# Chaotic Behaviors of Symbolic Dynamics about Rule 58 in Cellular Automata

Authors: Yangjun Pei; Qi Han; Chao Liu; Dedong Tang; Junjian Huang
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

The complex dynamical behaviors of rule 58 in cellular automata are investigated from the viewpoint of symbolic dynamics. The rule is Bernoulliστ-shift rule, which is members of Wolfram’s class II, and it was said to be simple as periodic before. It is worthwhile to study dynamical behaviors of rule 58 and whether it possesses chaotic attractors or not. It is shown that there exist two Bernoulli-measure attractors of rule 58. The dynamical properties of topological entropy and topological mixing of rule 58 are exploited on these two subsystems. According to corresponding strongly connected graph of transition matrices of determinative block systems, we divide determinative block systems into two subsets. In addition, it is shown that rule 58 possesses rich and complicated dynamical behaviors in the space of bi-infinite sequences. Furthermore, we prove that four rules of global equivalence class ε43 of CA are topologically conjugate. We use diagrams to explain the attractors of rule 58, where characteristic function is used to describe that some points fall into Bernoulli-shift map after several times iterations, and we find that these attractors are not global attractors. The Lameray diagram is used to show clearly the iterative process of an attractor.


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## Body

## 1. Introduction

Cellular automaton (CA) was first introduced in 1951 [1]. CA is a mathematical model consisting of large numbers of simple identical components with local interactions [2]. The simple components act together to produce complex global behavior. CA performs complex computation with high degree of efficiency and robustness. Three major factors have resulted in the revival of interest in the behavior of cellular systems [3]. First, the development of powerful computers and microprocessors has made the rapid simulation of CA possible. Second, the use of CA to simulate physical systems has attracted much interest in the scientific community. Third, the advent of VLSI as an implementation medium has focused attention on the communication requirements of successful hardware algorithms. In recent years, many applications of CA have been reported, especially in cryptography [4–9], image processing [10, 11], and associative memory [12, 13].In recent years, many researches were devoted to find properties of rules of CA, especially binary one-dimensional CA. Because the rules of binary one-dimensional CA are simple to study, the evolutions of these rules can be reflected directly by image. In 1980s, Wolfram proposed CA as models for physical systems which exhibit complex or even chaotic behaviors based on empirical observations, and he divided the 256 ECA (binary one-dimensional CA with radius 1) rules informally into four classes using dynamical concepts like periodicity, stability, and chaos [14–16]. Recently, [17–22] focused on ECA in detail. In [17], Chua et al. listed 256 Boolean function “cubes” defining all Boolean functions of three binary variables, and they elucidated that every binary cellular automata of any spatial dimension was a special case of a cellular automaton with the same neighborhood size. In [18], Chua et al. partitioned the entire set of 256 local rules into 16 different gene families. Chua et al. [19] mentioned that each rule has three globally equivalent local rules determined by three corresponding global transformations, namely, left-right transformation T†, global complementation T-, and left-right complementation T*. Each equivalence class is identified by εmκ, where κ is complexity index and m is index of κth class. In [20], the authors presented that 112 rules of 256 local rules were Bernoulli στ-shift rules. Each of the 112 Bernoulli στ-shift rules has an ID code BN[α,β,τ], where α denotes the number of attractors of rule N, β denotes the slope of the Bernoulli στ-shift map, and τ denotes the relevant forward time-τ. Hence, the space-time evolution of any one of the 112 rules on their attractors can be uniquely predicted by two parameters: β=±2σ and τ. For example, rule 58 has two attractors (β=2,σ=1,τ=1) and (β=1/2,σ=-1,τ=2). Wolfram considered that Bernoulli στ-shift rules were simple as periodic, but he did not find other complex dynamical behaviors. Recently, some authors [23–28] found that some rules of Bernoulli στ-shift rules are chaotic in the sense of both Li-York and Devaney. However, they did not involve and studied the dynamical behavior of rule 58. For 256 rules, every rule has its properties which are different from other rules. So, we need to study these special properties for every rule for practical application.In this paper, the complex dynamical behavior of rule 58 in cellular automata is studied in detail. It will be shown that rule 58 with bi-infinite bit strings possesses rich and complicated dynamical behaviors. The rest of the paper is organized as follows. In Section2, the basic concepts of one-dimension CA (1DCA) and symbolic dynamics are introduced. The Boolean functions of rule 58 are also presented, and expressions of two attractors are given. In Section 3, two subsystems of rule 58 are characterized. We prove that rule 58 is topologically mixing and chaotic in the sense of Li-York and Devaney on the Λƛ158 and Λ258, respectively. We also prove that four rules of global equivalence class ε43 of CA are topologically conjugate. In Section 4, characteristic function is used to describe that all points of rule 58 fall into Bernoulli-shift map after several times iterations, and the Lameray diagram is used to show clearly the iterative process of an attractor. We mention that rule 58 can be used in associative memory. Section 5 presents some conclusions.


