Transitivity and Chaoticity in 1-D Cellular Automata
Fangyue Chen1, Guanrong Chen2, Weifeng Jin3 1
School of Science, Hangzhou Dianzi University, Hangzhou, China 2
Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China 3
College of Pharmaceutical Sciences, Zhejiang Chinese Medical University, Hangzhou, China Email: [email protected], [email protected], [email protected]
Received November 22, 2012; revised December 30, 2012; accepted January 14,2013
ABSTRACT
Recent progress in symbolic dynamics of cellular automata (CA) shows that many CA exhibit rich and complicated Bernoulli-shift properties, such as positive topological entropy, topological transitivity and even mixing. Noticeably, some CA are only transitive, but not mixing on their subsystems. Yet, for one-dimensional CA, this paper proves that not only the shift transitivity guarantees the CA transitivity but also the CA with transitive non-trivial Bernoulli subshift of finite type have dense periodic points. It is concluded that, for one-dimensional CA, the transitivity implies chaos in the sense of Devaney on the non-trivial Bernoulli subshift of finite types.
Keywords: Bernoulli Subshift of Finite Type; Cellular Automata; Devaney Chaos; Symbolic Dynamics; Topological Transitivity
1. Introduction
1.1. Cellular Automata
Cellular automata (CA), formally introduced by von Neu- mann in the late 1940s and early 1950s, are a class of spatially and temporally discrete deterministic systems, characterized by local interactions and an inherently pa- rallel form of evolution [1]. In the late 1960s, Conway proposed his now-famous Game of Life, which shows the great potential of CA in simulating complex systems [2]. In the 1980s, Wolfram focused on the analysis of dyna- mical systems and studied CA in detail [3,4]. In 2002, he introduced the monumental work A New Kind of Science [5]. In fact, mathematical theory of CA was firstly deve- loped by Hedlund about two decades after Neumann’s seminal work [6]. Since 2002, Chua et al. provided a ri- gorous nonlinear dynamical approach to Wolfram’s em- pirical observations [7-10]. All elementary CA (ECA) rules are reorganized into four groups in terms of finite bit stings. There are 40 topologically-distinct period- rules , 30 topologically-distinct Bernoulli shift rules, 10 complex Bernoulli shift rules, and 8 hyper Bernoulli shift rules. Recently, the dynamical properties of Chua’s periodic rules and robust Bernoulli-shift rules with distinct parameters have been investigated from the viewpoint of symbolic dynamics [11-17].
k
k1, 2, 3, 6
By a theorem of Hedlund [6], a map f S: Z SZ
is an one-dimensional cellular automata (1-D CA) if and only if it is continuous and commutes with the shift map . For any 1-D CA, there exists a radius and a
local map
0
r
2 1
ˆ : r
f S
i f x S such that
,
ˆ i r i r f x
,
where notetions will be precisely defined below. In par- ticular, f is an ECA global map when r1 and S
0,1 . Each ECA can be expressed by a 3-bit Boolean function and coded by an integer , which is the de- cimal notation of the output binary sequence of the Boo- lean function [5,7,18].N
1.2. Definition of Chaos
Let X be a metric space and F: X X be a conti- nuous map. F is said to be topologically transitive or simply transitive if, for any non-empty open subsets and of
U V X there exists a natural number n such that Fn
U V ; F is topologically mixing or simply mixing if there exists a natural number such thatN
Vn
F U for all nN; F is sensitive to initial conditions (or simply sensitive) if there exists a
0
such that, for xX and for any neighborhood
B x of x, there exists a y B x
and a natural number n such that d F
n x ,Fn y
, whereis a distance defined on
d
X.
Let P F
xX n 0 ,Fn
x x
be the set ofperiodic points of F. P F
is said to be a dense sub- set of X if, for any xX and any constant , there exists a0
yP F such that d x y
, .Devaney if (1)Fis transitive, (2) is a dense sub- set of
P F X, (3)Fis sensitive [19].
