Characteristic representation of elementary cellular automata

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13 February 1995

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PHYSICS

LETTERS

A

4._

IX‘XVIER

Physics Letters A 198 (1995) 23-33

Characteristic representation of elementary cellular automata * Yoshihiko Kayama a, Hajime Anada b, Yasumasa Imamura a a Department ofGeneral Education, BAIKA Women’s College, Shukuno-sho 2-19-5, Ibaraki-shi, Osaka 567, Japan b Mathematical Science Division, Ist System Department, NIKON System Incorporated, OHI Oflce: 26-2, Futaba 2-chome. Shinagawa-ku, Tokyo 142, Japan

Received 15 October 1993; revised manuscript received 30 November 1994; acceptedfor publication 6 December 1994 Communicated by A.P. Fordy

Abstract We propose a characteristic representation of one-dimensional and two-state, three-neighbor cellular automaton rules, which can be interpreted as a numerical description of Wolfram’s classification scheme. Each rule can be considered to behave, in many situations, like some other rules. The similarity of behavior is quantified. The quantification is done by creating a function designed to give an average weight of configurations after many time steps, and then comparing this function, in a precise way, to some other rules by the value of coefficients expanded by the mle functions. We call such an expansion form of the function characteristic representation. The simulation result of the representation shows that Class IV rules are distinguished from Class III rules by the similarity to Class II rules. Criticality of Class IV rules is displayed by functions linearly combining typical Class II and III rule functions.

1. Introduction

Cellular automata (CAs) are discrete dynamical systems with a lattice of sites. Each site has a finite set of possible values that evolve synchronously in discrete time steps according to an identical rule. Wolfram [ 1,2] has sought systematically for interesting phenomena and possible behavior of one-dimensional automata. He found the four essential types of behavior: homogeneous (Class I), periodic (Class II), chaotic (Class III) and complex (Class IV). This classification scheme has been discussed using several characteristic parameters and approximation theories [ 3-131. The simplest CAs consist of a line of sites, Each site carries a value, 0 or 1. The value of a site is determined by the previous values of its nearest neighbors. They are called onedimensional and two-state, three-neighbor CAs or elementary CAs. Rules 54 and 110, according to Wolfram’s numbering [ 21, are thought to be in Class IV [ 8,9]. The typical patterns generated by them show long transients and soliton-like structures (Fig. 1). In the previous work [ 131 of one of the authors and collaborators, the characteristic parameters are calculated in all the rules of the simplest CAs and applied to give quantitative grounds for Wolfram’s phenomenological classification. Among the parameters discussed there, the spreading * Work supported in part by BAIKA Women’s College, Japan. 0375-9601/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(94)00996-l

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Y. Kayama et al. /Physics

L.etfers A 198 (1995) 23-33

(a) Fig. I. Space-time behaviors of (a) rule 54 and (b) rule 1 IO. They show long transients with 200 sites and 150 time steps, starting from pseudorandom initial configurations. Table 1 The values of Dis in Ref.

(b) and soliton-like

phenomena.

They are displayed

[ 131 have been sorted; rules 54 and 110 exist between Class II and III groups

rate of difference patterns, Diff, is the most powerful parameter to classify the rules. The calculated results show that Classes I, II and III are sufficiently categorized. The Class IV rules, however, cannot be distinguished from the Class III rules. The values of Diff are tentatively sorted in Table 1. It is remarkable that rules 54 and 110 have boundary values between Class II and III groups, which supports the criticality of Class IV rules discussed in Refs. [8-l 11. We cannot see, however, any difference between rule 41 and rules 54 or 110. Such a single parameter is insufficient to distinguish Class IV rules from others. Now we reconsider the traditional classification of Wolfram [2]. For example, rules 0 and 32 belong to Class I because the values of all sites of their space-time patterns become zero after many time steps. From the viewpoint of the action of a rule function, the functions of these rules are effectively identical. In other words the difference between them become inactive in the long run. If we can define effective rule functions reflecting the similarity of patterns, the patterns can be classified on the basis of this similarity. To make this classification, we use a definition of the effective function of each CA rule, which we call a characteristic function, and its representation given by a linear combination of all the elementary CA rule functions. The similarity of patterns

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Y. Kayama et al. /Physics Letters A I98 (1995) 23-33

is represented by the resemblance of expansion coefficients (components) between characteristic functions. We apply the characteristic functions to the elementary CA rules. Our simulation results show the feature of each Class. For example, the characteristic functions of all the Class I rules become equivalent to rule 0 (or 255). The Class IV rules can be distinguished from the Class III rules by comparing their Class II components. The Class IV rules are more similar to the Class II rules than the Class III rules. We discuss the meaning of this result and search for the cause of Class IV-like complex structures.

