1 Introduction↩︎

A cellular automaton (CA) is a model of computation and a dynamical system that consists of a grid of cells, where each cell can take one of a finite number of states [1]. The system evolves in discrete time steps, with the state of each cell in the next step being determined by a fixed set of rules based on its current state and those of its neighboring cells. Even if its time evolution rule is simple, a CA can exhibit very complex behavior, generate a variety of patterns, and effectively capture how complex global patterns can arise from simple, local interactions. Hence, it is often used as a mathematical model for a natural or a social phenomenon [2].

An elementary cellular automaton (ECA) is the simplest possible type of CA [3]. It is a one-dimensional two-state three-neighboring CA, that is, the cells are arranged in a single line, each cell can only be in one of two states, typically represented as 0 (off/white) and 1 (on/black), and the state of a cell in the next generation is determined by its own state and the states of its immediate left and right neighbors from the current generation.

Despite its simplicity, it can generate very complex behavior, including fractal and chaotic patterns. It can be used as a mathematical model for pattern formation in chemical or biological systems. It also gives a fundamental mathematical model, a special case of the totally asymmetric simple exclusion process (TASEP), for traffic flow [4].

Although CAs have been used as models for natural and/or social phenomena, the state of a cell in a CA is limited to a discrete set, such as \(\{0, 1\}\), which has the advantage of being easy to implement on a computer. However, many real-world phenomena inherently involve uncertainty, ambiguity, or continuous quantities, which cannot be adequately represented by only two states. A fuzzy cellular automaton (FCA) is a system that attempts to overcome this limitation of the discrete model by extending the CA’s state space from a discrete set to a continuous interval through a process called fuzzification[5]. For a two-state CA (Boolean CA), a systematic method of fuzzification is to express the update rule in disjunctive normal form and replace it as [6] \[\begin{align} \neg a \rightarrow 1-a,\; a \land b \rightarrow a\cdot b, \; a \lor b \rightarrow a+b -a\cdot b \;\mathrm{or}\, \min[a+b,1]. \end{align}\] While it is almost impossible to predict the long-term behavior of a Boolean CA like an ECA, there have been reports of cases where mathematical analysis of global behavior and fixed points is possible for fuzzification of some ECAs [7]–[10]. Furthermore, various applications of FCA to real-world systems have been actively discussed. Key areas include pattern recognition and information processing (e.g., noise removal, edge detection, and compression in image analysis; dataset generation via feature extraction in machine learning [11]–[13], and the modeling of complex systems (such as urban growth, traffic flow, and natural disasters like forest fires [14]–[16].

However, the aforementioned FCA based on fuzzy disjunctive normal form (FDNF) generally exhibits only monotonic behavior, even when the original CA displays complex dynamics. Consequently, when a CA that models natural or social phenomena is fuzzified, the characteristic features of those phenomena are lost. Meanwhile, in the current application of FCA to real-world systems, state transition rules (fuzzy transition rules) are typically determined empirically or through methods like genetic algorithms. This makes it extremely difficult to leverage the advantages of FDNF, such as its straightforward numerical computation programs and analytical techniques.

For a given Boolean CA, its fuzzification into a FCA may be defined by the following conditions:

1. The state of an FCA cell takes values in the continuous interval \([0,1]\).

2. The FCA’s fuzzy transition rule must coincide with the original CA rule when all cell states are restricted to \(\{0,1\}\).

While FCAs constructed using FDNF satisfy these conditions, they suffer from the disadvantage mentioned previously. In this paper, we propose a method to construct an FCA by generalizing FDNF through the use of fuzzification functions. The resulting FCA construction avoids this drawback. We analyze the FCAs obtained using this method from several perspectives and explore their potential applications. The remainder of this paper is organized as follows. Section 2 reviews the construction of FCAs via FDNF. Section 3 details the method for generating generalized fuzzy elementary automata (GFECAs) by applying fuzzification functions to ECAs. In Section 4, we present several examples of pattern formation from GFECAs. Section 5 analyzes a phenomenon resembling a phase transition–observed by varying parameters in a fuzzification function–using effective dimension related to the \(\ell^p\)-norm. Section 6 investigates the dynamics of minimal three-cell GFECAs, which display non-trivial temporal evolution, based on their orbits and attractors. Finally, Section 7 is devoted to concluding remarks.

