ DN, which evolves to the configuration without local sequence 0110 under Rule 18, contains no local sequence 1001 in its configuration, the total number of these elements, qN, estimated in appendix B as

0 1 0 1 1 47 1 1 2 4 2 46 3 16 3 8 4 48 4 16 5 16 5 48

6 16

Basin volume 132 136

Mean transient 2.841 3.801 length

qN --- (1/~,+) N, (5.2)

where ~+ = ( - 1 + ¢~-)/2 is the larger root of the equation x 2 + x - 1 = 0. Noting the relation among the quantities estimated above, it is found that

ON = V{ D N } - qN/2. (5.3)

T

0c] T(oc) .~/T(oc) d

I"26)[T(~)] Fig. 10. Correspondence of node structure on a cycle between the state transition diagram under Rule 18 (left) and that under Rule 126 (right). Structure of the corresponding skeleton diagram under Rule 126 is also shown (middle).

sequence 0110. The element a ~ D N is uniquely determined by the state 4. If & does not contain the local sequence 0110, the set is composed of two classes G[ax] and G[a2]. The total number of the distinct classes G[a] in the whole transition diagram under Rule 18 is nothing but V{DN). Since the set D N is composed of all elements which contain no isolated 0 site, the volume V{ D N } can be estimated in a similar way as for I2 L. It is shown in appendix B that

V ( D N } =--- (1/x)N_--__ (1.7549) N, (5.1)

where ~ (--- 0.5698) is the real root of the equation x 3 - x 2 + 2x - 1 = 0. The volume V{DN} is equal

This agrees with the result obtained by Martin et al. [6] by a different calculation*. Therefore the relative ratio of the number of the reachable configurations with pre-images of two classes to the total number of reachable configurations becomes (qN/2)/ON ---- ( r /Y+) N ------ (0.9220) N. Thus for almost all reachable configurations, the set of pre- images is composed of only one class, in the large N limit.

As shown in the previous sections, every configuration element a ~ D N is composed of block sequences Wa, W 2 . . . . . W k (k > 0), where each W i is the local sequence of l~ successive 1 sites sandwiched in between two 0 sites (/~ _> 1). There may be successive 0 sites between them. Here we will consider the average number of block sequences, K ( N ) , contained in the configurations of elements in Du in the large N limit. After some combinatorial calculations in appendix C, it is found that

K ( N ) = yN, (5.4)

where ,/ (=-0.1770) is the single real root of the equation 23x 3 - 23x 2 + 9x - 1 = 0.

With the help of the above estimations, we obtain two main features of the correspondence in

*The final expression (5.8) for ON was in error in their paper.


the global structure between the two rules in the thermodynamic limit (N --¢ o0):

i) As the system size N increases, the volume of the set D N increases as V(DN) - - ( l / r ) N. On the other hand, the volume of the whole configuration space B N increases as V(B N } = 2 N. Besides, every element a ~ B N is contained in some class G[a] with a ~ D N. The average number of ele- m e n t s in one class is es t imated as V ( B N } / V { D N } --- (2/(I/K)) N --- (1.1397) 8. As discussed in the previous sections, at every node in the transition diagram, the difference A V - V{ l°26)[T( a)]} - V{G[a]} is at most 1 irrespec- tive of the system size N, therefore for each class G[a] (a ~ DN) the relative ratio of the error AV to the total number of elements in the class G[a] decreases faster than 1/(1.1397) N= (0.8774) N. In other words, even if AV were always supposed to be 1 at every nodes, the total number of errors would be at most (l/K) N. Its relative ratio to the volume of the whole configuration space decreases exponentially, as the system size N increases, as (1/~ )N / 2 N ----- (0.8774) N *.

ii) As shown in eq. (4.12), for every class G[a] (a ~DN), the difference AV--V(IO26)[T(a)]}- V{G[a]} is given by the product of k values (F (I) - F (tI) x with I i > 1 (i = 1, k), where k is 1i+1 / i+ l J " " , the number of block sequences { W~ } contained in the configuration a. Besides, each value rE(I) - X. I~+1

