International Journal of Networking and Computing curve) are continuous everywhere but not differentiable anywhere. However, singular functions are continuous everywhere and differentiable almost everywhere [5–7]. The number of “0” states in the configuration of a two-dimensional nonlinear cellular automaton has been studied numerically, as shown in Figure 1 [3,8]. We can observe that the number of “0” states at each time step is described well by a singular function, which comprises a self-affine function on a unit interval. We show that the spatio-temporal pattern of the cellular automaton is related to this singular function, instead of directly calculating its fractal dimension. This result indicates that the pattern itself is also fractal. Based on these previous studies, we consider a characterization of the fractal structures of spatial or spatio-temporal patterns regardless of the linearity of a cellular automaton. In this study, we replace the spatial or spatio-temporal patterns of a nonlinear cellular automaton by the unions of squares. We show that the dynamics of the area of the union are represented by a singular function, and we obtain the fractal dimension (herein, we consider the box-counting dimension) of a spatial pattern by calculating the area of the union of squares. Further, we focus on cellular automata that are two-dimensional, symmetrical, and elementary. Numerical simulations show that there exist 1024 automata that create important patterns with fractal structures. It is also shown that some of the patterns are related to the singular functions. For six of the patterns, the number of “0” states at each time step is described by a singular function, which represents a generalization of a result for a particular cellular automaton [3,8]. In addition, for two of the patterns the cumulative sum of the number of “0” states is also described by this function. These results indicate that the singular function is strongly associated with cellular automata. The remainder of the paper is organized as follows. Section 2 describes the prerequisites con- cerning cellular automata and singular functions. In Section 3, we provide an overview of previously derived results [4], and present a more precise calculation. In Section 4, numerical results are pre- sented. We study the spatio-temporal patterns generated by symmetrical two-dimensional cellular automata, and establish their relation to the singular function. The results reveal that the cellular automata have a deep relation not only with fractal geometry, but also with singular functions. Finally, Section 5 presents a discussion and describes possible areas for future studies.
90000 Number of 0 at time t
80000
70000
60000
50000
Number 40000
30000
20000
10000
0 0 50 100 150 200 250 Time step t
Figure 1: The number of “0” states in a configuration of the cellular automaton for the time steps 0 ≤ n ≤ 256. The behavior is different for the odd and even n cases. The upper curve of the graph represents the odd n cases and the lower one represents the even n cases.
2 Prerequisites
In this section, we present some definitions and notations for symmetrical two-dimensional elemen- tary cellular automata and singular functions.
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2.1 Symmetrical two-dimensional elementary cellular automata Suppose that (X,F ) is a discrete dynamical system consisting of a space X and a transformation F on X. The n-th iterate of F is denoted by F n. Thus, F 0 is the identity map on X. Let A = {0, 1} 2 be a binary state set and AZ be the two-dimensional configuration space. We call the configuration 2 xo ∈ AZ such that 0 if (i, j) = (0, 0), (xo)i,j = (1) 1 if (i, j) ∈ Z2\{(0, 0)}, a single site seed. In this article, we consider the orbits from a single site seed as an initial configuration. We next define two-dimensional elementary cellular automata. 2 Definition 2.1. 1. A two-dimensional elementary cellular automaton (2dECA) (AZ ,F ) is given by xi,j+1 T (F x)i,j = f xi−1,j xi,j xi+1,j = f LCR (2) xi,j−1 B
2 for any coordinate (i, j) ∈ Z2 and x ∈ AZ , where f : A5 → A is a map depending on the five states of the von Neumann neighborhood. We name f a local rule.
2 2.A 2dECA (AZ ,F ) is a symmetrical 2dECA (Sym-2dECA) if the local rule f satisfies the following conditions: T B T T L B R fLCR = fLCR = fRCL , fLCR = fBCT = fRCL = fTCB . (3) B T B B R T L The first equality of (3) provides the left-right and top-bottom symmetry and the second gives the rotational symmetry of the local rule. We can easily prove that 232 2dECAs exist, among which 212(= 4096) are Sym-2dECAs. The local rule of a Sym-2dECA can be obtained from a combination of only 12 transitions for 0 0 0 1 0 1 0 0 0 1 0 1 000 , 000 , 001 , 000 , 101 , 101 , 010 , 010 , 011 , 010 , 111 , 111 . 0 1 1 1 1 1 0 1 1 1 1 1 (4)
2.2 A Sym-2dECA as a modified Ulam’s cell model
2 Here, we introduce a particular Sym-2dECA (AZ ,T ). 2 Definition 2.2. The rule of the Sym-2dECA (AZ ,T ) is defined by
(T x)i,j = xi,j + xi−1,jxi+1,j + xi,j−1xi,j+1 (mod 2) (5)
2 for each coordinate (i, j) ∈ Z2 and any x ∈ AZ . This CA is a nonlinear CA, because the second and third terms of the local rule are the products of two states. This local rule is also given in Table 1.