2 Fuzzification of ECAs via FDNF↩︎

Let us consider ECAs, which are one-dimensional Boolean CAs where the neighborhood of a cell consists of itself and its two immediate neighbors (a neighborhood size of three). There are \(2^8=256\) possible time evolution rules for ECAs, conventionally numbered from 0 to 255. An ECA governed by rule \(k\) is referred to as the rule \(k\) ECA or ECA \(k\) for short. When we denote by \(u_j^t \in \{0,1\}\) (\(j \in \mathbb{Z}\), \(t \in \mathbb{Z}_{\ge 0}\)) the value of the \(j\) cite at time step \(t\), the update rule of the rule \(k\) ECA is expressed by a function \(F_k:\, \{0,1\}^{\times 3} \rightarrow \{0,1\}\) as \[u_j^{t+1}=F_k(u_{j-1}^t,u_j^t,u_{j+1}^t) \label{eq:ECA}\tag{1}\] Since the function \(F_k\) is determined by the \(2^3=8\) values of 0 or 1, \(a_0,\,a_1,\,\cdots,\,a_7\), as shown in the Table 1, the rule number \(k\) is defined by \[k = a_0\cdot2^0 + a_1\cdot2^1 + a_2\cdot2^2 + a_3\cdot2^3 + a_4\cdot2^4 + a_5\cdot2^5 + a_6\cdot2^6 + a_7\cdot2^7\]

For example, \[\begin{align} 184=0\cdot 2^0+0 \cdot 2^1+0 \cdot 2^2+1\cdot 2^3+ 1\cdot 2^4+1 \cdot 2^5+0 \cdot 2^6+1\cdot 2^7, \end{align}\] and the Rule \(184\) ECA, which is a well-known traffic flow model, is defined from the Table 2.

Figure 1 illustrates the time evolution of rule 184 ECA for \(0\leq t \leq 50\) on a lattice of width \(50\) under a periodic boundary condition. The evolution proceeds downwards from the initial configuration (top row), where white and black cells denote states 0 and 1, respectively.

Now we consider the fuzzification of ECAs using FDNF [6]. From Eq. 1 and Table 1, the update function of an ECA is written in disjunctive normal form as \[\begin{align} \begin{aligned} F_k(x,y,z)=a_0 \left(( \neg x) \land (\neg y) \land (\neg z)\right) \lor a_1\left(( \neg x) \land (\neg y) \land z\right) \\\lor a_2\left(( \neg x )\land y \land (\neg z)\right) \lor a_3 \left(( \neg x )\land y \land z\right)\\\lor a_4 \left(x\land ( \neg y )\land ( \neg z )\right) \lor a_5 \left(x\land ( \neg y )\land z\right)\\ \lor a_6 \left(x \land y \land( \neg z)\right) \lor a_7 \left(x \land y \land z\right) \end{aligned} \label{eq:Rule95F95K} \end{align}\tag{2}\] Here, we use the following notation: \[\begin{align} A \lor a_j\left(\;\cdots \;\right) &=\left\{ \begin{array}{cl} A \lor \cdots &\;(a_j=1) \\ A &\;(a_j=0) \end{array} \right. \\ a_j\left(\;\cdots \;\right)\lor B &=\left\{ \begin{array}{cl} \cdots \lor B &\;(a_j=1) \\ B &\;(a_j=0) \end{array} \right. \end{align}\] The FDNF employs the following fuzzy mappings for Boolean operations: \(\neg x \mapsto 1-x\), \(x \land y \mapsto x \cdot y\), and \(x \lor y \mapsto \min[x+y, 1]\), and the update function \(F_k(x,y,z)\) turns into a polynomial of order 3 as \[\label{eq:fuzzy95polynomial} \begin{align} f_k(x,y,z)=a_0(1-x)(1-y)(1-z)+a_1(1-x)(1-y)z+a_2(1-x)y(1-z)\\ +a_3(1-x)yz+a_4x(1-y)(1-z)+a_5x(1-y)z+a_6xy(1-z)+a_7xyz \end{align}\tag{3}\] Since \(0\le f_k(x,y,z) \le 1\) for \((x, y, z) \in [0,1]^3\), and \(f_k(x,y,z)=F_k(x,y,z)\) for \((x,y,z) \in \{0,1\}^3\), \(f_k(x,y,z)\) serves as an update function of the fuzzy elementary ellular automaton (FECA). For instance, application of Eq. 3 to the rule 184 ECA yields \[\begin{align} f_{184}(x,y,z)&=(1-x)yz+x(1-y)(1-z)+x(1-y)z+xyz\notag \\ &=x-xy+yz \end{align}\]

Let us examine time evolution patterns of the FECAs. In contrast to the clear structure seen in Figure 1, the time evolution of rule 184 FECA (Figure 2) appears largely uniform. This is because the system is provably destined to become a monotone pattern when the number of cells is odd, and a checkerboard pattern when it is even[15].