## 2. Preliminaries

For simplicity, for some notations about symbolic dynamics, one can refer to [27–29].It follows from [30] that the Boolean function of rule 58 is
(1)[f58(x)]i=xi-1·x-i⊕x-i-1·xi+1∀x∈SZ, i∈Z, where “·,” “−,” and “⊕” stand for “AND,” “NOT,” and “XOR” logical operation, respectively. Sometimes, “·” is omitted for simplicity. The truth table of Boolean functions of rule 58 is shown in Table 1. The subsets, denoted by Λ158 and Λ258, are derived from the parameters of rules 58: β=2, σ=1, and τ=1 and β=1/2, σ=-1, and τ=2, respectively; that is,
(2)Λ158={x∈SZ∣[f58(x)]i=xi+1,∀i∈Z},Λ258={x∈SZ∣[f582(x)]i=xi-1,∀i∈Z},
where f2(·) denotes two iterations for a bit.Table 1
The truth table of Boolean function of rules 58.
x
i
-
1
x
i
x
i
+
1
x
i
-
1
x
-
i
⊕
x
-
i
-
1
x
i
+
1
000
0
001
1
010
0
011
1
100
1
101
1
110
0
111
0The subsetsΛ158 and Λ258 will be rigorously characterized from the viewpoint of symbolic dynamics.


