325
Progress of Theoretical Physics, Vol. 89, No.2, February 1993
Periodicity in One-Dimensional Finite Linear Cellular Automata
Shin-ichi TADAKI and Shiny a MATSUFUJI
Department 0/ In/ormation Science, Saga University, Saga 840
(Received October 26, 1992)
We investigate the periodic structures of the trajectories in one-dimensional finite linear cellular
automata with rule-90, whose time evolutions are described with matrices. The periodicity of the
trajectories is found numerically and by use of eigenvalue equations. The period strongly depends
on the number of sites. In some cases the null state becomes a limit point.
§ 1.
Introduction
Cellular automata are one of the simplest mathematical models for nonlinear
dynamics to produce complicated patterns of behaviour. Von Neumann had originally introduced cellular automata as an idealized biological system.!) Wolfram had
reintroduced cellular automata as a model to investigate complexity and randomness. Z)
He has investigated many fundamental features of them. 3 )-5) Since then many
authors have made efforts to clarify the properties of cellular automata and applied
them to natural systems. 6 )
There seems to be some approaches to study cellular automata. The first is to
treat cellular automata as simple models to simulate concrete natural systems, Ising
spin systems, lattice gases and so on. Second there are more mathematical
approaches. At these viewpoints cellular automata are the abstract systems to
model nonlinearly coupled complex phenomena. Studying the properties of cellular
automata may yield the proper concept to understand complex systems. There are
also more information theoretical viewpoints relating to the analogy to shift registers
and· pseudo-random sequences. 7 )-9)
One-dimensional cellular automata are described by the discrete time evolution
equation of site ai:
(1'1)
where ai takes k discrete values over Zk. The simplest model, elementary cellular
automaton, consists of sites with two internal states over Zz interacting with the
nearest neighbour sites (r=I). Wolfram had introduced a naming scheme for these
systems and classified the behaviours of cellular automata into four classes. Z),3) The
dependence of the behaviours on the time-evolution rule, however, is not clear. 10)
Most authors had worked on cellular automata within the scope of the infinite
number of sites. On the nonlinear dynamical system, however, the influence of the
boundaries and the finiteness of the system are not so clear. A few works had
concerned periodic boundary conditions (cylindrical automata).5),9),1l) It is also inter~
esting to investigate the structure of the trajectories, namely the state transitions, of
finite cellular automata. The purpose of this paper is to investigate the periodic
s.
326
Tadaki and S. Matsujuji
structure of the orbits of one-dimensional linear cellular automata with a concrete
time evolution rule.
§ 2_
Model and numerical results
We investigate the so-called rule-gO cellular automata following Wolfram's
naming scheme. 3 ) The time evolution of the i-th site ai(t)E{O, I} (i=l ~ N) is described as a sum modulo 2 of the nearest neighbour sites:
(2 -I)
We use the Dirichlet boundary conditions, ao= aN+l =0. According to the Wolfram's
classification, rule-gO cellular automata belong to the third class which shows the
chaotic behaviour. The time evolution is also expressed by matrix.
A(t+ 1)= UA(t) ,
(2-2)
where A(t)=t(al(t), a2(t), ... , aN(t)) describes the state of cellular automata and the
transfer matrix U is given by
j=i±l,
otherwise.
(2-3)
It is very special that the time-evolution rule can be expressed in a matrix form and
we call the case linear cellular automata.
We find the periodic structure numerically as shown in Table 1. For cellular
automata with even number of sites we find
Table 1. The empirical data for periodicity for the
cases with 3~31 sites rule 90 cellular automata.
N
N
4
U 6=/
6
8
U14=/
U 14 =/
1000
1100
10
U62=1
0100
1110
U l1 = U'
U 29 =U
12
U I2 6=/
14
U'o=/
1010
1011
U 15 =0
U 29 =U
16
U'o=/
0001
0011
1001
18
UI022=/
0010
0111
0110
U 27 = U'
U 125 = U
20
UI2/6=/
22
U4094=/
0101
1101
1111
U 23 =U7
U 2.53 =U
24
U2046=/
26
UI022=/
28
U'2766=/
29
U 59 =U'
U 61 =U
30
31
U'I=O
32
U62=/
U 62 =/
3
5
7
9
11
13
15
17
19
21
23
25
27
U'=O
U 5=U
7
U =0
U 1'=U
•
•
•
•
•
•
•
••
•
• •
Fig.1. The orbits of N=4 cellular automata. All
states are classified into 3 orbits except the null
state. Two of them are period 6 and the other
period 3. The null state is an isolated fixed
point.
