Chaotic Behaviors of Symbolic Dynamics about Rule 58 in Cellular Automata

Abstract

The complex dynamical behaviors of rule 58 in cellular automata are investigated from the viewpoint of symbolic dynamics. The rule is Bernoulli 𝜎_𝜏-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 𝜀³₄ 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.

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 𝜀^𝜅_𝑚, 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 𝜎_𝜏-shift rules. Each of the 112 Bernoulli 𝜎_𝜏-shift rules has an ID code 𝐵_𝑁[𝛼,𝛽,𝜏], where 𝛼 denotes the number of attractors of rule 𝑁, 𝛽 denotes the slope of the Bernoulli 𝜎_𝜏-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=±2^𝜎 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 𝜎_𝜏-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 𝜎_𝜏-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 Section 2, 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 ₋Λ_𝜆1⁵⁸ and Λ₂⁵⁸, respectively. We also prove that four rules of global equivalence class 𝜀³₄ 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 𝑓₅₈(𝑥)_𝑖=𝑥_𝑖−1⋅𝑥_𝑖⊕𝑥_𝑖−1⋅𝑥_𝑖+1 (1)∀𝑥∈𝑆^𝑍, 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, Λ₁⁵⁸=𝑥∈𝑆^𝑍∣𝑓₅₈(𝑥)_𝑖=𝑥_𝑖+1,Λ,∀𝑖∈𝑍₂₅₈=𝑥∈𝑆^𝑍∣𝑓²₅₈(𝑥)_𝑖=𝑥_𝑖−1,∀𝑖∈𝑍, (2) where 𝑓²(⋅) denotes two iterations for a bit.

Table 1 

The truth table of Boolean function of rules 58.

The subsets Λ₁⁵⁸ and Λ₂⁵⁸ will be rigorously characterized from the viewpoint of symbolic dynamics.

3. Dynamical Behaviors of 𝑓₅₈ on Two Subsystems

In this section, dynamical behaviors of 𝑓₅₈ 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 subset Λ₁⁵⁸⊂𝑆^𝑍 which satisfies 𝑓₅₈|_Λ158=𝜐|_Λ1⁵⁸ if and only if ∀𝑥=(…,𝑥₋₁,𝑥₀,𝑥₁,…)∈Λ₁⁵⁸; 𝑥_𝑖−1, 𝑥_𝑖 and 𝑥_𝑖+1 have the following relations:(i)if 𝑥_𝑖=1, then 𝑥_𝑖−2=1, 𝑥_𝑖−1=0, 𝑥_𝑖+1=0, and 𝑥_𝑖+2=1; 𝑥_𝑖−2=0, 𝑥_𝑖−1=0, 𝑥_𝑖+1=1, and 𝑥_𝑖+2=0; 𝑥_𝑖−2=0, 𝑥_𝑖−1=1, 𝑥_𝑖+1=0, and 𝑥_𝑖+2=1;(ii)if 𝑥_𝑖=0, then 𝑥_𝑖−2=0, 𝑥_𝑖−1=0, 𝑥_𝑖+1=0, 𝑥_𝑖+2=0, 𝑥_𝑖−1=1, and 𝑥_𝑖+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 (𝑆^𝑍,𝑓₅₈).
The dynamical behaviors of 𝑓₅₈(𝑥) on the set Λ₁⁵⁸ are shown as follows.
Let 𝑃₁⁵⁸={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^′=(𝑏^′₀𝑏^′₁𝑏^′₂)∈𝑃₁⁵⁸,∀1≤𝑗≤2 such that 𝑏_𝑗=𝑏^′_𝑗−1}. Furthermore, subshift Λ_𝜛1⁵⁸ of 𝜐 is defined as Λ_𝜛1⁵⁸={𝑟=(…𝑟₋₁,𝑟₀,𝑟₁…)∈𝑃₁^𝑍58∣𝑟_𝑖∈𝑃₁⁵⁸,𝑟_𝑖𝑟_𝑖+1∈𝜛₁⁵⁸,∀𝑖∈𝑍}. The transition matrix 𝐵₁⁵⁸ of the 𝜐|_Λ𝜛158 is 𝐵₁⁵⁸=.1000000010000010110000010 (3)
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 𝐴^𝑛_𝑖𝑗>0.