F (II) ~ takes three different values ( - 1, 0, or + 1), li+11 as shown in eq. (4.10). If N is large, the average number of block sequences in the configuration of the element in D N is given by K(N) = ,/N. Thus the difference AV is given by the product of K ( N )

*We must be careful about the possibility that, as N increases, only the cycles with a small basin volume would increase exponentially. Then the average basin volume would not increase. In this case, the easy statistical treatment for each cycle is not justified. However, this never happens in the real system, because the elements of cycles under Rule 126 are contained in the skeleton diagram, which consists of ( l /K) N dements. Since the total number of distinct cycles in the transition diagram is not greater than ( l / r ) N, the average number of basin volume for each cycle increases faster than 2N/K N = (1.1397) N. According to the data under small N, the number of cycles seems to increase very slowly as shown in fig. 2.

factors. If one of them is zero, the product (i.e., AV) vanishes. If each K'(N) factor takes one of the three values ( - 1, 0, + 1) with the equal probability 1, the probability to get the non-zero AV is (2/3) g(N) -=- (2/3) vN = (0.9307) 8. The equal probability is considered to be a reasonable assumption in the large N limit. Therefore in the large N limit, for almost all class G[a] (a ~ DN), the difference between the two volumes V(G[a]} and V(I026)[T(a)]) is zero. It leads to the exact corre- spondences in the global structure between the two rules. From these two properties, we conclude that in the thermodynamic limit (N = oo) the correspondence in the global structure between Rules 18 and 126 can be obtained exactly by the structure of the skeleton diagram and the local relation V{G[et]) = V{I(126)[T(a)]}, (et E DN) , at every node in the transition diagram. It is rather astonishing that the whole structure of the transition diagram is constructed by these simple structures, since the skeleton diagram consists of a small number of elements compared with the volume of the whole configuration space.

6. Conclusion and discussions

The present analysis of the simplest one-dimensional cellular automata has shown that there is a correspondence in global structure between the systems under two different rules, Rule 18 and Rule 126. Through this analysis, we have investigated how local interaction rules determine the global structure of the discrete phase space in these systems. We have found that the transition diagram under Rule 18 yields the main structure of the transition diagram under Rule 126, which is named skeleton diagram, by a transformation of configurations. This property leads to the exact one-to-one correspondence of cycle structure between the two transition diagrams. Furthermore the quantitative correspondence of node structure in the transition diagrams has been presented through combinatorial analyses. This local node relation gives rise to other properties of the corre-

H. lto / Global structure in finite cellular automata 333

spondence in global structure. The corresponding cycles in the two transition diagrams have almost the same basin volumes. Thus the mean cycle length, averaged with the weight of each basin volume, takes almost the same value in the two transition diagrams. The structure of transient trees which fall into the zero-state (limit poin0 also shows quite a good correspondence not only qualitatively but also quantitatively. For the transient trees which fall into cycles with longer peri- ods, however, the correspondence of node structure cannot reflect the similarity of this structure so clearly. The statistically averaged quantities over the whole cycle compositions, such as the distribution of the transient length and the mean transient length, show dominantly clear correspondence, under the system with size N in which the cycle of the zero-state has a large basin volume. Moreover we have shown that as the system size N increases, the above correspondence becomes more complete, that is, the ratio of discrepancy in the correspondence between two transition diagrams to the whole configuration space decreases exponentially. Therefore in the thermodynamic limit (N = oo), the correspondence becomes exact.

It seems worthwhile to relate an approach to the local properties of cellular automata shown by Grassberger [11]. He found that the local properties of the system under both Rule 18 and Rule 126 can be understood by considering two ground states for each rule*. These ground states simulate the evolution according to the additive rule, Rule 90. In the evolution from disordered initial configurations, there exist many kinks which divide the system into domains of the ground states.