Table 1: Local rule of the CA given by (5) as a Sym-2dECA. This rule is a particular case of h10ci in Table 2 T 000101000101 LCR 000 000 001 000 101 101 010 010 011 010 111 111 B 011111011111
(T x)i,j 000110111001
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125 107 (a) T xo (b) T xo
125 107 2 Figure 2: Configurations T xo and T xo of the CA (AZ ,T ). A black dot represents the state “0,” and a white dot “1.” A fine structure can be observed on the boundary.
Figure 2 presents patterns of a nonlinear two-dimensional CA for the time steps 125 and 107. The spatial patterns of the CA started from a single site seed xo, as defined in Equation 1. A black dot represents the state “0,” and a white dot “1.” The patterns shown in Figure 10 h10ci have also evolved from the time steps 8 to 15. In these pictures, a black dot represents the state “0” and a white dot represents the state “1.” The CA is related to Ulam’s cell-model, which is employed to study the formation of crystalline patterns [9]. Ulam studied the model to construct a simpler model that creates fractal-like crystalline patterns. In this model, the existing cells never die during the time evolution. For a particular 2n−1 ∞ initial configuration xo, {T xo}n=1 is exactly the same as the spatio-temporal pattern of Ulam’s cell model, so that the “0” states never turn to the “1” states. The spatio-temporal patterns of 2n ∞ an even number of iterations {T xo}n=0 are different from those of an odd number of iterations 2n−1 ∞ {T xo}n=1. The relation between the two spatio-temporal patterns has been demonstrated 2n−1 4n−2 numerically [3]. The number of “0” state cells in T xo is the same as in T xo. Thus, we consider only the former case. Kawaharada demonstrated the relation between the one-dimensional 2 elementary cellular automaton Rule150 and the CA (AZ ,T ) [3]. The subdynamics of the CA are equivalent to Rule 150.
2.3 Singular function Here, we introduce a self-affine function on the unit interval. This function is an example of a fractal curve. Let α be a parameter such that 0 < α < 1 and α 6= 1/2.
Definition 2.3 ( [5, 10]). The singular function Lα : [0, 1] → [0, 1] is defined as follows: αLα(2x) (0 ≤ x < 1/2), Lα(x) := (6) (1 − α)Lα(2x − 1) + α (1/2 ≤ x ≤ 1).
The functional equation has a unique continuous solution on [0, 1]. The graph of Lα with α = 1/4 is presented in Figure 3. It is known that the function is strictly increasing on [0, 1] and that its derivative is zero almost everywhere. Kawamura [11] derived the set in which the derivative of the function is zero. In this paper, “singular function” refers to the function defined by the functional equation (6). This function Lα has been studied by many researchers [7, 10, 12]. Now, we introduce an additional definition [5]. We consider flipping a coin with the probability α ∈ (0, 1) for heads and a probability 1 − α for tails, where we assume that α 6= 1/2. The binary P∞ n expansion of x ∈ [0, 1] is determined by flipping the coin infinitely, x = n=1 xn/2 , where each
357 Relation between spatio-temporal patterns
xn is 0 or 1 as determined by the n-th toss. Then, the singular function Lα : [0, 1] → [0, 1] is also defined by the distribution function of x:
Lα(t) := prob{x ≤ t} (0 ≤ t ≤ 1). (7)
1 line 1
0.8
0.6
0.4
0.2
0 0 0.2 0.4 0.6 0.8 1
Figure 3: Graph of the singular function Lα with the parameter α = 1/4 defined by eq. (6).
3 Analysis of the fractal structures of a particular Sym-2dECA
2 In this section, we study the fractal structure of the Sym-2dECA (AZ ,T ) (5). We present the prefractal sets {Kn}, and characterize the sets by the singular function. We also show the existence of the “limit set” and calculate the box-counting dimension of the boundary of this limit set.