Definition 4 (see [31]). A square {0,1} matrix 𝐴 is aperiodic, if there exists 𝑁, such that 𝐴^𝑛_𝑖𝑗>0 and 𝑛>𝑁, ∀𝑖,𝑗.

Definition 5 (see [31]). Suppose that 𝑔∶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 𝑔^𝑛(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 𝐴_𝑖𝑗=1. A square {0,1} matrix is irreducible if and only if the corresponding graph is strongly connected. If Λ_𝐴 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 graph 𝐺¹₅₈ of the matrix 𝐵₁⁵⁸ in Figure 1. We find that 𝐺¹₅₈ 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: ₋ℎ₁⁵⁸={𝑟₀} and ₋𝜆₁⁵⁸={𝑟₁,𝑟₂,𝑟₃,𝑟₄}.

Figure 1 

The corresponding graph 𝐺158 of the matrix 𝐵158.

Let ℎ₁⁵⁸={(𝑟𝑟^′)∣𝑟=(𝑏₀𝑏₁𝑏₂),𝑟^′=(𝑏^′₀𝑏^′₁𝑏^′₂)∈𝑃₁⁵⁸,∀1≤𝑗≤2, such that 𝑏_𝑗=𝑏^′_𝑗−1} corresponding to ₋ℎ₁⁵⁸ and 𝜆₁⁵⁸={(𝑟𝑟^′)|𝑟=(𝑏₀𝑏₁𝑏₂),𝑟^′=(𝑏^′₀𝑏^′₁𝑏^′₂)∈𝑃₁⁵⁸,∀1≤𝑗≤2, such that 𝑏_𝑗=𝑏^′_𝑗−1} corresponding to ₋𝜆₁⁵⁸. We can obtain the transition matrix 𝐶₁⁵⁸ corresponding to 𝜐|_Λ−ℎ158and transition matrix 𝐷₁⁵⁸ corresponding to 𝜐|_Λ−𝜆158, where 𝐶₁⁵⁸=[1],𝐷₁⁵⁸=.0010000111000010 (4)

Based on the above definition and analysis, we give the following results.

Proposition 7. Consider the following:(a)₋Λ_ℎ1⁵⁸∪Λ_−𝜆1⁵⁸=Λ₁⁵⁸, where ₋ℎ₁⁵⁸∪₋𝜆₁⁵⁸=𝑃₁⁵⁸ and ₋ℎ₁⁵⁸∩₋𝜆₁⁵⁸=∅.(b)𝜐∶Λ_−𝜆1⁵⁸→Λ_−𝜆1⁵⁸ and 𝜐∶Λ_𝜆1⁵⁸→Λ_𝜆1⁵⁸ are topologically conjugate.(c)𝜐∶Λ_−𝜆1⁵⁸→Λ_−𝜆1⁵⁸ is topologically mixing. (d)𝑓₅₈∶Λ_−𝜆1⁵⁸→Λ_−𝜆1⁵⁸ is topologically mixing.(e)The topological entropy 𝑒𝑛𝑡(𝑓₅₈|_Λ−𝜆158)=𝑒𝑛𝑡(𝜐|_Λ−𝜆158)=0.2812.

Proof. (a) Let 𝑥={…,𝑥₋₁,𝑥^∗₀,𝑥₁}. It is obvious that if 𝑤^′≺𝑥, 𝑤^′∈₋ℎ₁⁵⁸, then ∀𝑤^′′∈₋𝜆₁⁵⁸, 𝑤^′′⊀𝑥, thus 𝑥∈Λ_−ℎ1⁵⁸; conversely, if 𝑤^′≺𝑥, 𝑤^′∈₋𝜆₁⁵⁸, then ∀𝑤^′′∈₋ℎ₁⁵⁸, 𝑤^′′⊀𝑥, thus 𝑥∈Λ_−𝜆1⁵⁸; namely, ∀𝑥∈Λ₁⁵⁸, 𝑥∈Λ_−ℎ1⁵⁸, or 𝑥∈Λ_−𝜆1⁵⁸. Hence, ₋Λ_ℎ1⁵⁸∪Λ_−𝜆1⁵⁸=Λ₁⁵⁸.
The proofs of (b), (c), (d), and (e) can be referred to in Proposition  2 in [28].