*For the two rules, an exact correspondence between the special configurations is shown in table 7 of ref. [5]. That is, any configuration composed of 00's and 10's for Rule 18 yields the corresponding one in Rule 126 by transforming every 10 block into 11. One can find that such a correspondence is supported by the transformation T in our discussion. However, the above correspondence holds only when the system size N is even, and the volume of such special configurations (2 'v/2) is quite small compared with that of the whole configuration space (2N). It must be noted that the correspondence has been discussed for the whole configuration space in this paper.

Because kinks perform some random-walk-like diffusion and the pair annihilation, the local properties in the limit of long times can be well explained by those of the ground states, whose fraction of volume in the whole configuration space is quite small. Using this property, we can show a correspondence between the two rules in the local properties in the limit of long times. Moreover, as shown in the previous sections, the elements, not contained in the skeleton diagram, are the unreachable configurations by the time evolution according to Rule 126. Therefore, the structure of the skeleton diagram has sufficient information as to the study of the local properties in the limit of long times. However, we must know every state transition between all elements to understand the global structure, especially, the basin volume of each cycle and transient tree structure, because every element plays a role at least as an element of the basin of some cycle and almost all elements are not contained in the skeleton diagram.

As shown in the previous sections, the existence of the skeleton diagram under Rule 126 plays a crucial role to explain the correspondence in global structure between the two rules. The skeleton diagram determines the structure of the main stream in the time evolution under Rule 126. This structure completely determines how the 2-dimensional spatio-temporal spaces are tiled by triangular patterns as shown in Fig. 1 *. The other elements which are not contained in the skeleton diagram join the main stream after a one-step time evolution. These elements have local configurations which are absorbed into the base of triangular patterns. It is rather surprising to find that the evolution of the system is determined by such a simple structure. In time evolution of cellular automata, some different elements evolve to the same element. This property gives rise to the irreversible

*In fig. l(b), the initial configuration in the first line is arranged so that this is the transformation of the initial configuration in fig. l(a) by the mapping T. We can find that the successive evolution patterns, the filings of the triangular patterns, correspond to each other in these two figures.


nature of the system. We have shown that the numbers of pre-images of corresponding elements (nodes) are exactly the same, i.e., V(G[a]} = V{It126)[T(a)]}, in the transition diagrams under Rules 18 and 126 in the thermodynamic limit ( N = oo). Therefore we conclude that the two systems have exactly the same irreversible nature in this limit.

Next we relate the present study to the other non-linear rules which belong to the class III in Wolfram's classification [1, 2]. Three rules, Rules 18, 146 and 182, show considerably similar properties in the local properties as well as in the global structure, because of simple relations among these local evolution rules [12]. Moreover two rules, Rules 122 and 126, show similar properties as well. The similarity in the local properties between Rules 18 and 126 was shown by Grassberger [11]. The similarity in the global structure has been shown in this study. Thus we know that the above five different rules show similar properties*.

It would be rather difficult to apply the present arguments in a straightforward way to other systems which may have other boundary conditions or be under other rules. However some parts of the arguments may be applicable.

Finally we discuss the remaining problem and the future extension. In the present study the global structure in finite cellular automata has been analyzed. The global structure in cellular automata under local evolution rules shows the irregular and complex dependence on the system size N [6, 8]. It is very difficult to formulate the statistical principles in these systems. On the other hand, local properties such as density of 1 valued sites, size distribution of triangular patterns and various entropies, etc., show quite smooth and stiff statistical properties which are peculiar to each rule (self-organization [1, 2]). This property re- flects the absence of long-range correlations and a

uniformity in the system due to local rules. These two properties are the different aspects of the same system. Therefore the relation between the global structure and the local property under local rules gives some important problems: for example, "Can we know the character of global structure from the knowledge about the local property?" or "Can we control the global structure and the local property by varying local evolution rules?" The answers to these problems will give an instructive draft when we make models of physical systems using cellular automata. We need to acquire more empirical laws in order to solve the problems.