2. Characteristic

representation

Let xk be the value of the kth site in a one-dimensional CA. The value of each site is specified (two-state). If there are N total sites, a configuration can be denoted by

x=

(x,,x2

The Hamming

w(X)

as 0 or 1

(2.1)

XN) .

)...,

distance of X from 0 = (0,. . . , 0) is called weight,

=5x,.

(2.2)

k=l Let {X},,, denote the configuration space which is the set of all the distinct configurations The time evolution of a site value is determined by iteration of the mapping, ‘k(‘+I)

= f(xp),,

. . . ,xi’),

. . . ,x;:‘J

represented

by (2.1)

5 fr(.q).

(2.3)

Here f is called a rule function specifying each CA. The subscript r determines a neighborhood of the rule function ((2r + 1)-neighbor). In the following, we abbreviate the subscript because our discussions are restricted to three-neighbor CAs. A rule function is determined by the succeeding eight values, f(0,0,0),~(0,0,1),...,f(1,1,1)~ or f(O),f(l),. defined by

5

f(i)

(2.4)

. . , f( 7) for short. There are 28 = 256 rules of the elementary

CAs and rule numbers

X 2’.

are

(2.5)

i=O

According to the notation of the previous paper [ 131, if f (‘I denotes a rule function applying it r times gives the tth rule function, f”‘(Xk)

= f”‘(f’“(.

. .“p)(q)

which is a rule for the (2t + 1)-neighbor f(‘)(Xk)

= f(f)(xk_r,.

of the first time step,

. . .)),

(2.6)

of xk,

. . ,xk,. . . ,Xk+r) .

A mapping on the configuration

space {X},,

fct) : {X},

(2.7) ---f {X},,

is defined with a proper boundary

treatment

by f”‘(X)

=f(‘)((X*,X2,...

9XN)) = (f”‘(xdJ’t’(X2)

,...,f”‘(Xd).

(2.8)

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Y. Kayama et al. /Physics Letters A 198 (1995) 23-33

In this article, we adopt periodic bound~y conditions. The image of f(‘), f”‘( (X>,), Since the following is a monotonically decreasing sequence of ffff ((X},)‘s, {X}, > f”?(X),)

3 f’2’({X}N)

1.. I 3 f”‘({X}N)

2..

is a subset of (Xl,.

. ,

(2.9)

there exists a limit set limj-,oo f (‘I ( {X}N). Because all asymptotic configurations are elements of the limit set, asymptotic space-time patterns are described by the trajectories in it. If such asymptotic trajectories of two different rules are equivalent or the total probability of the appearance of different trajectories between them goes to zero following N ---f 00, the differences between their rule functions are negligible and of course these rules belong to the same class. In such a case, some representation of rule functions reflecting this similarity might exist. Let us give an example. Class I rule functions, e.g. rule 32, map almost all initial configurations to 0 or 1 = (1, . . . ,1) after many time steps. Two exceptional configurations, ( l,O, l,O, . . . , 1,0) , 0,1)) are contained in the limit set under conditions of an even number of total sites and and (O,l,O,l,... periodic boundary. A point 0 and a limit cycle between the above two configurations are asymptotic trajectories. The reason for the limit cycle not being important for the classification is that the probability of its appearance 2/2N is negligible for sufficiently large N. Rule 32 is characterized by the asymptotic configuration 0 and classified in Class I. On the other hand, the difference between the rule 0 and 32 functions is the value of f”) ( l,O, 1). Because the above two configurations hardly appear, the difference is inactive. Effectively the rule 32 function is ~uivalent to the rule 0 function. This example of effective similarity suggests the creation of functions reflecting the differences between the probabilities of appearance of asymptotic configurations. Let fR(‘) denote the rule function of rule R. After t time steps, the domain of the mapping ft’ to the (t+ 1)th time step is fg’({X}N). Because f!’ : {X}, --+ {X}, is a many-to-one mapping in general, we define the as Q(X(~)). Moreover n~(i,X(‘)) denotes how multiplicity of the mapping for an element Xct) of fg’((X},) many times each block of three sites, (0, 0, Of, (0, 0,l) , . . . , ( I, 1,l) (i = 0,1,. . . ,7, respectively), appears in a configuration X(j) . Then the following equations are trivial, I

Iz

?YzR( x”‘) = 2N,

c

nR(i , X(‘)) = N.