It has been proved that additive 1-D CA are chaotic [20]. For general dynamical systems, it has been proved that (1) and (2) together imply (3) [21], and for 1-D CA, (1) implies (3) [22]. In the next section of this paper, it will be proved that, for 1-D CA with Bernoulli subshift of finite type (BSFT), (1) also implies (2).
1.3. Symbolic Dynamical Systems and SFT For a finite alphabet , a word over is a finite sequence of elements of . Denote by the set of all words of length . If
0,1,S
0, ,a a
,k 1
1
n
a
S S
n
S n x is
a finite or infinite word and I
i j, is an interval of integers on which x is defined, then denote i j,
i, , j
x x x . Moreover, is said to be a subword of
a
x, denoted as ax, if axI for some interval
IZ , where Z is the set of all integers. The set of bi-infinite configurations is denoted by SZ, and a dis- tance d onSZis defined by
| |
, 2
i i i
, i x y ,
d x y
where x
,x1,x x0, ,1
, 1 0,
, , 1,
Zy y y y S , and 0 if i i
x yi i ,
x y , or 1 otherwise. It is known that SZ is a compact, perfect and totally disconnected metric space [23].
For a map f S: Z SZ, a set X SZ is said to be if
-inv nt
f aria f X
X . If X is closed and-inv nt,
f aria then
X f,
is called a subsystem of thedynamical system . For example, let be a set of some words of length
over , and,
SZ f
n S be
the set of the bi-infinite configurations consisting of all the words in . Then,
,
is a subsystem of
SZ,
, where is the left shift L
x i xi1 (or the right shift R
x i xi1) defined onZ S , and is called the determinative block system of
. The subsystem
,
0 a1
, 1,
i b
, or simply is called a subshift of finite type (SFT) with respect to .
,
E
0, ,bn1
,,bFurthermore, can be described by a finite directed graph, , where each vertex is labeled by a word in , and is the set of edges connecting the vertices in . Two vertices
and
are connected by an edge 0 n1
if and only if1
k k . One can think of each element of as a bi-infinite path on the graph . Whereas a directed graph corresponds to a square transition matrix with if and only if there is an edge
, ,
a a
2,,
ij A
, G
b b
0 n1
1 n
1
E, , a a
a b k
m m
n
ijA A
b
G
from vertex to vertex b j , where
1
m is the
number of elements in , and (or ) is the code of
the corresponding vertex in , . Thus,
i j
i j, 0,1,,m
is precisely defined by the transition matrix A. Remarkably, a
0,1 square matrix A is irreducible if, for any , there exists an n such that ; aperiodic if there exists an such that for all, where
,
i j
n ij
0 n ij
A
0 n ij
A
n
,
i j A is the
i j, entry of the power matrixn
A . If is an SFT of
Z,
S , then it is transitive if
and only if A is irreducible; it is mixing if and only if
A is aperiodic. Equivalently, A is irreducible if and only if for every ordered pair of vertices b i and b j there is a path in G starting at b i and ending at
j
b [23,24].
2. Transitivity and Chaoticity
In this section, it is proved that, for any 1-D CA res- tricted on its Bernoulli-shift subsystem, the shift tran- sitivity implies the CA transitivity, and transitive non- trivial Bernoulli subshift of finite type (BSFT) has dense periodic points. Consequently, for 1-D CA, transitivity implies chaos in the sense of Devaney on the non-trivial BSFT.
2.1. Shift Transitivity Implies CA Transitivity
Definition 2. A closed invariant subset SZ of a 1-D CA f is called a Bernoulli-shift subsystem if there exists an integer pair
q p,
with pr q 0, such that fp
x q
x ,x , where is the radius of rthe local map fˆ of the CA f and is the left (or right) shift map.
For simplicity, only consider as the left shift in the following discussion.
Proposition 1. If is a Bernoulli-shift subsystem of a 1-D CA f with fp
x q
x ,x ,prq0, then there exists a
2pr1
-sequence set such that
x SZ xi pr i pr , ,
i Z
.