Figures 3 and [fig:90FECA] show the results of applying FDNF to rules 30 and 90, respectively. The upper diagrams depict the original ECA with complex patterns, while the lower ones show their fuzzified counterparts. The preceding results show that fuzzifying an ECA via FDNF yields only trivial or monotonic dynamical systems. Consequently, a different approach is required to construct a meaningful FCA model that exhibits complex behavior.

3 Fuzzification function and generalized fuzzy elementary cellular automata↩︎

Let us remind that a Boolean FCA constructed from a CA should satisfy the two conditions: (i) the value of its cell is in \([0,1]\), and (ii) time evolution rule coincides with that of the CA when its initial values are in \(\{0,1\}\). Hence we define

Condition (2) of Definition 1 means that \[(x,y,z) \in \{0,1\}^3 \Rightarrow f_k(x,y,z) = F_k(x,y,z).\] To generalize the update functions \(f_k\) from those constructed via FDNF, we define a set of fuzzification functions as follows.

An important property of the fuzzification functions is found from the following proposition, which follows directly from the definitions of \(f_k\) and \(\Omega\).

Several examples of fuzzification functions are shown in Figs. 4 \((a)\)-\((f)\).

If we use the function shown in Fig. 4 \((a)\) for the fuzzification functions \(u,\,v,\,w,\,g\) in Eq. ?? , then \(\tilde{f}_k=f_k\) and the original FECA via FDNF is recovered. On the other hand, if we use the function shown in Fig. 4 \((b)\), the update function \(\tilde{f}_k\) tends to take values \(0\) or \(1\) and its time evolution pattern is similar to that of the CA. If we use the other fuzzification functions shown in Fig. 4 \((c)\)-\((f)\), it is difficult to predict what kinds of patterns will emerge. In the next section, we will examine pattern formation using several types of fuzzification functions.

4 Pattern formation by GFECAs↩︎

As Proposition 1 indicates, the choice of fuzzification functions yields infinitely many types of GFECA. This section presents some of the representative evolution patterns that emerge.

1. FECA converging to ECA behavior↩︎

Let \(g\) be the identity function, and define \(u(x) = v(x) = w(x)\) as: \[\begin{align} u(x) = \begin{cases} 0 & (0 \le x \le \frac{1}{3}) \\ 3x - 1 & (\frac{1}{3} < x \le \frac{2}{3}) \\ 1 & (\frac{2}{3} < x \le 1) \end{cases} \label{eq:FECAtoECA} \end{align}\tag{4}\] This fuzzification function is illustrated in Fig. 4 \((b)\).

Using Eq. (4 ), we obtain rule 90 and 184 GFECAs. Figures 5 and [fig:184FECA95to] show time evolution patterns of the rule 90 and 184 GFECAs respectively thus obtained. We used random initial values in the interval \([0,1]\) with periodic boundary conditions. Although initial values vary from 0 and 1, they eventually converge to binary values over time, reproducing the original ECA behavior. In the extreme case where \(u(x)\) is defined as a step function: \[\begin{align} u(x) = \begin{cases} 0 & (0 \le x \le \frac{1}{2}) \\ 1 & (\frac{1}{2} < x \le 1) \end{cases}, \end{align}\] the update function \(\tilde{f}_k\) takes only \(0\) or \(1\), causing the time evolution pattern for \(t \ge 1\) to coincide with that of the corresponding ECA for any initial condition.