Theorem 8. 𝑓₅₈ is chaotic in the sense of both Li-Yorke and Devaney on ₋Λ_𝜆1⁵⁸.

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 ₋Λ_𝜆1⁵⁸.

Remark 9. Though, ∀𝑛>0, (𝐶₁⁵⁸)^𝑛>0, we cannot believe that 𝑓₅₈|_−ℎ158 is topologically mixing. Because 𝐶₁⁵⁸ is not a square {0,1} matrix. The topological entropy ent(𝑓₅₈|_Λ−ℎ158)=0. So, 𝑓₅₈ is not chaotic in the sense of both Li-Yorke and Devaney on ₋Λ_ℎ1⁵⁸.

Remark 10. Carefully observing Figure 1, 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 subset Λ₂⁵⁸⊂𝑆^𝑍 which satisfies 𝑓²₅₈|_Λ258=𝜍|_Λ2⁵⁸ if and only if, ∀𝑥=(…,𝑥₋₁,𝑥₀,𝑥₁,…)∈Λ₂⁵⁸, 𝑥_𝑖−1𝑥_𝑖𝑥_𝑖+1 cannot equal to 010, ∀𝑖∈𝑍.

Proof. The global map of rule 58 is [𝑓₅₈(𝑥)]_𝑖=𝑥_𝑖−1𝑥_𝑖⊕𝑥_𝑖−1𝑥_𝑖+1, so 𝑓²₅₈(𝑥)_𝑖=𝑓₅₈(𝑥)_𝑖−1⋅𝑓₅₈(𝑥)_𝑖⊕𝑓₅₈(𝑥)_𝑖−1⋅𝑓₅₈(𝑥)_𝑖+1=𝑥_𝑖−2𝑥_𝑖−1⊕𝑥_𝑖−2𝑥_𝑖⋅𝑥_𝑖−1𝑥_𝑖⊕𝑥_𝑖−1𝑥_𝑖+1⊕𝑥_𝑖−2𝑥_𝑖−1⊕𝑥_𝑖−2𝑥_𝑖⋅𝑥_𝑖𝑥_𝑖+1⊕𝑥_𝑖𝑥_𝑖+2. (5)
Then, the proof is similar to proof of necessity and sufficiency in Proposition 1, so the details are omitted.

Let 𝑃₂⁵⁸={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 𝐵₂⁵⁸=.1100000001000000000111100000001000000011000000011 (6)
We give corresponding graph 𝐺²₅₈ of the matrix 𝐵₂⁵⁸ in Figure 2. It is obvious that 𝐺²₅₈ is a strongly connected graph. So, 𝐵₂⁵⁸ is irreducible.

Figure 2 

The corresponding graph 𝐺258 of the matrix 𝐵258.

Based on the above analysis, we have the following results.

Proposition 13. Consider the following:(a)𝜍|_Λ2⁵⁸ is topologically mixing.(b)𝑓²₅₈|_Λ258 is topologically mixing.(c)The topological entropy 𝑒𝑛𝑡(𝑓²₅₈|_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. 𝑓²₅₈ 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. Consider 𝑓(𝑥)∈Λ₂⁵⁸, ∀𝑥∈Λ₂⁵⁸.

Proof. We check that 𝑥₁,𝑥₂∈Λ₂⁵⁸, where 𝑥₁={…,𝑎₋₂,𝑎₋₁,𝑎₀,𝑎₁,𝑎₂,…}. Let 𝑓²₅₈(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(𝑎_𝑖−1𝑎_𝑖𝑎_𝑖+1)=𝑏_𝑖=𝑐_𝑖,…. So, we get 𝑓²₅₈(𝑦₁)=𝑦₂. Therefore, 𝑦₁,𝑦₂∈Λ₂⁵⁸. Hence, 𝑓(𝑥)∈Λ₂⁵⁸, ∀𝑥∈Λ₂⁵⁸.

Table 2 

The iterative process of bit string 𝑥₁.