## 3. Dynamical Behaviors off58 on Two Subsystems

In this section, dynamical behaviors off58 will be researched. We find that rule 58 is chaotic in the sense of Li-York and Devaney on its two subsystems.Proposition 1.
For rule 58, there exists a subsetΛ158⊂SZ which satisfies f58|Λ158=υ|Λ158 if and only if ∀x=(…,x-1,x0,x1,…)∈Λ158; xi-1, xi and xi+1 have the following relations:(i)
ifxi=1, thenxi-2=1,xi-1=0,xi+1=0, andxi+2=1;xi-2=0,xi-1=0,xi+1=1, andxi+2=0;xi-2=0,xi-1=1,xi+1=0, andxi+2=1;(ii)
ifxi=0, thenxi-2=0,xi-1=0,xi+1=0,xi+2=0,xi-1=1, andxi+1=1.
The explicit proof of this proposition is presented in the Appendix.Remark 2.
From the definition of subsystem, we know that(Λ158,f58) is subsystems of (SZ,f58).
The dynamical behaviors off58(x) on the set Λ158 are shown as follows.
LetP158={r0,r1,r2,r3,r4} be a new state set, where r0=(000), r1=(010), r2=(011), r3=(101), r4=(110), and ϖ158={(rr′)∣r=(b0b1b2),r′=(b0′b1′b2′)∈P158,∀1≤j≤2 such that bj=bj-1′}. Furthermore, subshift Λϖ158 of υ is defined as Λϖ158={r=(…r-1,r0,r1…)∈P158Z∣ri∈P158,riri+1∈ϖ158,∀i∈Z}. The transition matrix B158 of the υ|Λϖ158 is
(3)B158=[1000000010000010110000010].
In order to give our results, in the following, some definitions need be introduced.Definition 3 (see [31]).
A square{0,1} matrix A is irreducible, if for every pair of indices i and j there is an n such that Aijn>0.Definition 4 (see [31]).
A square{0,1} matrix A is aperiodic, if there exists N, such that Aijn>0 and n>N, ∀i,j.Definition 5 (see [31]).
Suppose thatg:X→Y is a continuous mapping, where X is a compact topological space. g is said to be topologically mixing if, for any two open sets U,V⊂X, ∃N>0, such that gn(U)∩V≠∅, ∀n≥N.Definition 6 (see [24]).
Let(X,f) and (Y,g) be compact spaces, one says that f and g are topologically conjugate if there is homeomorphism h:X→Y, such that h∘f=g∘h.
A square{0,1} matrix corresponds to a directed graph. The vertices of the graph are the indices for the rows and columns of A. There is an edge from vertex i to vertex j if Aij=1. A square {0,1} matrix is irreducible if and only if the corresponding graph is strongly connected. If ΛA is a two-order subshift of finite type, then it is topologically mixing if and only if A is irreducible and aperiodic [31].
We give corresponding graphG581 of the matrix B158 in Figure 1. We find that G581 is not a strongly connected graph. Therefore, f58|Λ158 is not topologically mixing. But the subgraphs r0 and r1, r2, r3, and r4 are strongly connected graph, respectively. So, we can divide P158 into two subsets: ℏ158={r0} and ƛ158={r1,r2,r3,r4}.Figure 1
The corresponding graphG581 of the matrix B158.Leth158={(rr′)∣r=(b0b1b2),r′=(b0′b1′b2′)∈P158,∀1≤j≤2, such that bj=bj-1′} corresponding to ℏ158 and λ158={(rr′)|r=(b0b1b2),r′=(b0′b1′b2′)∈P158,∀1≤j≤2, such that bj=bj-1′} corresponding to ƛ158. We can obtain the transition matrix C158 corresponding to υ|Λℏ158and transition matrix D158 corresponding to υ|Λƛ158, where
(4)C158=[1],D158=[0010000111000010].Based on the above definition and analysis, we give the following results.Proposition 7.
Consider the following:(a)
Λℏ158∪Λƛ158=Λ158, whereℏ158∪ƛ158=P158andℏ158∩ƛ158=∅.(b)
υ:Λƛ158→Λƛ158andυ:Λλ158→Λλ158are topologically conjugate.(c)
υ:Λƛ158→Λƛ158is topologically mixing.(d)
f58:Λƛ158→Λƛ158is topologically mixing.(e)
The topological entropyent(f58|Λƛ158)=ent(υ|Λƛ158)=0.2812.Proof.
(a) Letx={…,x-1,x*0,x1}. It is obvious that if w′≺x, w′∈ℏ158, then ∀w′′∈ƛ158, w′′⊀x, thus x∈Λℏ158; conversely, if w′≺x, w′∈ƛ158, then ∀w′′∈ℏ158, w′′⊀x, thus x∈Λƛ158; namely, ∀x∈Λ158, x∈Λℏ158, or x∈Λƛ158. Hence, Λℏ158∪Λƛ158=Λ158.
The proofs of (b), (c), (d), and (e) can be referred to in Proposition  2 in [28].Theorem 8.
f
58 is chaotic in the sense of both Li-Yorke and Devaney on Λƛ158.Proof.
It follows from [32] that the positive topological entropy implies chaos in the sense of Li-Yorke, and topological mixing implies chaos in the sense of Li-Yorke and Devaney, since rule N=58 possesses very rich and complicated dynamical properties on Λƛ158.Remark 9.
