Abstract
Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 14, which is Bernoulli -shift rule and is a member of Wolfram’s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of rule 14, whether it possesses chaotic attractors or not. In this paper, the complex dynamical behaviors of rule 14 of one-dimensional cellular automata are investigated from the viewpoint of symbolic dynamics. We find that rule 14 is chaotic in the sense of both Li-Yorke and Devaney on its attractor. Then, we prove that there exist fixed points in rule 14. Finally, we use diagrams to explain the attractor of rule 14, where characteristic function is used to describe that all points fall into Bernoulli-shift map after two iterations under rule 14.
1. Introduction
Cellular automata (CA) was first introduced by von Neumann 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. Therefore, many applications of CA have been reported, especially in cryptography [3, 4] and image processing [5, 6].
Here, we will only consider Boolean automata for which the cellular state . A cellular automata consists of a number of cells which evolve by a simple local rule (identical rule). The value of each cell in the next stage is determined by the values of the cell and its neighbor cells in the current stage under the local rule [4]. The identical rule contained in each cell is essentially a finite-state machine, usually specified in the form of a rule table, with an entry for every possible neighborhood of a cell consists of itself and the adjacent cells [7]. The cellular array is -dimensional, where is used in practice. In this paper, we will concentrate on . For a one-dimensional CA, a cell is connected to local neighbors (cells) on either side where is referred to as the radius. A one-dimensional CA has cells linked in a line or in a circle. Denote the value in the th cell at the th stage by . For 2-state 3-neighborhood CA (), the evolution of th cell can be represented as a function of the present states of th, th, and th. The local function is a deterministic function to determine the next-stage value of the th cell, . For example, the rule 14 is a one-dimensional CA, and its rule table is shown in Table 1. Thus, we have , , , , , , , and .
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 [8–10]. However, some authors [11–14] found that some rules of Bernoulli -shift rules are chaotic in the sense of both Li-York and Devaney, where these rules were said to be simple as periodic by Wolfram. Rule 24 is belonging to Bernoulli -shift rules. Therefore, we need to research the rule 24 and to find its some new dynamical properties. The rest of the paper is organized as follows. In Section 2, the Boolean function of rule 14 is also presented, and the expressions of its attractors are given. In Section 3, the dynamical behaviors of rule 14 are studied. In Section 4, characteristic function is used to describe that all points fall into Bernoulli-shift map after several iterations under rule 14, and Lameray diagram is used to show clearly the iterative process of an attractor. Section 5 presents some conclusions.
2. Preliminaries
For simplicity, some notations about symbolic dynamics can refer to [11, 14].
In this paper, we will use some notations about CA as follows.
Chua et al. [15] mentioned that each rule has three globally equivalent local rules determined by three corresponding global transformations, namely, left-right transformation , global complementation , and left-right complementation . Each equivalence class is identified by , where is complexity index, and is index of th class. In [16], the authors presented that 112 rules of 256 local rules were Bernoulli -shift rules. Each of the 112 Bernoulli -shift rules has an ID code , where denotes the number of attractors of rule , denotes the slope of the Bernoulli -shift map, and denotes the relevant forward time-. Hence, the space-time evolution of any one of the 112 rules on their attractors can be uniquely predicted by two parameters: and . For example, the attractors of rule 14 are (, , ) and (, , ).
It follows from [17] that the Boolean function of rule 14 is for , , where “”, “−”, and “” stand for “AND”, “NOT”, and “XOR” logical operation, respectively. Sometimes, “” is omitted for simplicity. The truth table of Boolean function of rule 14 is shown in Table 1.
The subsets denoted by , are derived from the parameters of rule 14: , , and and , , and , respectively, that is,
3. Dynamical Behaviors of on Two Subsystems
In this section, we investigate the complexity and chaotic dynamic behaviors of . In order to give our results, some definitions need be introduced.
Definition 3.1 (see [18]). A square matrix is irreducible if for every pair of indices and , there is an such that .
Definition 3.2 (see [18]). A square matrix is aperiodic if there exists , such that , , .
Definition 3.3 (see [18]). Suppose that is a continuous mapping, where is a compact topological space. is said to be topologically mixing if, for any two open sets , , such that , .
Definition 3.4 (see [13]). Let and be compact spaces, we say and are topologically conjugate if there is a homeomorphism , such that .
Based on the above definitions, we investigate the complexity behaviors of on two subsystems as follows.
Proposition 3.5. For rule 14, there exists a subset which satisfies if and only if , and , , and cannot be 0 simultaneously, .
Proof. (Necessity) Suppose that there exists a subset such that , then , and we have .