Corollary 16. Consider 𝑓²₅₈|_Λ258=(𝑓₅₈|_Λ258)².

Proposition 17. Consider the following:(a)𝑓₅₈|_Λ258 is topologically mixing.(b)The topological entropy 𝑒𝑛𝑡(𝑓₅₈|_Λ258)=0.2812.
The explicit proof of this proposition is presented in the Appendix.

Theorem 18. 𝑓₅₈ is chaotic in the sense of both Li-Yorke and Devaney on Λ₂⁵⁸.

Remark 19. It is obviously that (Λ₂⁵⁸,𝑓₅₈) are subsystems of (𝑆^𝑍,𝑓₅₈). Hence, there are two subsystems for (𝑆^𝑍,𝑓₅₈).
Next, we will discuss the relationship on four rules of global equivalence class 𝜀³₄.

Remark 20. From [20, 28–30], the following results can be obtained:(1)𝑓₅₈∶𝑆^𝑍→𝑆^𝑍 and 𝑓₁₁₄∶𝑆^𝑍→𝑆^𝑍 are topologically conjugate;(2)𝑓₅₈∶𝑆^𝑍→𝑆^𝑍 and 𝑓₁₆₃∶𝑆^𝑍→𝑆^𝑍 are topologically conjugate;(3)𝑓₅₈∶𝑆^𝑍→𝑆^𝑍 and 𝑓₁₇₇∶𝑆^𝑍→𝑆^𝑍 are topologically conjugate.

Remark 21. 𝑓₅₈, 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…𝑥_𝐼−1𝑥_𝐼 on the unit interval [0,1]: 𝑥=𝑥₀𝑥₁…𝑥_𝐼−1𝑥_𝐼⟼𝜙≜0_•𝑥₀𝑥₁…𝑥_𝐼−1𝑥_𝐼,𝑥_𝑖∈{0,1}, (7) where 𝜙=∑^𝐼_𝑖=02^−(𝑖+1)𝑥_𝑖 is the decimal form of Boolean string 𝑥=[𝑥₀𝑥₁…𝑥_𝐼−1𝑥_𝐼]. The CAs’ characteristic function 𝜒_𝑁 of rule 𝑁 is defined as 𝜒_𝑁∶𝑄[0,1]⟶𝑄[0,1],thatis,𝜙_𝑛=𝜒¹_𝑁𝜙_𝑛−1,𝜙_𝑛=𝜒²_𝑁𝜙_𝑛−2, (8) where 𝑄 denotes rational numbers.

Let 𝐼=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.

(a)

(b)

Figure 3 

All points fall into Bernoulli-shift map after several iterations under rule 58, where 𝐼=13.

If we choose different values of 𝐼 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 ₋Λ_ℎ1⁵⁸ and Λ₂⁵⁸; Figure 4(c) shows that the period of attractor is 2, and the attractor belongs to ₋Λ_𝜆1⁵⁸; Figure 4(d) shows that the attractor belongs to both ₋Λ_𝜆1⁵⁸ and Λ₂⁵⁸, where the solid lines represent that the attractor belongs to ₋Λ_𝜆1⁵⁸ 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.

(a)

(b)

(c)

(d)

Figure 4 

Some attractors of rule 58, where 𝐼=5, and the white lattice stands for 0 and black for 1.

Figure 5 

Period-5 isle of Eden is shown, where 𝐼=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.

(a)

(b)

Figure 6 

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, where 𝜏=1, and (b) shows the iterative process of an attractor, where 𝜏=2.

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 Bernoulli 𝜎_𝜏-shift rule, are carefully investigated from viewpoint of symbolic dynamics. We derive the conditions according to Bernoulli 𝜎_𝜏-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 ₋Λ_𝜆1⁵⁸. 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 𝜀³₄ 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.