Proof: If f Sˆ : 2r1S is the local map of f, then one can get the p times iteration of fˆ , fˆp:S2pr1 S.
Thus, f p
x q
x ,x , if and only if
,
ˆp p
i pr i pr
qi q
i i
f x f x x x ,
for all iZ. Let
2 1
0 2 0 2
,
, , , ,
,
pr
, ,
pr p
i pr i pr
a a S a a
x x i Z
r
and
, i pr i,
Z
pr
x S x i Z
,
Definition 3. The Bernoulli-shift subsystem in Proposition 1 is called the Bernoulli subshift of finite type (BSFT), and is called the determinative block system of . If BSFT is an infinite set, then it is said to be non-trivial.
Based on Definitions 1, 3 and Proposition 1, an obvi- ous result is the following proposition.
Proposition 2. Consider two BSFTs, 1 and 2, of a 1-D CA f with
, , 1, 2, 0i
p q
f x x x i prq
.
.
Then, if and only if .
1 2 1 2
Theorem 1. Let be a BSFT of a 1-D CA
with If the shift
x ,xf
p q
f x : is
transitive, then f : is also transitive.
Proof: Since the transitivity of on is equiva- lent to the existence of a such that
x
Orb x , where
2
, , ,
Orb x x x x
is the orbit of starting from x and Orb
x is its closure [14,15]. It can be verified that for any, there exists at least an such that
2pr1 -seq
i
uence
Z
a a0, 1,,a2pr
i i, 2pr
0, 1, , 2pr
x a a a .
Conversely, for any , the .
iZ
2pr1 -sequence
i i, 2pr
x
For the x above, consider the orbit
2
, , ,
f
Orb x x f x f x
and let Orbf
x . It is clear that is closed and . Because is -invariant and closed, one
f
f
has and fp x q
x ,x . Obviously, fis transitive on
e determinative block system of
.
Let denote th .
On one and, based on Proposition 2 and , one has . On the other hand, since x llows
that ,
h
, it fo
i i, 2pr
x i Z, but the
2 1
-sequence onsisting of i i, 2prpr
set c x
iZ
is is implies
and
. Th
. Thus, , i.e., f is transi-
rk 1.
gives a convenient method to check if a C
tive on .
Rema
1) Theorem 1
A f is transitive on a BSFT, since is transitive on SFT if and only if the transition matrix orresponding to the SFT is irreducible [23,24].
2) Theorem 1 shows that the shift transitivity implies the CA transitivity on the BSFT,
c
but the inverse im-
plication may not be correct, with a counter example of ECA f39. One has
2 39
f , where
0 ,1 and 0
, 0, 0, 0,
, 1
,1,1,1,
, s o on- tains two points. Itc is clear that f39 is transitive but
is not. In case BSFT is an infinite ansitiv
se
is sti
t, whether the CA transitivity implies the shift tr ity on the BSFT
ll an open problem.
3) When a BSFT is a finite set on which f is transitive, then it is a set of f-periodic points,
x f x, , ,fk1
x for some
Z
xS and
1 0
k , it is said that is
all two claims proved in [22]: 1 trivial.
Remark 2.
Rec ) transitive 1-D CA
is tive; 2) a 1-D CA is transitive but not se
always sensi f
nsitive on a SFT if and only if
x, x , ,k2 x
for some Z
xS and k20. It is easy to be ver d that k1k2 and t
q.
2.2. Transitivity Implies De ifie
p a
hey are
nsity of Periodic Points
com- mon multiples of nd
Theorem 2 Let be a BSFT of a 1-D CA
with
f
q
x ,xp
f x . If the shift : is transitive, then th eriodic points of f,
e set of p
0,
,P f y n fn y y is dense in .