2. Functions with a gap at \(\boldsymbol{\frac{1}{2}}\)↩︎

We define a class of GFECAs by setting the functions \(g, u, v,\) and \(w\) identically to \(g(x)\): \[\begin{align} g(x) = \begin{cases} ax & (0 \le x \le \frac{1}{2}) \\ ax - a + 1 & (\frac{1}{2} < x \le 1) \end{cases}, \label{eq:gap} \end{align}\tag{5}\] where \(a\) is a parameter in the range \(1 \le a \le 2\). As illustrated in Fig. 4 (e), this function introduces a discontinuity gap of size \(a-1\) at \(x = \frac{1}{2}\). For the specific case where \(a=1\), the gap vanishes, and the GFECA becomes equivalent to the FECA via FDNF.

Figure 6 shows how the time-evolution patterns of the rule 102 GFECA change as a function of the parameter \(a\). The simulations use a system of \(50\) cells over a time interval of \(0 \le t \le 200\), with the parameter \(a\) set to \(1.0, 1.2, 1.4, 1.6, 1.8,\) and \(2.0\). For \(a=1.0\), the system evolves into a uniform state as expected. As \(a\) increases, however, characteristic patterns emerge.

In the range \(1.1 \le a \le 1.6\), the patterns are notably similar to those of the standard rule 102 ECA (Fig. 7). For \(a \ge 1.6\), this distinct structure disappears, giving way to a random-like pattern. We observe similar behavior in case of the rule 195 GFECA.

Figure 8 shows the rule 184 GFECA obtained by the same fuzzification functions and parameters. For \(a=1.0\), the system evolves into a uniform state as expected too. As \(a\) increases, a labyrinthine pattern emerges around \(1.1 \le a \le 1.3\), and changes to a random-like pattern for \(1.4 \le a\). The similar behavior is observed in the rule 226 GFECA.

5 FCAs by mixing of ECA rules↩︎

By omitting condition (2) from Definition 1, we can define a more general class of one-dimensional, three-neighbor FCAs. A key method for constructing such automata involves creating a convex combination of the evolution rules from different GFECAs. Specifically, given a set of update functions \(\tilde{f}_{k_j}\) (\(j=1,2,...,l\)) of GFECAs, we can define a new, composite function \(\tilde{f}\) as: \[\begin{align} \tilde{f}(x,y,z):=\sum_{j=1}^l\alpha_j \tilde{f}_{k_j}(x,y,z) \end{align}\] where \({}^\forall j,\,\alpha_j \ge 0\),and \(\sum_{j=1}^l\alpha_j =1\). This construction guarantees that the resulting function \(\tilde{f}\) satisfies the condition (1) of the definition 1. It is important to note that this method constitutes a global mixing of rules. This is distinct from local mixing schemes, where different update rules might be applied at various points in space and time.

Let us consider a simple case where the fuzzy update function is constructed from a convex combination of two standard ECA update functions, \(f_{k_1}\) and \(f_{k_2}\). For a mixing parameter \(\alpha \in [0, 1]\), this new function is defined as: \[\tilde{f}_{k_1k_2}^{(\alpha)}(x,y,z) := \alpha f_{k_1}(u(x), u(y), u(z)) + (1-\alpha) f_{k_2}(u(x), u(y), u(z)). \label{eq:mixFECA}\tag{6}\] Here, the function \(u(x)\), as defined in Eq. 4 , maps the fuzzy cell states to crisp values suitable for the ECA rules.

Since the ECA functions \(f_{k_1}\) and \(f_{k_2}\) have codomains of \(\{0, 1\}\), it follows that the new function is correctly bounded, i.e., \(0 \le \tilde{f}_{k_1k_2}^{(\alpha)} \le 1\). The parameter \(\alpha\) smoothly interpolates between the two base rules. At the extreme values of \(\alpha\), the mixed rule collapses to one of the original ECA rules: \[\tilde{f}_{k_1k_2}^{(0)}(x,y,z) = f_{k_2}(u(x), u(y), u(z)) \quad \text{and} \quad \tilde{f}_{k_1k_2}^{(1)}(x,y,z) = f_{k_1}(u(x), u(y), u(z)).\] This allows us to observe how the dynamics of the system change as the rules are mixed.

Figure 9 illustrates the dynamics that emerge from mixing the ECA rules \(k_1=184\) and \(k_2=90\). The simulation was performed on a lattice of \(50\) cells for \(200\) time steps (i.e., \(0 \leq t \leq 200\)). The initial condition was set by assigning the leftmost cell a value of \(0.6\) while all other cells were set to \(0\). The panels in the figure, from left to right, show the time evolution patterns corresponding to values of the mixing parameter \(\alpha \in \{0, 0.2, 0.4, 0.6, 0.8, 1.0\}\).