Appendix

Proof of Proposition 1.
Necessity. Suppose that there exists a subset Λ₁⁵⁸∈𝑆^𝑍 such that 𝑓₅₈|_Λ158=𝜐|_Λ1⁵⁸. Then, ∀𝑥=(…,𝑥₋₁,𝑥₀,𝑥₁,…)∈Λ₁⁵⁸, we have [𝑓₅₈(𝑥)]_𝑖=𝑥_𝑖+1, ∀𝑥∈𝑍.(1)If 𝑥_𝑖=1, then 𝑥_𝑖−1𝑥_𝑖⊕𝑥_𝑖−1𝑥_𝑖+1=0⊕𝑥_𝑖−1𝑥_𝑖+1; according to Table 1, we get 𝑥_𝑖−1=0, 𝑥_𝑖+1=0, 𝑥_𝑖+2=1; 𝑥_𝑖−1=0, 𝑥_𝑖+1=1, and 𝑥_𝑖+2=0; 𝑥_𝑖−2=0, 𝑥_𝑖−1=1, 𝑥_𝑖+1=0, 𝑥_𝑖+2=1.(2)If 𝑥_𝑖=0, then 𝑥_𝑖−1𝑥_𝑖⊕𝑥_𝑖−1𝑥_𝑖+1=𝑥_𝑖−1⊕𝑥_𝑖−1𝑥_𝑖+1; according to Table 1, we get 𝑥_𝑖−2=0, 𝑥_𝑖−1=0, 𝑥_𝑖+1=0; 𝑥_𝑖−2=0, 𝑥_𝑖−1=0, and 𝑥_𝑖+1=1; 𝑥_𝑖−1=1, and 𝑥_𝑖+1=1.(3)Now, we prove that (001)⊀𝑥, 𝑥∈Λ₁⁵⁸. The proof is by contradiction. Suppose that (001)≺𝑥, 𝑥∈Λ₁⁵⁸. We can check that 𝑦∈Λ₁⁵⁸, where 𝑓(𝑦)=𝑥 and 𝑥=(…,𝑥₋₃,𝑥₋₂,0,0^∗,1,𝑥₂,𝑥₃,…). The following is considered in Table 1.
 ① Let 𝑦=(𝑦_−𝑘,…,𝑦₋₃,0,0,0^∗,0,1,𝑦₃,…,𝑦_𝑘). Firstly, 𝑦₋₃=0, 𝑦₋₄=0,…𝑦_−𝑛=0,…; otherwise, there exists (100)≺𝑦 which does not belong to Λ₁⁵⁸. Secondly, there certainly exists (𝑦_𝑖𝑦_𝑖+1𝑦_𝑖+2)=(100)≺𝑦, where 3≤𝑖≤𝑘. ② Let 𝑦=(𝑦_−𝑘,…,𝑦₋₃,𝑦₋₂,0,1^∗,0,𝑦₂,𝑦₃,…,𝑦_𝑘). We cannot get 𝑓(𝑦₋₂01)=0 in Λ₁⁵⁸. ③ Let 𝑦=(𝑦_−𝑘,…,𝑦₋₃,𝑦₋₂,1,1^∗,0,𝑦₂,𝑦₃,…,𝑦_𝑘). We cannot get 𝑓(𝑦₋₂11)=0 in Λ₁⁵⁸.
Based on the above analysis, these arrive at contradictions.
Therefore, we get that 𝑥_𝑖−1, 𝑥_𝑖, and 𝑥_𝑖+1 have the following relations:(i)if 𝑥_𝑖=1, then 𝑥_𝑖−2=1, 𝑥_𝑖−1=0, 𝑥_𝑖+1=0, 𝑥_𝑖+2=1; 𝑥_𝑖−2=0, 𝑥_𝑖−1=0, 𝑥_𝑖+1=1, and 𝑥_𝑖+2=0; 𝑥_𝑖−2=0, 𝑥_𝑖−1=1, 𝑥_𝑖+1=0, and 𝑥_𝑖+2=1.(ii)If 𝑥_𝑖=0, then 𝑥_𝑖−2=0, 𝑥_𝑖−1=0, 𝑥_𝑖+1=0, and 𝑥_𝑖+2=0; 𝑥_𝑖−1=1 and 𝑥_𝑖+1=1.
Sufficiency. Suppose that there exists a subset Λ₁⁵⁸⊂𝑆^𝑍, and, ∀𝑥∈Λ₁⁵⁸, the relations between 𝑥_𝑖−1, 𝑥_𝑖, and 𝑥_𝑖+1 satisfy the conditions (i) and (ii) in Proposition 1, ∀𝑖∈𝑍.(i)If 𝑥_𝑖=1, we have [𝑓₅₈(𝑥)]_𝑖=𝑥_𝑖−1𝑥_𝑖⊕𝑥_𝑖−1𝑥_𝑖+1=0⊕𝑥_𝑖−1𝑥_𝑖+1. Therefore, 𝑓₅₈(𝑥)_𝑖=0,𝑥_𝑖−1=0,𝑥_𝑖+1=0,1,𝑥_𝑖−1=0,𝑥_𝑖+1=1,0,𝑥_𝑖−1=1,𝑥_𝑖+1=0. (A.1)(ii)If 𝑥_𝑖=0, we have [𝑓₅₈(𝑥)]_𝑖=𝑥_𝑖−1𝑥_𝑖⊕𝑥_𝑖−1𝑥_𝑖+1=𝑥_𝑖−1⊕𝑥_𝑖−1𝑥_𝑖+1. Therefore, 𝑓₅₈(𝑥)_𝑖=0,𝑥_𝑖−1=0,𝑥_𝑖+1=0,1,𝑥_𝑖−1=1,𝑥_𝑖+1=1. (A.2)Hence, [𝑓₅₈(𝑥)]_𝑖=𝑥_𝑖+1.