Periodicity in One-Dimensional Finite Linear Cellular Automata
••
•
••
•
•
10000
•
•
•
••
•
10111
10010
01000+00101
11100+01101
10100 --. 0001 0
10110 --. 00111
11101
00001
01001
01011
01100
00100
01.10~
01111--.11001
10001~
00110
11111
10101
01110
00L)
11l:)
Fig. 2.
110
010 ---.101
001
000
~
111
011
Fig. 3. The orbit of N=3 cellular automata. All
states are drawn into the null state within less
than 3 steps.
00011
10011 +11110
11010
•
•
• • •
100
••••
• •
•
•
•
11000
327
(2·4)
where 1 is a unit matrix. It means that
every state except the null one (all sites
are zero) is on the orbits with period less
than lIN. These orbits for N=4 cellular
automata are shown in Fig. 1. For cellular automata with odd number of sites
except N = 2n - 1, we find another type of
periodicity
The orbits of N=5 cellular automata.
(2·5)
Some states belong to the orbits with
period less than lIN and the others except the null state are drawn to the orbits after
some time steps less than J[N. The orbits for N =5 is shown in Fig. 2. It is very
interesting that in the N=2n-1 (nEZ) case, every state is drawn into the null state
after at least N steps,
(2·6)
Namely the configuration space has only one.basin with the null state at the center.
The orbits for N =3 cellular automata are shown in Fig. 3.
§ 3.
Eigenvalue analysis
To investigate the periodic structures analytically, we consider the eigenvalue
equation
UA=-AA,
(3·1)
which reads the secular equation:
(3·2)
As a site ai takes binary values, all calculations are carried out over Zz. By virtue
of the recursion relation D N(A)=AD N- 1(A)- DN-Z(A), we obtain the explicit form of
DN(A) as polynomials in Galois field of order 2, GF(2), namely all coefficients of Ai are
over Zz:
(3·3)
328
S. Tadaki and S. Matsujuji
where jrnax=L(N-4)/2J , L J is a Gaussian symbol and
.
.N
Ct=(-l)J
-<2i+3)(k +
~
j)
.(N- j- 2) .
.
=(-l)J.
;+1
;+2
k=l
For later use, we obtain another form of
N -(
eirnax-k-
l)k
(3-4)
ef:
(jrnax+2)!
(N - jrnax+ k-2)!
(N -2jrnax-4)! eN
(3 5)
(jrnax+2- k)!
(N - jrnax-2)! (N -2jrnax+2k-4)! irnax'
Table II. Examples of DN(A).
N
3
D'(A)=A'
4
D4(A)=A4+A2+ 1
D 5(A)=A 5+A
5
6
7
D6(A)=A6+A4+1
D7(A)=A7
9
D8(A)=A 8+ A6+ A4+ 1
D9(A)=A 9+A5+A
10
D10(A)=A10+A8+A4+A2+ 1
11
D"(A) =A" + A'
D12(A)=A 12 + A10+A 8+A 2+ 1
8
12
13
14
D"(A)+A"+A 9+A
D14(A)=A 14 +A 12 +A8+ 1
16
D 15(A)=A 15
D16(A)=A 16 + A14+ A12+ A8+ 1
17
D17(A)=A 17 +A"+ A9+A
18
D18(A)=A18+A16+A12+ A10+ A8+A 2+ 1
19
20
D19(A)=A 19 + A"+ A'
D20(A)=A 20 + A18 + A16 + A10+A8+A4+A2+ 1
21
D21(A)=A 21 +A 17 + A9+A5+A
22
D22(A)=A22+ A20+ A18+A8+A6+A4+ 1
23
D 2'(A)=A 2'+A 7
D 24 (A)=A 24 + A22+ A20+ A16 +A 6+ A4+ 1
15
24
26
D 25(A)=A 25 + A21 +A 17 +A 5+ 1
D26(A)=A 26 + A24+A20+A18+A16+ A4+ A2+ 1
27
D27(A)=A27 +A 19 + A'
28
D28(A)=A28+A26+A24+ A18+A16+A2+ 1
29
D29(A)=A29+A25+A17 + A
30
31
D'O(A)=A,o+A 28 +A 24 + A16 + 1
D31(A)=A 31
32
D'2(A)=A'2+ A,o+A28+A24+A16+ 1
25
Periodicity in One-Dimensional Finite Linear Cellular Automata
329
Some examples of DN().) are shown in Table II.
First we consider the peculiar case that the number of sites is N = 2n -1. It is
easily found the coefficient of the last term of the polynomial,
Cfmax= -2 n- 1 ,
(3·6)
where jrnax=2 n- I -3. In this case Eq. (3'5) is simplified to
If (2 z (n-1) --.: z'Z)
N
N
-(
l)k II .=1(2k+ I)!