A magnified view, shown in Figure 10, focuses on the behavior around \(\alpha \approx 0.66\), revealing a sharp, phase-transition-like shift in the dynamics. Similar critical transitions were also observed in other rule combinations, typically near \(\alpha \approx 0.33\) and \(\alpha \approx 0.66\).

To quantify the complexity of the evolving patterns, we introduce an order parameter \(d_{\alpha}\). This parameter is calculated for an analysis domain \(D\) containing \(M\) cells, where the state of a cell at position \((i, j)\) is given by \(\phi_{ij}\).

First, we compute the mean square of the cell states, \(C_{\alpha}\), to serve as a normalization constant: \[C_\alpha := \frac{1}{M} \sum_{(i,j) \in D} \phi_{ij}^2\] Next, using \(C_{\alpha}\), we define a normalized moment of order \(2p\), a quantity related to \(\ell^p\)-norm, for any real number \(p \ge 1\): \[N_{\alpha}(p) := \frac{1}{M} \sum_{(i,j) \in D} \left( \frac{\phi_{ij}}{\sqrt{C_\alpha}} \right)^{2p}\] Finally, the effective dimension \(d_{\alpha}\) is defined in the limit as \(p \to \infty\): \[d_\alpha:=\lim_{p\to \infty} \left( 2-\frac{2}{p-1}\log_M N_{\alpha}(p) \right)\] The resulting value, \(d_{\alpha}\), is a single scalar that characterizes the spatial complexity of the observed patterns, as illustrated in Fig. 11. To build intuition for this measure, we will now examine its behavior in three idealized cases.

1. Point-like (0-d) pattern
Consider a case where a single cell has a value \(a > 0\) and all other \(M-1\) cells are zero. The normalization constant and the moment are calculated as: \[\begin{align} C_{\alpha} &= \frac{1}{M} (a^2) = \frac{a^2}{M} \\ N_{\alpha}(p) &= \frac{1}{M} \left( \frac{a}{\sqrt{C_\alpha}} \right)^{2p} = \frac{1}{M} \left( \frac{a}{a/\sqrt{M}} \right)^{2p} = \frac{1}{M} (\sqrt{M})^{2p} = M^{p-1} \end{align}\] Substituting this into the definition for the effective dimension yields: \[\begin{align} d_\alpha = \lim_{p\to \infty} \left( 2-\frac{2}{p-1}\log_M (M^{p-1}) \right) = 2 - \frac{2(p-1)}{p-1} = 0 \end{align}\]

2. Line-like (1-d) pattern
For a one-dimensional structure (e.g., a line) in the 2D domain, the number of active cells is approximately \(\sqrt{M}\). Assuming \(\sqrt{M}\) cells have the value \(a\) and the rest are zero: \[\begin{align} C_\alpha &= \frac{1}{M} (\sqrt{M} \cdot a^2) = \frac{a^2}{\sqrt{M}} \\ N_\alpha(p) &= \frac{1}{M} \sum_{i=1}^{\sqrt{M}} \left( \frac{a}{\sqrt{C_\alpha}} \right)^{2p} = \frac{\sqrt{M}}{M} \left( \frac{a}{a/\sqrt[4]{M}} \right)^{2p} = M^{-1/2} (M^{1/4})^{2p} = M^{\frac{p-1}{2}} \end{align}\] This configuration correctly results in a dimension of 1: \[\begin{align} d_\alpha = \lim_{p\to \infty} \left( 2-\frac{2}{p-1}\log_M (M^{\frac{p-1}{2}}) \right) = 2 - \frac{2}{p-1} \cdot \frac{p-1}{2} = 1 \end{align}\]

3. Space-filling (2-d) pattern
Finally, if all \(M\) cells have a uniform value \(a\), then \(C_\alpha=a^2\) and \(N_{\alpha}(p)=1\). This gives \(\log_M N_\alpha(p) = 0\), which yields \(d_\alpha=2\).

These examples demonstrate that \(d_\alpha\) provides a quantitative index that corresponds to the intuitive spatial dimension of a pattern.