If , then , so we get ; , ; , ; , ; if , then , so we get , ; if , then or .
(Sufficiency) Suppose that there exists a subset , , , and cannot be 0 simultaneously, . This yields that . Thus, , namely, . Therefore, we have proven that , .
Remark 3.6. A condition of bit strings according to Bernoulli -shift evolution under is obtained in Proposition 3.5. From Proposition 3.5, we can know that a bit string belonging to is evolved on the basis of shifting the bit string to the right by 1 bit and then complementing (changing 0 to 1 and changing 1 to 0) to obtain the next bit string after one iteration under .
Remark 3.7. From the definition of subsystem, we know that are subsystems of .
Proposition 3.8. .
Proposition 3.9. and are not topologically conjugate.
is topologically mixing if and only if is topologically mixing.
The topological entropy .
Proof. (a) Suppose that and are topologically conjugate, we give a proof by contradiction. Since and are topologically conjugate, then there exists a homeomorphism such that , so we have . Because of , then we obtain that is identical map, which leads to a contradiction.
(b) Let any nonempty open sets .
(Sufficiency) Assume that is topologically mixing, then there exists , such that , . Since is a homeomorphism, is also a nonempty open set in , hence there exists , such that , . Therefore, , . Then, we show that there exists a such that , . Let .
(i)If , , then
(ii)If , , then
(Necessity) Suppose that is topologically mixing, then there exists an , such that , . Since is a homeomorphism, is also a nonempty open set in , hence for open sets and , there exists an , such that . Let .(i)If , , then
(ii)If , , then
(c) .
Hence, . The proof is complete.
Because of , the following corollaries are immediate.
Corollary 3.10. and are not topologically conjugate.
Corollary 3.11. is topologically mixing if and only if is topologically mixing.
Corollary 3.12. The topological entropy .
Remark 3.13. By Proposition 3.5, we know that the determinative block system of is , where , , , , , , and . We have .
Let such that . is considered as the excluded block system [19] , where . Obviously, , . Therefore, it is easy to calculate the transition matrix of the as where
Obviously, is a square matrix. A square matrix corresponds to a directed graph. The vertices of the graph are the indices for the rows and columns of . There is an edge from vertex to vertex if . A square matrix is irreducible if and only if the corresponding graph is strongly connected. If is a two-order subshift of finite type, then it is topologically mixing if and only if is irreducible and aperiodic [18].
We give the corresponding graph of matrix in Figure 1, where vertices are the elements of set . It is obvious that is a strongly connected graph.
Carefully observing Figure 1, we find that there are several strongly connected subgraphs: , , , , , , , , and so forth. The elements of will be composed by all vertices of those strongly connected subgraphs, respectively. For example, if , is composed by vertices of subgraph , then we have , and , and all vertices of the subgraph will occur in if , . If and is composed by vertices of subgraph , then we have , and , and all vertices of the subgraph will occur in if . Therefore, we can deduce the elements of set via the corresponding strongly connected graph of matrix .
Based on the above analysis, we have the following proposition.
Proposition 3.14.
(a) and are topologically conjugate.
(b) is topologically mixing.
(c) The topological entropy .
Proof. (a) Define
where , . In fact, by the definition of , we have , ; thus, . Then, it is easy to check that is homeomorphism and . Therefore, and are topologically conjugate.
(b) Because , , the transition matrix of subshift of finite type is irreducible and aperiodic. By [18, 19], is topologically mixing.
(c) Because two topological conjugate systems have the same topological entropy, and the topological entropy on equals , where is the spectral radius of the transition matrix of the subshift . So .
Remark 3.15. By Corollaries 3.11 and 3.12, we know that is topologically mixing and .
Theorem 3.16. is chaotic in the sense of both Li-Yorke and Devaney on .
Proof. It follows from [19] that the positive topological entropy implies chaos in the sense of Li-Yorke, and a system with topologically mixing property has chaotic properties in different senses such as Devaney. Therefore, rule possesses very rich and complicated dynamical properties on .
Proposition 3.17. For rule 14, there exists a subset which satisfies if and only if , and , , and have the following relations:(i)if , then , , and ; , ,(ii)if , then , ; ; , , and .
Let be a new state set, where , , , , and .
Remark 3.18. From the definition of subsystem, we know that is subsystem of .
The transition matrix of subshift is
We give the corresponding graph of matrix in Figure 2, where vertices are the elements of set .
Proposition 3.19.
(a) is topologically mixing.
(b) The topological entropy .
Theorem 3.20. is chaotic in the sense of both Li-Yorke and Devaney on .