Note added

I have become aware that the other type of correspondence between Rule 18 and Rule 126 has been introduced in ref. [18]. That is, every element (a, b) ~ S~ 26) yields the corresponding one (a ' , b') ~ S~ s), where a ' = f ( a ) and b' = f(b). The function f is a transformation operator of the configuration elements. The details are explained in ref. [18]. In section 3, we have found that the elements in the graph S~ s) are transformed into those in S(N 126) by the mapping T. On the other hand, the function f yields a part of the graph S~ s) from S~ 26). The transformation f also induces the contraction of state elements. It has not been investigated yet whether such an extensive analysis as the present study can be possible by using the transformation f. I am grateful to P. Grassberger for letting me know about ref. [18].

Recently an interesting work on the local properties of cellular automata was presented by Gutowitz et al. [19]. Such an approach is considered to be complementary to our discussion of the global structure.

*This similarity can be seen in tables in ref. [5]. The remaining non-linear rules in the simplest cellular automata, not discussed here, are Rules 22 and 54. Recent analysis by Grassberger [15, 16] showed the complex nature in the system under Rule 22.

Acknowledgements

I am grateful to Professor Y. Wada for useful discussions and a careful reading of the

H. Ito / Global structure infinite cellular automata 335

manuscript. I also express my sincere thanks to T. Ikegami for stimulating discussions. Numerical calculations were carried out on FACOM M-190 of LICEPP in the University of Tokyo. This work was partially supported by the Japan Society for the Promotion of Science and a Grant-in-Aid for Scientific Research from the Ministry of Educa- tion, Science and Culture.

Appendix A

Derivation of FL (i) and F~ II)

From fig. 7 with (C~, C2)--(111,111), it can be found that the number F~ I) (L > 2) is given by the number of strings of L symbols 0 or 1 in which the first and last two sites are 10 and 01, respectively. There should not be sequences 000 or sequences 111. We introduce two subsidiary sequences, h L is the number of strings with length L which begin with 10 and contain neither 000 nor 111 in their first L - 3 positions, and terminate with 11101. #t. is the number of strings with length L which begin with 10 and contain neither 000 nor 111 in their first L - 2 positions, and terminate with 10001. We find the relations

F (I) + XL+ 1 + 2FL (I) + ~L= L+I ~L+I,

FL (I) + #L = ~L+I q- )kL+2 "q- ~kL+3,

F(I) = ~L+I q- ~tL+2 (L > 6).

(A.1)

These equations can be solved by introducing the generating functions

Eq. (A.1) gives

2A(z) + B(z) = I (A(z ) + B(z) + C ( z ) - 3),

1 1 = +-j+ C ( z ) 7 z 2,

(1 z_~) 1 A ( 2 ) = 7 + S ( z ) z

(A.3)

We solve them to obtain

-- (23 + 322 -I- 4z + 3) A(z) = (z 2 + z _ 1)(z 2 + z + 1)" (A.4)

Substitution of (A.4) into (A.2) gives

FL(I) ~ 2~5 [ ( ~ ) L-1 -- ( 1--~_ ) L-I ]

1 Vrj-sin [ ~ r ( L - 1)] ( L > 2), (A.5)

where y+ = ( - 1 + ~/5-)/2 and y_ = ( - 1 - ¢3-)/2 are the two roots of x2+ x - 1 = 0. The first ten values for F(L t) (L > 2) are 0, 1, 1, 1, 3, 4, 6, 11, 17 and 27.

When (Ct, C2) = (111,000), case II in fig. 7, the quantity F~ u) can be given by the number of strings of L symbols in which both the first and the last two sites are 10. There should be no sequences 000 or 111. Similar calculations lead to

oo A(2) = E p(I) ~,m ~m+6 ~ ,

m~0

B(z) = )'-', #,.+6z m, (A.2) m--0

r(ii) ~_~

+'~-1 sin [3z~r(L- 1)] (L_> 2), (A.6)

o o

c(2)= E x.+6:. m--0

where the first ten values for F~ u) (L > 2) are 1, 0, 1, 2, 2, 4, 7, 10, 17 and 28.