(2.10)

I=0

X’%f:I’({X},f

Using these parameters, the probability of appearance of each block at the tth time step is Pi”(i)

I

= ~ 2N x N

El

m~(Xqn&Xq.

(2.11)

X”‘Ef;‘(lX),) It is easy to extend the above definition to Pi*‘(i) for all initial configurations X(O) E (X},. p’*‘(i) = 8 because all existence rates of block i in all con~gurations of (X}, are equal. R The average weight of Xcr+‘) is given by

= NLPi”(i)ft’(i). (WR(X(f+‘9)N

We obtain

(2.12)

i=O

At the first time step, Pi” (i) = i yields (wR(X(‘)))N = gN&(i1.

(2.13)

i=O

By modifying the rule function fk”, we try to construct the effective function discussed in the rule 32 case as a function ft’ satisfying

Y. Kayama et al. /Physics Letters A 198 (1995) 23-33

21

Namely, 7:’ is defined so as to take into account the change of the average weight of the configuration X(‘) for t 2 1. Comparing Eqs. (2.12) and (2.14), we can think of making a naive choice f:‘(i) = 8Pi”(i)fi”(i). The following example of rule 255, however, shows that this candidate for f?‘(i) is not acceptable. Rule 255 is defined as (f:::(o)J:::(l),... Then fi$

=(l,l,...,l).

(2.15)

=(O,O,...,g)

(2.14)

,&j(7))

for t Z 1 becomes

(~~:5:(0>,~~::(1>,...,~~::(7))

because P${(i)= 0 (i= 0,.1.,6)and P&$(7)= 1. On the other hand, the average weight does not change: (We&@)),,, = N for t 2 1, which shows that the expected result in this example is f$ = j$$ and such a function as (2.16) is not acceptable for the effective function of f.l:. The 8~~f~(~)~~“(~) must be shared properly among the f:‘(i). Although the components of fg’ are not specified as 0 or 1 any longer, they can be restricted between 0 and 1. As is discussed later, we give them a probabihty interpretation. From the above discussion, the following conditions are imposed on the effective function: (i) The mapping 2:’ constructed from yt’ is defined on {X},. (ii) The value of a characteristic function is limited between 0 and 1: 0 < f;‘(i)

< 1

for i=O, 1,. . . ,7.

(2.17)

-(I) (iii) The average weight of a configuration contained in the image of the mapping f R is equal to the average weight of Xcr+‘) (Eq. (2.14)). The first and second conditions mean that fg’ is the extension of the CA rule functions on its values of com~nents; $‘,

The image of j’y’ is a subset of the unit N-cube. Now we propose the following definition of

for Pi"(i)> $ ,

for Pi"(i)< $ ,

(2.18)

which we call the tth characteristic function. fro’ is identical with the rule function fh’) because there is no effect of the time evolution. As we explained in the Class I examples (rules 32 and 255), the modi~cation of a rule function must describe its characteristic. In this sense, the values of f;‘(i) for Pi"(i)2 i should reflect it. Consequently we leave such &i’(i) as they are. The first term for Pi"(i)< i is the naive choice discussed before. The second term compensates for the loss of Pi"(i) . Thetotal excess of the naive choice over f$ ) (i) for Pi”(i) > ’ r8’ [8~~f’(~) c jforPt'(j))1/8

-

Ilf:‘(j>,

(2.191

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Y. Kayama et al. /Physics

Letters A I98 (1995) 23-33

(a)

b)

Fig. 2. Space-time pattern of probabilistic rule function f = 1 f232+ 1fw which is analogous to the patterns of Class IV plotted with 200 sites and I50 time steps. Initial configurations are pseudorandomly generated.

are averaged by the total excess of the probability I@‘(j)

1

rules.

They are

of appearance, (2.20)

- g1,

jforP~‘(j))l/8

and the deficiency (2.18)

is calculated

satisfies Eq. (2.14) [ 1 - p”‘(i)) 8

c Pk”(i)
R

by the product with the loss Z$- Pj” (i) . It is easy to show that the definition Pi” ( i) = 1 and the following

if we notice C(, =

[f$“(i)

c

equation,

- ‘1 8 ’

(2.21)

Pi’(i)al/8

In fact, the summation

of the second term for Pi” (i) < $ becomes

c [$ - lp(i)]

iforPF’(i)
G

=

[8@‘(j)

(2.22)

- llf;“(j).

jforf~‘(j)bl/8

Then we have

.&r i=o

(0 i 0

R

=

c iforP;‘(i)>l/8

f;‘(i)

+

x iforPi’(i)
8P;‘)(i)f;)(i)

+

c

[WA’)(i)