Though,∀n>0, (C158)n>0, we cannot believe that f58|ℏ158 is topologically mixing. Because C158 is not a square {0,1} matrix. The topological entropy ent(f58|Λℏ158)=0. So, f58 is not chaotic in the sense of both Li-Yorke and Devaney on Λℏ158.Remark 10.
Carefully observing Figure1, we find that there are several strongly connected subgraphs: r2→r3→r4→r2, r1→r3→r1, r1→r3→r2→r4→r3→r1, and r0→r0. The elements of Λ158 are composed by all vertices of those strongly connected subgraphs, respectively. For example, x∈Λ158 and x is composed of vertices of subgraph r2→r3→r4→r2; then we have r0⊀x and r1⊀x, and all vertices of the subgraph will appear in x, if |x|=3k, k=1,2,….Proposition 11.
For rule 58, there exists a subsetΛ258⊂SZ which satisfies f582|Λ258=ς|Λ258 if and only if, ∀x=(…,x-1,x0,x1,…)∈Λ258, xi-1xixi+1 cannot equal to 010, ∀i∈Z.Proof.
The global map of rule 58 is[f58(x)]i=xi-1x-i⊕x-i-1xi+1, so
(5)[f582(x)]i=[f58(x)]i-1·[f58(x)]i¯⊕[f58(x)]i-1¯·[f58(x)]i+1=[(xi-2x-i-1⊕x-i-2xi)·xi-1x-i⊕x-i-1xi+1¯]⊕[xi-2x-i-1⊕x-i-2xi¯·(xix-i+1⊕x-ixi+2)].
Then, the proof is similar to proof of necessity and sufficiency in Proposition1, so the details are omitted.LetP258={r0,r1,r2,r3,r4,r5,r6}, where r0=(000), r1=(001), r2=(011), r3=(100), r4=(101), r5=(110), and r6=(111).Remark 12.
The transition matrix of subshift(Λ258,ς) is
(6)B258=[1100000001000000000111100000001000000011000000011].
We give corresponding graphG582 of the matrix B258 in Figure 2. It is obvious that G582 is a strongly connected graph. So, B258 is irreducible.Figure 2
The corresponding graphG582 of the matrix B258.Based on the above analysis, we have the following results.Proposition 13.
Consider the following:(a)
ς|Λ258is topologically mixing.(b)
f582|Λ258is topologically mixing.(c)
The topological entropyent(f582|Λ258)=ent(ς|Λ258)=0.5624.Proof.
(a) Because(B258)n>0, ∀n≥4, the transition matrix B258 of subshift of finite type ς is irreducible and aperiodic. By [32, 33], ς|Λ258 is topologically mixing.Theorem 14.
f
58
2 is chaotic in the sense of both Li-Yorke and Devaney on Λ258.Proof.
It follows from [32] that topological mixing implies chaos in the sense of Li-Yorke and Devaney.Proposition 15.
Considerf(x)∈Λ258, ∀x∈Λ258.Proof.
We check thatx1,x2∈Λ258, where x1={…,a-2,a-1,a0,a1,a2,…}. Let f582(x1)=x2. Suppose that f58(x1)=y1 and f58(x2)=y2, where y1={…,b-2,b-1,b0,b1,b2,…} and y2={…c-3,c-2,c-1,c0,c1,…}. Table 2 shows the iterative process of bit string x1. Observing Table 2, we have f58(a-1a0a1)=b0=c0,…,f58(ai-1aiai+1)=bi=ci,…. So, we get f582(y1)=y2. Therefore, y1,y2∈Λ258. Hence, f(x)∈Λ258, ∀x∈Λ258.Table 2
The iterative process of bit stringx1.
x
1
⋯
a
-
2
a
-
1
a
0
a
1
a
2
⋯
y
1
⋯
b
-
2
b
-
1
b
0
b
1
b
2
⋯
x
2
⋯
a
-
3
a
-
2
a
-
1
a
0
a
1
⋯
y
2
⋯
c
-
3
c
-
2
c
-
1
c
0
c
1
⋯Corollary 16.
Considerf582|Λ258=(f58|Λ258)2.Proposition 17.
Consider the following:(a)
f58|Λ258is topologically mixing.(b)
The topological entropyent(f58|Λ258)=0.2812.
The explicit proof of this proposition is presented in the Appendix.Theorem 18.
f
58 is chaotic in the sense of both Li-Yorke and Devaney on Λ258.Remark 19.
It is obviously that(Λ258,f58) are subsystems of (SZ,f58). Hence, there are two subsystems for (SZ,f58).
Next, we will discuss the relationship on four rules of global equivalence classε43.Remark 20.
From [20, 28–30], the following results can be obtained:(1)
f58:SZ→SZ and f114:SZ→SZ are topologically conjugate;(2)
f58:SZ→SZ and f163:SZ→SZ are topologically conjugate;(3)
f58:SZ→SZ and f177:SZ→SZ are topologically conjugate.Remark 21.
f
58, f114, f163, and f177 are topologically conjugate, respectively. Therefore, if we know that one of four rules is chaotic in the sense of both Li-Yorke and Devaney in its attractors, we can deem that others of four rules are chaotic in the sense of both Li-Yorke and Devaney in their attractors, respectively. The phenomenon also presents that the global equivalence class introduced by Chua et al. [22] is useful and important for research of rule of cellular automata.