Cjmax,
Cjmax-k-
k-O
1
-"
••• ,
.
Jrnax.
(3'7)
To investigate (C/ mod 2)=0, we introduce new functions SF(x) and!x of an integer
x defined as
SF(x)=n,
if x=(odd number)X2 n
(3·8)
,
_ (X!)
¥ .
(3·9)
!x=SF
It is easily found the identity holds
SF (
IIJ=I(2z(n-I)(2k+ I)!
iZ»)_
(3·10)
-!k .
In the case k=2m the function enjoys !zm= -1. In the region kE{O, 1, ... , 2m}, the
minimum value of !k is given only at k=2m -1 as !zm-I = - m, and the second minima
are given at k=2m-2 and k=2m-3 as !2m-z=!zm-3=-m+1 (see Fig. 4). Then in the
region kE{O, 1, ... ,jrnax} we find !k?:'-n+2 and
(3·11)
Therefore for N =2n-1 cellular automata, all coefficients except that of ).N vanish:
DN().)=).N.
(3'12)
This concludes Eq. (2·6) and all states
are irreversibly drawn into the null state
within steps less than N.
For linear cellular automata with
even number of sites, jrnax=N/2-2 and
we find
o
-1
-2
-3
-4
Ct:' = (_ l)N /z_z(N/ 2)
Jmax
N/2
-5
-6
=( _l)N'z-z .
(3 '13)
The constant term remains and the time
evolution is described with a regular
matrix, 7rN=O. The time evolution is
reversible and the equation of periodicity has the form of Eq. (2·4). It is
-7
-8
-9
-10
k
2 10
Fig. 4.
The behaviour of function
I".
s.
330
Tadaki and S. Matsujuji
not easy to evaluate the concrete period of cellular automata in this case.
instance, N =4 cellular automata, we have from Table II
,14=,12+1.
(3·14)
Note that the signs are not concerned in GF(2).
,16~,14+,12=1
For
Then we get
,
(3·15)
and ll4=6. Unfortunately we have no general procedure to obtain periods from
secular equation DN(,1)=O.
For odd site cellular automata except N =2n -1, the power of the lowest order
term ,1irN determines the maximum length to the periodic orbits (see Eq. (2·5)). By
virtue of algebra in GF(2), the new recursion relation
D N(,1)=,1 2D N- 2(,1) + DN-4(,1) ,
(3 ·16)
holds.
And we find
D 2n +l(,1)=,1{,12n +«N -l)mod 2),1 2(n-2)+
jmax
~
(I C/lmod
2),1 2(n-4-2j)}
j=O
(3 ·17)
where jrnax=L(n-4)/2J. Therefore the period and the length of the path to the orbit
of N=2n+l sites cellular automata are described by those of N=n sites as
(3 ·18)
ll2n+! =2lln+ 1,
(3·19)
§ 4.
Concluding remarks
We investigate the periodic structure of one-dimensional rule-90 cellular automata with Dirichlet boundary condition. We find three types of behaviour. The first
is periodic one which appears in cellular automata with even number of sites. The
dynamics of the case is reversible. The second appears in the case of odd number of
sites. There are some periodic orbits and irreversible paths to the orbits. The
peculiar behaviour happens to the case N =2n -1. All states are drawn to the null
state within N steps. These features are very different from those of rule-30 cylindrical automata, whose periods are estimated as 10g(llN)cx(N + 1) by Wolfram. 9 )
We analyze the eigenvalue equations to investigate the behaviour of cellular
automata. Martin, Odlyzko and Wolfram had investigated the trajectories of rule-90
cylindrical automata by characteristic polynomials which describe the states of cellular
automata. 5 ) They had also found three types of behaviour of the orbits, whose
dependence on the number of sites is different from the cases with the ·Dirichlet
boundaries. Our analysis is simpler than theirs and will be advantageous to be
extended to other cases, rule-150 cellular automata (time evolution is expressed in
matrix form) and periodic boundary condition and so on. The analysis of the global
structure of the ·orbits and the distribution of the maximum length to the periodic
Periodicity in One-Dimensional Finite Linear Cellular Automata
331
orbits are our future problems.
It is also very interesting to investigate the so-called hybrid cellular automata,
which are mixture of two rules. By fine-tuned mixture of rule-gO and rule-150 cellular
automata, it is shown to be able to produce a pseudo-random sequence whose period
is 2N -1.7) The behaviour of randomly mixed cellular automata, however, is an open
question.
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