Let us use the above effective dimension \(d_{\alpha}\) to investigate the transition in the time-evolution patterns of the GFECA 90 and GFECA 184 mixing rules as a function of the parameter \(\alpha\). Figure 12 plots \(d_{\alpha}\) calculated over the time interval \(450 \le t \le 500\). For \(0\le \alpha \le 0.6\), \(d_{\alpha} \approx 1.0\), while for \(\alpha \ge 0.7\), \(d_{\alpha} \approx 1.6\). Between these regions, \(d_{\alpha}\) increases in a step-like manner, though with significant fluctuations. A smaller, secondary jump is also observed around \(\alpha \approx 1/3\).

This behavior is likely caused by the kinks in the fuzzification function \(u(x)\) at \(x=1/3\) and \(x=2/3\). These features in \(u(x)\) appear to induce the corresponding changes observed in \(d_{\alpha}\) around \(\alpha \approx 1/3\) and \(\alpha \approx 2/3\).

Therefore, the effective dimension \(d_{\alpha}\) serves as a useful order parameter for quantitatively describing these pattern changes. Furthermore, we note that the function \(N_\alpha(p)\), or its logarithm \(\log N_\alpha(p)\), contains more detailed information and could be used to investigate subtler pattern features.

6 GFECA dynamics with small degrees of freedom↩︎

Cellular automata with a small number of cells typically do not exhibit complex patterns. For example, an ECA with only three cells exhibits periodic time evolution over its \(8\) possible states, with the period’s maximum length of \(8\) depending on the boundary conditions. However, a GFECA can exhibit complex time evolution if its fuzzification functions are chosen properly. As an example, consider a GFECA with three cells where the update function Eq. ?? uses the identity function for \(g(x)\), and the functions \(u(x)\), \(v(x)\), \(w(x)\) are identical and given by \[\begin{align} u(x) = \begin{cases} 3x & (0 \le x \le \frac{1}{3}) \\ -3x + 2 & (\frac{1}{3} < x \le \frac{2}{3}) \\ 3x - 2 & (\frac{2}{3} < x \le 1) \end{cases} \label{eq:chaos} \end{align}\tag{7}\] The fuzzification function Eq. 7 is shown in Fig. 4 (\(f\)). Let \(x_t\), \(y_t\) and \(z_t\) be the values of the three cells at time \(t\). Then, the dynamics of rule \(k\) GFECA are governed by the following system of equations: \[\begin{align} \begin{cases} x_{t+1}=\tilde{f_k}(z_t,x_t,y_t) \\ y_{t+1}=\tilde{f_k}(x_t,y_t,z_t) \;(t=0,1,2,...)\\ z_{t+1}=\tilde{f_k}(y_t,z_t,x_t) \\ \end{cases} \label{eq:3site} \end{align}\tag{8}\] Starting from a generic initial state \((x_0,y_0,z_0)\), the orbit generated by this system evolves within the three-dimensional unit cube, \([0,1]^{3}\). The characteristics of this orbit are determined by the specific rule \(k\). In particular, its asymptotic points often form a stable manifold, providing a basis for classifying different GFECAs.

For any given update function \(\tilde{f_k}(x, y, z)\), permuting its inputs \(x, y, z\) can generate up to six distinct variations. For instance, a new rule \(k'\) can be defined from rule \(k\) as follows: \[\begin{align} f_{k'}(x, y, z) := f_k(y, z, x) \end{align}\]

When all such variations are treated as symmetrically equivalent, the \(256\) ECA rules can be classified into \(80\) distinct groups. Examples of these groups include the single-member group \(\{1\}\), the three-member group \(\{2, 4, 16\}\), and the six-member group \(\{138, 196, 176, 140, 162, 208\}\).

This equivalence is based on the functional form. The other rules in a group are generated by permuting the inputs of an update function. For example, consider the update function for rule \(2\): \[\begin{align} f_2(x,y,z)=(1-x)(1-y)z \end{align}\] The other functions in its group, for rules \(4\) and \(16\), can be derived by swapping the inputs of \(f_2\): \[\begin{align} f_4(x,y,z) &= x(1-y)(1-z) \quad (\text{by swapping } x \leftrightarrow z \text{ in } f_2) \\ f_{16}(x,y,z) &= (1-x)y(1-z) \quad (\text{by swapping } y \leftrightarrow z \text{ in } f_2) \end{align}\] Similarly, permuting the inputs of \(f_{138}(x,y,z)=xyz+(1-x)z\) generates the other rules in its group, such as \(f_{196}\) and \(f_{176}\).