Proof of Proposition 17. (a) Let any two open sets 𝑈,𝑉⊂Λ₂⁵⁸. By Proposition 13 (b), we know that 𝑓²₅₈|_Λ258 is topologically mixing; then there exists 𝑁₁>0, such that (𝑓²₅₈|_Λ258)^𝑛(𝑈)∩𝑉≠∅, ∀𝑛≥𝑁₁. Then we consider the following two cases.
Case 1. Consider 𝑛=2𝑘, 𝑘∈𝑍⁺; then 𝑓_58Λ258^𝑛(𝑈)=𝑓∩𝑉_58Λ2582𝑘(𝑈)=𝑓∩𝑉_258Λ258𝑘(𝑈)∩𝑉≠∅,∀𝑛≥2𝑁₁. (A.3)
Case 2. Consider 𝑛=2𝑘+1, 𝑘∈𝑍⁺. It is obvious that Λ₂⁵⁸ is surjective. Suppose that there exist 𝑦,𝑦₁∈Λ₂⁵⁸ such that 𝑓₅₈(𝑦)=𝑓₅₈(𝑦₁). Thus, 𝑓²₅₈(𝑦)=𝑓²₅₈(𝑦₁) holds, which implies 𝑦=𝑦₁. Therefore, 𝑓₅₈|_Λ258 is injective. Since Λ₂⁵⁸ is a compact Hausdorff space and 𝑓₅₈|_Λ258 is one-to-one onto, and continuous, 𝑓⁻¹₅₈|_Λ258 exists and is continuous. Therefore, 𝑓₅₈|_Λ258 is a homeomorphism, which implies that 𝑓₅₈(𝑈) is an open set. Thus, 𝑓_58Λ258^𝑛(𝑈)=𝑓∩𝑉_58Λ2582𝑘∘𝑓₅₈(𝑈)∩𝑉≠∅,∀𝑛≥2𝑁₁+1. (A.4) Hence, for any two open sets 𝑈,𝑉, ∃𝑁=2𝑁₁+1, such that (𝑓₅₈|_Λ258)^𝑛(𝑈)∩𝑉≠∅, ∀𝑛≥𝑁; namely, 𝑓₅₈|_Λ258 is topologically mixing.
(b) Consider ent(𝑓₅₈|_Λ258)=(1/2)ent(𝑓²₅₈|_Λ258)=0.2812.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by Research Project of Chongqing University of Science and Technology (CK2013B15), in part by Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grants nos. KJ131401 and KJ131416), in part by Natural Science Foundation Project of CQ CSTC (Grants nos. cstc2012jjB0095 and cstc2012jjA1459), in part by the National Natural Science Foundation of China (51275547), in part by Achievement Transfer Program of Institutions of Higher Education in Chongqing (KJ121413), and in part by Teaching & Research Program of Chongqing University of Science and Technology (X2012-29-S).