3.1 Prefractal sets {Kn} representing the fractal structure of the Sym- 2dECA
2 We construct the prefractal sets {Kn} on the two-dimensional Euclidean space R . To simplify, we will identify each “0” state of a configuration as a square, in the following manner. 2 2 In a configuration x ∈ AZ , if xi,j = 0 for a certain coordinate (i, j) ∈ Z then we replace the state “0” with a 45 degree rotated square with the vertices (i − 1, j), (i + 1, j), (i, j + 1), and (i, j − 1). Then, we have a combination of squares instead of lattice points (see Figure 4). In the configuration x, if xi,j = 0, xi±√1,j = 1, and xi,j±1 = 1, then the center state “0” is converted to one square with a side length of 2.
Figure 4: Result when xi,j = 0, xi±1,j = 1, and xi,j±1 = 1.
Next, we define an increasing sequence {Kn}, which comprises the union of these squares.
Definition 3.1. A family of prefractal sets {Kn} is given for each natural number n.
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k 1. For n = 2 (k = 0, 1, 2, 3,... ), the set K2k is a square centered at the origin with the coordinates of its vertices being (2k+1, 0), (−2k+1, 0), (0, 2k+1), and (0, −2k+1).
k k+1 2. For n with 2 < n < 2 , we define Kn in the following manner. Using the binary expansion of n, we have
n = 2lN + 2lN−1 + ··· + 2l1 + 2l0 , (8)
N where {lj}j=0 is an increasing sequence and lN = k. We place a K2lN−1 centered at each of the four vertices of K2k . We call the resulting figure K2k+2lN−1 . Next, we place a K2lN−2 centered at each of the 12 vertices of K2k+2lN−1 , and we call the resulting figure K2k+2lN−1 +2lN−2 . For J l K2k+2 N−1 +···+2lN −J (J ∈ {0, 1,...,N}), the number of vertices is 4 · 3 . Inductively, we can define Kn for every n uniquely.
From previous observations [3,8], we notice that for the Sym-2dECA Kn represents the structure 2n−1 k 2n−1 of T xo. If n = 2 for each natural number k, then the pattern of T xo is simply in a basic k k square. If 2k < n < 2k+1, then the pattern is the union of T 2 and some parts of T (2n−1)−2 . Figure 2n−1 5 shows the configurations T xo and prefractal sets Kn for n = 32, 48, 52, and 54. We can 2n−1 observe that Kn is a good representation of T xo.
63 95 103 107 T xo T xo T xo T xo
K32 K32+16 K32+16+4 K32+16+4+2
Figure 5: Figure for K54 = K25+24+22+21 .
107 Example 3.1. We construct K54 for the structure T xo (b) in Figure 2 using the following pro- cedure (see Figure 5). From the binary expansion of 54, we have that 54 = 25 + 24 + 22 + 21. We 6 6 place K25 on a square centered at the origin, with the coordinates of its vertices as (2 , 0), (−2 , 0), 6 6 (0, 2 ), and (0, −2 ). Next, we place K24 centered at each of the four vertices of K25 . Then, we obtain the figure K25+24 . Inductively, we place K22 centered at each of the 12(= 4 · 3) vertices of 2 K25+24 , and place K21 centered at each of the 36(= 4 · 3 ) vertices of K25+24+22 . Finally, we obtain K25+24+22+21 .
3.2 Area of Kn and the singular function Lα
2 As in Example 3.1, for the Sym-2dECA (AZ ,T ) the prefractal set Kn represents the structure of 2n−1 a configuration T xo. Hereafter, we will estimate the area of Kn, and illustrate the relation between the CA and the singular function. PN lj N For a natural number n, n = j=0 2 with an increasing sequence {lj}j=0 by the binary expan- m+1 sion. Define m = lN (we have n < 2 ). From the definition of {Kn}, Kn+2m+1 is constructed by
359 Relation between spatio-temporal patterns
placing K2m+1 first, and then four Kn sets centered on each vertex of K2m+1 . Define S(n) as the area of Kn. Then we have that 3 S(n + 2m+1) = S(2m+1) + 4 × S(n) = S(2m+1) + 3S(n). (9) 4 Example 3.2. We calculate S(54). Figure 6 shows the areas S(6), S(22), and S(54). By Definition k 2 3 3.1, K(2 ) for k ∈ N is a simple square. First, we have that S(6) = S(2 ) + 4 × 4 S(2), because 2 3 we notice from Figure 6(a) that S(6) is divided by a square S(2 ) and four copies of 4 S(2). Next, 4 3 S(22) in Figure 6(b) is represented by a square S(2 ) and four copies of 4 S(6), and we have that S(54) = S(25) + 4 × 3 S(22) by) a similar argument. 4 (
S(4) S(16) S(32)
4 (3/4) S(6) 4 (3/4) S(2) 4)(3/4) S(22)
(a) S(6) (b) S(22) (c) S(54)
) ( ) ) ( ( ) ( S(2+4) = S(4)+4 (3/4)S(2) Figure 6: Area of K(n).