Here are some typical examples of asymptotic behavior and stable manifolds.

A class of GFECAs has orbits that converge to a single fixed point. Some rules converge to the same point regardless of the initial state. For example, orbits under rule \(8\) always converge to \((0,0,0)\), and those under rule \(238\) always converge to \((1,1,1)\). In other cases, the fixed-point attractor depends on the initial conditions. For instance, rule \(150\) converges to a fixed point, but which point it is depends on the starting values. This behavior is also found in rules \(60\), \(90\), and \(233\).

A common behavior is the convergence of an orbit to a periodic cycle. For example, consider the rule \(3\) GFECA, which has the update function: \[\begin{align} f_3(x,y,z)=(1-x)(1-y) \end{align}\] As shown in Fig. 13, the orbits of the GFECA in the cube \([0,1]^3\) converges to a period-two orbit consisting of the points \((0,0,0)\) and \((1,1,1)\). The figure’s left panel plots the points from \(t=0\) to \(1000\), and the right panel plots the final stable orbit from \(t=9000\) to \(10000\). Similar behavior is observed in GFECA with rule \(1\), \(5\), \(7\), and \(17\).

Other rules converge to longer cycles. For instance, rule \(210\) converges to a period-six orbit (Fig. 14). Similar periodic convergence is observed in the GFECAs with rule \(154\), \(156\), and \(198\).

Certain classes of GFECAs possess higher-dimensional stable manifolds. Figure 15 illustrates this by plotting orbits from \(t=9000\) to \(10000\), each starting from a generic initial state. In panels \((a)\)–\((c)\), the attractors asymptotically converge to one-dimensional line segments. In contrast, panels \((d)\)–\((f)\) display the characteristic, complex structures of higher-dimensional attractors.

The system can also exhibit chaotic orbits, as demonstrated in Fig. 16. This behavior is readily explained by the update function for rule \(85\). Since \(f_{85}(x,y,z)=1-y\), the fuzzified update function is: \[\begin{align} \tilde{f_{85}}(x,y,z)=1-u(y), \end{align}\] where \(u(y)\) is the inherently chaotic fuzzification function from Eq. 7 . Consequently, the GFECA under rule \(85\) inherits this chaotic dynamics directly.

This study focused on a three-cell GFECA—the minimal configuration for a three-neighbor FCA—using only the fuzzification function in Eq. 7 . Increasing the number of cells would create dynamical systems with more degrees of freedom, likely leading to more complex evolutionary patterns and asymptotic behaviors. Furthermore, exploring a wider variety of fuzzification functions is a promising avenue for discovering richer and more diverse dynamical systems within the GFECA framework.

7 Concluding Remarks↩︎

This paper introduces a general method for constructing a FCA from any given Boolean CA. Since a multi-state CA can be replaced by equivalent Boolean CAs (for example, see [17]), the method is also applicable to multi-state CAs. Unlike the heuristic FDNF method, our approach not only retains the characteristic patterns of the original CA but also generates more complex patterns by using diverse fuzzification functions. We applied this method to ECAs, creating GFECAs. We demonstrated that by selecting the fuzzification function, these GFECAs exhibit controlled pattern transformations based on the function’s parameters. Furthermore, our method facilitates the mixing of ECA rules, which leads to observable phase transitions as the mixing ratio is altered. These transitions were quantitatively confirmed using the effective dimension related to \(\ell^p\)-norm. Notably, the resulting GFECAs display non-trivial temporal evolution even in minimal systems of just three cells. In essence, we have developed a general fuzzification method that preserves the complex behaviors of CAs, is mathematically tractable, and is expected to expand their utility as a modeling tool for natural and social phenomena.

Future research will focus on integrating our FCA framework with machine learning to explore new applications. This integration will enable the systematic and automatic discovery and calibration of fuzzy transition rules, for instance, by embedding FCAs within neural networks to learn transition functions or by informing deep learning architectures. By establishing an effective method for modeling localized interactions under uncertainty, this work could be applied to diverse fields such as materials science, biology, and sociology. Addressing these challenges and exploring these applications will be the primary focus of our future work.

This work was supported by Arithmer inc. and JSPS KAKENHI Grant Number JP23K22408.

References↩︎