Proof: Let sy
be the BSFT, a ter-
minative block stem. For any , there ex
nd be its de
and > 0
x
ists a positive integer M
pr
such that1 1
< 2 i
,2i M
and for
a0,,a2M
xM M, x ,it follows that
a2M2pr,,a2M
, a0,,a2pr
. is transitive on , there exists a path from Since
a2M2pr,,a2M
to
a0,,a2pr
in the graph
, . Let
G E
2M2pr, , 2M, 0, , k0, 0, , 2pr
b a a b b a a
be the sequence corresponding to this path. Then, its any
2pr1
-subsequences belong to .Now, construct a cyclic configuration
, , , ,
yb b b b , where
0, , 2M, 0, , k0
b a a b b .
and m
y y, where m by
is the length of b. Thus, y and
mp
f y mq y
M M, M M, e. ,
y x , i . yP f
a n d d x
,y .Therefore, the sets of periodic se in
d som n [21], th g
points is den
By Theorem 2 an e results i e followin
he rem 3. If
P f
.
two theorems are obtained.
T o is transitive on a non-trivial BSFT of
is chaotic in the sense of
D ns
2. s
and Chua’s ny sub- is rule’s a 1-D CA f , then f is sensitive on the BSFT.
Theorem 4. A 1-D CA f
evaney on its tra itive non-trivial BSFT.
3. An Example of Tran itive ECA Rule Rule 26 is a member of Wolfram’s class IV hyper Bernoulli-shift rules, which defines ma systems with rich and complex dynamics. Th
local map is fˆ
0, 0,1
fˆ
0,1,1
fˆ
1, 0, 0
1 and
ˆ , , 0
f a b c for all other triples
a b c, ,
0,13, and the correspondingProposition 3
global map is denoted by There exists a BSFT of
26.
f
, 26
f
2, 2
,Z i i
x S x
such that f2
x 2
x x and , where
26 , S 0,1
0, 0, 0,1, 0 , 0, 0,1, 0,1 , 0,1, 0, 0,1 ,
0,1,1 , 0,1,1, 0, 0 , 0,1
1, 0, 0,1, 0 , 1, 0,1,1, 0 , 1,1, 0, 0,1 , 1,1, 0,1,1 .
0,1, ,1, 0,1 ,
Proof: Firstly, is a closed set because SZ at
2
is an open set. Th n be easily verified th
for and
en, it ca any
26
f x x , fˆ262
a ai for an aty
ai2,ai1,a ai, i1,ai2
. This implies th
2 2
a
26
f x x for x .
It is clear that the transition matrix corresponding to is
Proposition 4. The transition matrix
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 1
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
A
A is irreduci- ble, so f26
: Le
is transitive on Proof t
. , where
C A I I
denote the
is the identity matrix
and let elements of the
po t
o is aperiodic.
,
1 , 1
Cn i j 0
ij
wer matrix Cn. It can be easily verified hat
0, 10,
n ij
C 1i j, for all 7, s Recall that a m trix
n C
a A is irreducible if and only if
AI is aperiodic 3,24][2 . Hence, A is irreducible and so f26 is transitive on .
Theorem 5 f26 is chaotic in the sense of Devaney on
.
3. Conclusion
As
been widely f na ay
ta,” Universi 1966.
[2] M. Gardner
a particular class of dynamical systems, CA have used
ty
, “ or m
Des rich
he of Illino
od pi a
or
eling an te their appa
nd com
ant No. hin
y i
d sim re lex e
CityU1
tions
ulating many
. nt simplicity,
volutions, but
g Kong Re- 117/10) and dical University
Automa- London,
of John Con- physical phenome
1-D CA can displ p
ese Me
of Self-Reproducing s Press, Urbana and
many properties of their temporal evolutions are unde- cidable [25,26]. Although checking the transitivity based on its definition is very difficult [27], and it alone is not sufficient for chaos to exist in general dynamical systems, this work has rigorously proved that the shift transitivity implies the CA transitivity, and the CA with transitive non-trivial BSFT are chaotic in the sense of Devaney, partly answer the open question whether Devaney chaos in 1-D CA is equivalent to transitivity [28].
4. Acknowledgments
This research was jointly supported by the NSFC (Grants No. 11171084 and No. 60872093), the Hon
search Grants Council (Gr Foundation of Zhejiang C (Grant No. 17211076).
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