Appendix B

Quantities in the large N limit

I) First we calculate aN, which is the number of strings with length N in which there is no sequence 101 and the periodic boundary condition is not assumed. We readily obtain

- z ( z 2 + 2) (B.1) E OlnZn ~ Z 3 - - Z 2 "~- 2z - 1 "

n = l

This equation gives

r 2 + 2 N

= 1.267 x (1.7549)N ( N >> 1), (B.2)

where • (-= 0.5698) is the single real root of the equation x 3 - x 2 + 2X -- 1 = 0.

Next, we estimate the number of strings of N symbols 0 or 1, in which there is no isolated 0, 101. We take the periodic boundary condition for site values. To satisfy the periodic boundary condition, we must exclude the following two kinds of sequences:

i) the first two sites are 01 and the last site is 1, ii) the first site is 1 and the last two sites are 10.

It can be found that the total number of the excluded sequences is given by the quantity 2~ N_ 1, where ~N is the number of strings with length N in which the first and the last sites are 1 and there are no sequences 101. It can be shown that

Thus it is shown that the total number of strings with length N, which contain no sequences of isolated 0 under the periodic boundary condition, is given by

aN-- 2liS-X = ( l / r ) N (N>> 1). (B.5)

This is V{ D u }, the volume of the set D N.

II) We first consider the number of strings with length N, which contain neither the sequence 101 nor 1001 when the periodic boundary condition is not assumed. It is denoted by fiN" It Can be found that

oo

E n m l

- z ( z 5 + 3 z 2 - 2z + 2) (z 2 + z - 1)(z 4 - 2z 3 + 3z 2 - 2z + 1 ) '

(B.6)

then

f l N = 1 5 + 7 ¢ 3 - ( 1 )N 20 ~ ( N >> 1), (B.7)

where y + = ( - 1 + ~ - ) / 2 is the larger root of the equation x 2 + x - 1 = 0.

Next, let */N be the number of strings with length N in which both the first and the last sites are 1, and there is neither the sequence 101 nor 1001. We can find that

- z 3 ( z 2 + 1) (B.3) //,,z" = z3 _ z2 + 2z - 1 '

n ~ 3

_ z 2 ( z 3 _ z + l )

~ , z " = (z 2 + z _ l ) ( z 2 - z + l ) ' nffi2

( B . 8 )

which gives

x3(x2 + 1 ) ( 1 ) N+t

- 2

which gives

~/N = 2~/5- ( N >> 1). (B.9)

= 0.1336 × (1.7549) N (N>> 1). (B.4) We can show that the number of strings with


length N in which there is neither the sequence 101 nor 1001 under the periodic boundary condition for site values is given by

f l u - (2~/N-1 + 371U-2) =-- ( l /T+) u (N>> 1).

(B.10)

The quantity is the qu in the text.

Appendix C

Estimation of average number of block sequences

The set D N consists of all configurations with length N, which contain no isolated 0 site. Every configuration element a ~ D u is composed of some block sequences W1, W 2 . . . . . W k ( k > 0), where each block sequence W~ is composed of I i successive 1 sites sandwiched in between two 0 sites (t,. > 1), and there may be 0 sites between them as shown in fig. 6. In this appendix, we will estimate the average number of block sequences in the configurations of elements in D u. We denote it by K'(N).

First, we evaluate the number of strings with length N in which there are K distinct block sequences. It is denoted by # u ( K ) . Noting the periodic boundary condition on site values, we can find that every configuration with length N and with K block sequences belongs to one of the following four cases.