## 4. Using Diagrams to Explain Attractors of Four Rules

From a definition on global characteristic function in [20], the Boolean string x can be associated with a real number 0•x0x1…xI-1xI on the unit interval [0,1]:
(7)x=[x0x1…xI-1xI]⟼ϕ≜0•x0x1…xI-1xI,fsdfsdfsdfsdfsfsdfsdfsdsdfdxi∈{0,1},
where ϕ=∑i=0I2-(i+1)xi is the decimal form of Boolean string x=[x0x1…xI-1xI]. The CAs’ characteristic function χN of rule N is defined as
(8)χN:Q[0,1]⟶Q[0,1],thatis,ϕn=χN1(ϕn-1),ϕn=χN2(ϕn-2),
where Q denotes rational numbers.LetI=13. Figure 3 shows characteristic functions of rule 58. Figure 3(a) describes that some points of rule 58 fall into Bernoulli-shift map after twenty-six iterations, respectively, where τ=1. Figure 3(b) describe that some points of four rules fall into Bernoulli-shift map after twenty-seven iterations, respectively, where τ=2.All points fall into Bernoulli-shift map after several iterations under rule 58, whereI=13.
(a)
(b)If we choose different values ofI for the four rules, we can get different initial binary configuration for the evolution of four rules. The different initial binary configuration may lead to different attractor periods. If the value of I is fixed, we find that the attractor period may be different. Let I=5; then we can obtain Figure 4, which shows some attractors of rule 58. Figures 4(a) and 4(b) show that the period of attractor is 1, and the attractor belongs to Λℏ158 and Λ258; Figure 4(c) shows that the period of attractor is 2, and the attractor belongs to Λƛ158; Figure 4(d) shows that the attractor belongs to both Λƛ158 and Λ258, where the solid lines represent that the attractor belongs to Λƛ158 and the dotted lines represent that the attractor belongs to Λ258. Let I=4; then we can obtain Figure 5, where the solid lines stand for the real evolution of binary configuration 11100, and the dotted lines stand for the attractor belonging to Λ258.Some attractors of rule 58, whereI=5, and the white lattice stands for 0 and black for 1.
(a)
(b)
(c)
(d)Figure 5
Period-5 isle of Eden is shown, whereI=4, and the white lattice stands for 0 and black for 1.Next, we use the Lameray diagram [33] to present our attractors. The diagrams show clearly the iterative process of attractors. In terms of the attractor of Figure 4(d), we get that the values of characteristic function of the attractor are 0.7031, 0.4219, and 0.8438, respectively. Figure 6(a) shows the iterative process of an attractor, where τ=1, and Figure 6(b) shows the iterative process of an attractor, where τ=2. Then, we can associate the period-3 attractor of rule 58 in Figure 6(a) as a period-3 point of a continuous map f:[0,1]→[0,1] which we know that it is chaotic because “period-3 implies chaos” [34]. We can also consider Figure 6(b) by the above method. Chaos implies that rules 58 have infinite period orbits on its subsystems.The evolution of characteristic function of the period-3 attractor, where the values of characteristic function of the attractor are 0.7031, 0.4219, and 0.8438, respectively. (a) shows the iterative process of an attractor, whereτ=1, and (b) shows the iterative process of an attractor, where τ=2.
(a)
(b)In recent years, associative memory was researched in many papers [12, 13]. It is obvious that rule 58 can be used in associative memory. By strongly connected graph of rule 58, we can know the elements on its attractors. Then, we can choose a bit string which belongs to an attractor as memory pattern. Since there are infinite orbits, the storage capability is very large. For example, we can choose 010101 as a memory pattern in Figure 4(c). The associative memory model provides a solution to problem where time to recognize a pattern is independent of the number of patterns stored.


## 5. Conclusions

In this paper, the dynamical behaviors of rule 58 in cellular automata, which is Bernoulliστ-shift rule, are carefully investigated from viewpoint of symbolic dynamics. We derive the conditions according to Bernoulli στ-shift evolution for rule 58. Then, in terms of the transition matrices of determinative block systems of subsystems of rule 58, we obtain the values of topological entropy of subsystems. According to corresponding strongly connected graph of transition matrices of determinative block systems of subsystems Λ158, we divide determinative block systems into two subsets. Then, we find that rule 58 is topologically mixing on Λƛ158. Furthermore, we find that f58|Λ258 is topologically mixing. So, rule 58 is chaotic in the sense of both Li-Yorke and Devaney. Then, we prove that four rules belonging to global equivalence class ε43 of CA are topologically conjugate. We use diagrams to explain the attractors of rule 58, where characteristic function and the Lameray diagram are used to describe that some points fall into Bernoulli-shift map after several times iterations and to show clearly the iterative process of an attractor, respectively.



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