- llf;‘(i>

iforPf’(i)>l/S

7 =

c ia0

8~~‘)(~)~~)(~),

(2.23)

which proves Eq. (2.14). Note that (2.18) is not the only definition of $’ (i) that satisfies the preceding conditions. As is discussed later, our empirical result suggests that this is a good choice for the effective functions of all the rule functions. In order to show the reiation or the simil~ity of the characteristic function with rule functions, we define an expansion form of this function. As we discuss in the following, all elementary rule functions, from rule 0 to

29

Y. Kayama et al. /Physics Letters A 198 (1995) 23-33

Rule 46

Rule 110

Fig. 3. Deformation of space-time patterns from rules 46 to I IO. 40 sites and 100 time steps are taken. Initial configurations are

pseudorandomly generated.

rule 255, can be regarded as base vectors of 256-dimensiona space and the characteristic function is a vector of this space. The ch~acteristic function can be expanded in the base vectors and the expansion coefficients denote the similarity to the rule functions. If we take a function space (ffc : {X}s --+ [0, I]}, the characteristic function and all the elementary rule functions are contained in it. Let ZZi= {(i --+ I ), (i + 0)} (i E {X),) be a space of elementary events and pi be a probability measure defined by each component f(i). Because the events (i + I ) and f i -+ 0) are complementary, pi (i --f 1) = f(i) and pi(i --+ 0) = 1 - f(i). Now we suppose two-dimensional Euclidean space RF and an ordinary inner product (XI, ~1) - (x2, y2) = x1 x2 + _yly2, where (XI, .VI 1, (x2, ~2) E RF. If we identify f(i) with a two-dimensional vector (f(i), 1 - f”(i) ), it is an element of iRy and can be represented by unit base vectors ei‘If = ( 1,O) and ei*) = (0,1) as follows, = (e!‘) . J(i))ej”

J(i)

On a space fll+ function

+ (eJ2).

@Ei (8-times

f. Simultaneously,

space fl$

f(i))ej2’.

( 2.24 )

product of E;), a product measure J-&, pi is defined by a11 f(i)

f” is identified

with a tensor nL

@IRK.The inner product between functions

(J’i!) E n{.f(i)

.2(i)

+ [l -f(i)].

@(f(i)

‘s, that is, by a

, I - f’(i) ) of a 28-dimensional

product

f’ and 2 is given by

[I --g(i)]}.

(2.25)

i=O

It is important

that 28 = 256 base vectors I)

e’

c3 ii

,i2,

I

1

can be identified with the elementary CA rule functions. Because they construct a complete orthonormal of n& @Rf, the following expansion form of the characteristic function is trivial,

system

255

(2.26)

The coefficient (f,. . ft’l denotes the similarity of the tth characteristic function with the rule r function. We call this expanded form the c~ffracteristic representat~5n of the rule function. We have empirically estimated the coefficients for all the independent even number rules of the elementary CAs. An initial configuration of 1600 sites is pseudorandomly generated so that its density is 0.5. Each rule

30

Y.Kayama et al. /Physics Letters A 198 (1995) 23-33

Table 2

Coefficientsof characteristic representationsof some elementary CA rules; Pi) (i) of FQ. (2.11) is approximatelycalculated from Eqs. (2.18) and (2.26) by sampling of 800 initial configurations of 1600 sites and 800 time steps; initial configurations am pseudorandomly generated and their density is 0.5; the vertical rule numbers indicate 88 independent elementary CA rules for base vectors and the ho~zontal rule numbers initial rules of characteristic functions; the coe~cients expanded by other dependent CA rules are summed up to the representatives; for example, the coefficients for rule 255 are added to those for rule 0

applied to the configuration with periodic .^^^. boundary conditions. After 800 time steps, Pcsoo) (i), 7, are determined. The determinations of P c8w)(i) for 800 initial configurations are averaged. The characteristic function and its expansion coefficients are determined by Eqs. (2.18) and (2.26). Some typical examples are listed in Table 2. All the Class I rules have the same components of characteristic functions with the rule 0 function. For most of the Class II rules, characteristic functions are similar to themselves or typical Class II rule functions, like rules 2,4 and 34. The Class III rules mostly stay in the same class. It is remarkable, however, that the Class IV rules 54 and 110 have larger values of coefficients to the Class II rules than the is iteratively

i=o,...,

Y. Kayama et al. /Physics Letters A 198 (199.5) 23-33

Table 2 Continued

Class III rules (Table 3). The difference suggests the cause of complex structures of the Class IV rules to us. Space-time configurations of the Class II rules have limited patterns that can be regarded as isolated and stable states. There is no communication of information between them. On the other hand, the Class III rules, as described by the spreading rate of difference patterns, are characterized by global information flow. As the similar criticality of the Class IV rules are discussed in Refs. [8-l 11, we can suppose that if the Class II and the Class III rules are mixed appropriately, Class III gives informational exchanges between stable states of Class II and they become unstable states, giving so-called soliton-like behavior of Class IV. For example, we