Case 1: Both the first and the last sites are 0 The configurations in this class are composed

through the following manner. First, we have L successive 1 sites, and divide them into K distinct blocks each of which contains at least one 1 site ( K < L < N). There are L_ICr_x different divi- sions. Let us sandwich the successive 1 sites by two 0 sites for every block sequence. The number of O's used in this procedure is 2K. Name each block sequence W~ ( i = 1 . . . . . K). The suffix i

ascends as the next right block sequence is named as shown in fig. 6. In order to compose configurations with length N which contain the block sequences (W~), N - L - 2 K more 0 sites must be added by inserting one or more 0 sites between some neighboring block sequences. Note that 0 sites can be added at the left side of the block sequence W1, and at the right side of W K. There are ( N - L - K ) ! / [ K ! ( N - L - 2 K ) ! ] ways of such insertions. Thus the total number of configurations which belong to Case 1 is given by

= N-2 ( N - t - K ) !

E L=K L - x C x - 1 K I ( N - L - 2K)!

(O_<K_<3) . (C.1)

Case 2: The first site is 1, but the last site is 0 The configurations in this class can be com-

posed in the same manner as those in Case 1 except that 0 sites must not be added at the left side of the block sequence W v Transplanting the 0 in the left side in W 1 to the right end of the sequence, the configurations in Case 2 can be obtained. Thus the number of configurations which belong to Case 2 is given by

N-2K (N - L - K - 1)! = E t . - I C K - x ( K _ I ) ! ( N _ L _ 2 K ) !

L - K

( 0 < K < 3 ) . (C.2)

Case 3: The first site is 0, but the last site is 1 The number of configurations which belong to

this case, #~ ) (K) , is equal to #~ ) (K) .

Case 4: The first and the last sites are 1 The configurations which belong to this case are

composed through the following manner. Let us


divide L successive 1 sites into K + 1 distinct blocks each of which contains at least one I site to make block sequences { W~ } (i = 1 . . . . . K + 1) by sandwiching the successive 1 sites by two 0 sites for each block sequence. N - L - 2K more 0 sites are added by inserting one or more successive 0 sites between some block sequences, but not at the left side of W x or at the right side of Wx+ x. Removing both the first 0 site at the left side of W 1 and the last 0 site at the right side of Wr+ 1, we can obtain the configurations in Case 4. Thus the number of configurations in Case 4 can be

given by N-2K (N - L - K - 1)!

# ~ ) ( K ) = E t.-xCx(K=-I~..C--~:-ff-_-TK)! L I K + I

( 3 K + 1 < N ) . (C.3)

Therefore we find the total number of configurations with length N which contain K distinct block sequences under the periodic boundary

condition to be

# , , ( K ) = +

+ +

N ( N - 2 K - 1 ) ! N-2x = + E A,(I,:,L)

L=K+I

( K < N/3) , (C.4)

where

f N ( K , L )

N ( L - 1) ! (N - L - K - 1)! K ! ( L - K ) ! ( K - I ) ! ( N - L - 2 K ) ! "

(c.s) The average number of block sequences contained in the configurations of the dements in D N is

defined by N/3

E K. # , , (K) K ( N ) = K-1 (C.6) ~-/3

E #, ,(K) K--1

Eq. (C.6) is estimated by introducing assumptions. Let K e be the value of K which gives the maximum value of the function # N (K ) . First we assume that if N is large, K'(N) is equal to Ke. Suppose that L~ is the value of L which gives the maximum of fN(K, L) with fixed K and N. Using the Stirling's formula, it can be shown that L~ = ( N - K) /2 in N >> 1. Now we use the second assumption: the value K~ also maximizes fN( K, Le). The two assumptions give K( N) = yN in N>> 1, where y (----0.1770) is the single real root of the equation 23x 3 - 23x 2 + 9x - 1 = 0. Moreover we have also performed the numerical estimation of h ' (N) , using eq. (C.6), to find that the results show an excellent agreement with the estimation above.

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intriguing properties of global structure in some classes of finite cellular automata

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