Theorem 3.21. For rule 14, there exist fixed points of .
Proof. A example is given as follows. Let By Table 1, we have Therefore, there exists a set of fixed points of rule 14.
Remark 3.22. By Theorem 3.21, we know that rule 14 has garden of Eden [20] (An bit string is said to be a garden of Eden of rule if and only if it does not have a predecessor under the local rule transformation ) in the finite case. By definition of garden of Eden, the incoming and outgoing bit strings are the same in a garden of Eden, while the basin of attraction of an attractor must contain, at least, one point not belonging to . However, in this paper, we consider garden of Eden as a kind of special attractor. Then, we find that , , , , , , , or . Furthermore, we guess that all initial bit strings after sufficient iterations will belong to under rule .
4. Characteristic Function of Rule 14
First, we give a definition on global characteristic function [16]. Given any local rule , and binary configuration for CA, where , then we can uniquely associate the Boolean string with the binary expansion of a real number on the unit interval , where is the decimal form of Boolean string . The CA’ characteristic function of rule is defined as where denotes rational numbers. By [19], the express of characteristic function of rule 14 is where
In order to transform every concrete bit string into abstract decimal digit, we use characteristic function . Next, we plot the characteristic function diagram for rule 14. We choose . Figure 3 describes that all points fall into Bernoulli-shift map after eight iterations under rule 14, which shows that is global attractor of . The phenomenon also shows that the prediction in Remark 3.22 is correct. Two sets and can also be found in Figure 3. The dots in the two straight lines, whose slope is , are in the set ; the dots in the two straight lines, whose slope is 2, are in the set .
Two isles of Eden are shown in Figure 4. We find that the bit string is isle of Eden, when the length is even. Some attractors are shown in Figure 5. Figure 5(a) shows that the period of attractor is 1; Figure 5(b) shows that the period of attractor is six, where , , and ; Figure 5(c) shows that the period of attractor is six, where , , and ; Figure 5(d) shows that after two iterations, the initial string is attracted. We find that the possible periods of attractors are 1 and 6 if . By Proposition 3.5, the attractors of Figures 4 and 5(c) belong to . By Proposition 3.17, the attractors of (a), (b), and (d) belong to . There is a interesting phenomenon, that is, Figure 5(d), where one iteration accords with Proposition 3.5; however, the two iterations accord with Proposition 3.17, and the attractor belongs to . The phenomenon shows that these two propositions can switch in some situation. The bit string of attractor in Figure 5(c) corresponds with strongly connected subgraph in Figure 1.
(a)
(b)
(c)
(d)
Next, we use Lameray diagram [21] to present our attractors. The diagrams show clearly the iterative process of attractors. The evolution of characteristic function of period-5 attractor is shown in Figure 6, where , and the values of characteristic function are 0.0938, 0.1875, 0.3750, 0.7500, and 0.5313, respectively. The five points fall into the time-1 map () in Figure 5. Then we can associate this particular period-5 attractor of rule 14 as a period-5 point of a continuous map which we know that it is chaotic because “period-5” implies chaos based on Li-York theorem [22] and Sarkovskii theorem [23]. According to Sarkovskii theorem, period-5 implies that the number of periods of are infinite.
In recent years, associative memory was researched in many papers [24, 25]. It is obvious that rule 14 can be used in associative memory. By strongly connected graph of rule 14, we can get the elements on its attractors. Then, we can choose a bit string which belongs to an attractor as memory pattern. For example, bit string 110000 in Figure 5(b) can be chosen as a memory pattern. Since there are infinite orbits, the storage capability is very large. The associative memory model provides a solution to problem where time to recognize a pattern is independent of the number of patterns stored.
5. Conclusion
In this paper, the dynamical behaviors of rule 14 of global equivalence class in ECA, where rule 14 is Bernoulli -shift rule, are carefully investigated from, the viewpoint of symbolic dynamics. We derive the conditions according to Bernoulli -shift evolution for rule 14. Then, we prove that rule 14 is chaotic in the sense of both Li-Yorke and Devaney on their attractors, respectively. We use diagrams to explain the attractor of rule 14, where characteristic function and Lameray diagram are used to describe that all points fall into Bernoulli-shift map after several iterations and to show clearly the iterative process of an attractor, respectively.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 60973114, Grant 61170249, and Grant 61003247, in part by the Natural Science Foundation project of CQCSTC under Grant 2009BA2024, in part by the State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, under Grant 2007DA10512711206, and in part by Teaching & Research Program of Chongqing Education Committee (KJ110401).