32

Y. Kayama

et al. /Physics L.ettersA 198 (1995) 23-33

Table 3 The values of rules 54, the coefficients expanded

1IO and all the Class III rules in Table 2 have been magnified; the values of the row ” ---t II” are total values of

take such functions

as

by the Class II rules; the Class IV rules, 54 and 110, have larger “ +

II” values than other Class III rules

2.27) where rules 4 and 232 are typical rules of Class II and rule 90 is one of III. These functions patterns like Fig. 2 which are analogous to the patterns of the Class IV rules.

generate complex

3. Discussion Reconsidering Wolfram’s classification scheme, we have proposed the characteristic function obtained by modifying the initial rule function to take into account the effects of the time evolution through changes of the probabilities of appearance of three-site blocks. Each value of the characteristic function f(x, y, z ) can be interpreted as the probability that the site y will have a value 1 at the next time step; such a system constructs a Markov process ’ . Its expansion over all the elementary rules is called the characteristic representation. The expansion coefficients denote the similarity of the characteristic function to the rule functions. As noted in the previous section, the empirical result shows that the rules included in the same Wolfram class have similar coefficients and that the coefficients of the Class IV rules to the Class II rules are larger than those of the Class III rules to the Class II rules. We suppose that disturbances of Class II-like stable states by Class III-like chaotic aspects cause complex structures of the Class IV rules [8-l 11. In fact, such examples have been presented as characteristic functions of linear combinations of the Class II and the Class III rule functions. The characteristic representation, therefore, offers not only a useful tool for categorizing the CA rules but also an understanding of the cause of complex patterns of the Class IV rules. I If wedefine

r-times action of f(x,y,z)

as f(‘)(v)

= f(,j’(...f(.v).

..)).

j;“‘(.v)

dependsonly

on f(‘-‘j(Y).

Y. Kayama et al. /Physics

Letters A 198 (1995) 23-33

33

It is wo~hwhile to describe the difference between mew-field approaches [ 121 and our fo~alism clearly. The &h-order mean field theory aims to obtain larger block probabilities than n sites from the values of smaller block probabilities. The classification is done by the resemblance of the large block probabilities. On the other hand, the characteristic function is determined by the probabilities of three-site blocks and the initial rule function. The classification is done by the resemblance of the coefficients of the characteristic representations. Our formalism is analogous to the idea of feedback of phenomenal changes to a gene or of the wave function reno~ali~tion of higher order interactions in particle physics. Dynamical systems are generated from the characteristic functions. It is interesting to use them in the classification of CAs. A pattern construction process can be shown by the continuous deformation of proper elements of f( X,y, z ). For example, rule 46 changes into rule 110 by varying the value of f( 1, 1,O) from 0 to 1 (Fig. 3). Moreover, this formalism can be applied to more complex systems such as five-neighbor CAs El41. References [ 1J S. Wolfram, Rev. Mod. Phys. 55 ( 1983) 601. 121 S. Wolfram, Physica D 10 ( 1984) I. [ 31 F?Gmssberger, Phys. Rev. A 28 ( 1983) 3666. 141 P. Grassberger, Physica D IO ( 1984) 52. 15I N.H. Packard, Complexity of growing patterns in cellular automata, in: Dynamical systems and cellular automata. eds. J. Demongeot, E. Goles and M. Tchuente (Academic Press, New York, 1985). 161 S. Wolfram, ed., Theory and applications of cellular automata (World Scientific. Singapore, 1986). [ 71 P Grassberger, .I. Stat. Phys. 45 ( 1986) 27. [ 81 W. Li and N. Packard, Complex Systems 4 ( 1990) 28 1. 191 C. Langton, Physica D 42 (1990) 12. 1IO] W. Li, N. Packard and C. Langton, Physica D 45 ( 1990) 77. [ 111H. Chat6 and I? Manneville, Physica D 4.5 ( 1990) 122. [ 121 H.A. Gutowitz, Physica D 45 (1990) 136. [ 131 Y. Kayama, M. Tabuse, H. Nishimura and T. Horiguchi, Chaos Solitons Fractals 3 ( 1993) 6.51, [ 141 Y. Kayama and Y. Imamura, BAIKA Women’s College preprint, in preparation.