Using the same procedure, it is easy to see that S(n + 2m+i) = S(2m+i) + 3S(n) for any natural PN N−j lj i i+1 2 number i. As a result, S(n) = j=0 3 S(2 ), and S(2 ) = 2(2 ) for all i. Therefore, we have PN N−j 2(lj +1) m+1 2(m+2) that S(n) = 2 j=0 3 2 . Scaling by S(2 ) = 2 · 2 gives the following:
N S(n) X = 3N−j22(lj −m−1). (10) S(2m+1) j=0 Moreover, we have the following theorem: m+1 N Theorem 3.1. Consider a rational x ∈ [0, 1) of the form p/2 , and let {lj}j=0 be a strictly PN lj increasing sequence of exponents in the binary expansion of p such that p = j=0 2 and x = PN lj −m−1 PN N−j 2(lj−m−1) j=0 2 . Then, j=0 3 2 = L1/4(x).
PN N−j 2(lj−m−1) PN−1 N−j−1 2(lj −m) It is clear that if we set g(x) := j=0 3 2 , then g(2x − 1) = j=0 3 2 . Thus, we have that
N−1 3 1 X g(2x − 1) + = 3N−j22(lj −m−1) + 3N−N 22(lN −m−1) = g(x). (11) 4 4 j=0
This relationship is the functional equation of the singular function L1/4 (6) for rational x ∈ [0, 1). 2n−1 Remark 3.1. In the previous papers [3,8], it was suggested that the number of “0” states in T xo was related to the singular function L1/4. Theorem 3.1 proves these results.
3.3 Box-counting dimension of fractal sets generated by {Kn} First, we define the box-counting dimension and the limit set. Fractals are mathematical objects whose fractal dimension is not an integer. A fractal dimension is a measure of the complexity of a pattern. The box-counting dimension is useful for calculating or estimating these dimensions. Let U be a nonempty bounded subset of RD, which is a D-dimensional Euclidean space, and let Nδ(U) be the smallest number of copies of a ball of radius δ necessary to cover U. Thus, Nδ(U) < ∞.
360 International Journal of Networking and Computing
Definition 3.2 ( [14]). The box-counting dimension dimB of U is defined by
log Nδ(U) dimB(U) := lim , (12) δ→0 − log δ if the limit exists. This definition means that for the grid size δ we count the number of intersecting cells. As the grid becomes finer, we can observe how this number changes. If the limit exists, then this is the box-counting dimension. Here, we calculate the box-counting dimension of the boundary of 2n−1 N T xo for the Sym-2dECA (5). For n, we have that a = (··· a3a2a1a0) = {ai}i∈N ∈ A such P∞ i Pm i that the binary expansion of n is i=0 ai2 . Let nm = i=0 ai2 (am = 1). Furthermore, for ∞ a given a = {ai}i∈N, let {mk}k=0 be an increasing sequence determined by amk = 1. Then, we have an infinite sequence {K } based on Definition 3.1 where n = Pmk a 2i. We define the nmk mk i=0 i limit set of {K /2mk+1}, which is a subset of the two-dimensional Euclidean space 2, as follows, nmk R where set K /2mk+1 denotes the contracted set of K by the rate 1/2mk+1, and is defined by nmk nmk {(i/2mk+1, j/2mk+1) ∈ 2 | (i, j) ∈ K ⊂ 2}. R nmk R
2nm −1 Definition 3.3 ( [15]). The limit set for the sequence of configurations {T k xo} for the 2 Sym-2dECA (AZ ,T ) is defined by
Kn lim mk , (13) k→∞ 2mk+1 if it exists.
Next, we discuss the box-counting dimension of the boundary of the limit set {K /2mk+1} nmk (k → ∞). First, we observe the step from K to K through the example: 4 = 100| → 12 = nmk nmk+1 2 1100|2 → 28 = 11100|2 → · · · . In this example, m0 = 2, m1 = 3, and m2 = 4.
K K K K nmk 4 12 28
Figure 7: Figures for K : K (n = n = 4), K (n = n = 12), and K (n = n = 28), nmk 4 m0 2 12 m1 3 28 m2 4 sequentially.
We denote the length of the boundary of Kn by l(n). For the example in Figure 7, we have that 3 l(12) = 4 4 l(4) + (l(8) − l(4)) and l(28) = 3l(12) + (l(16) − l(8)). In general,