A complete description of the dynamics of legal outer-totalistic affine continuous cellular automata

This paper presents an investigation into the evolution and dynamics of the simplest generalization of binary cellular automata: Affine continuous cellular automata (ACCAs), with [0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,1]$$\end{document} as state set and local rules that are affine in each variable. The focus lies on legal outer-totalistic ACCAs, an interesting class of dynamical systems that exhibit some behavior that is not observed in the binary case. A unique combination of computer simulations (sometimes quite advanced) and a panoply of analytical methods allows to lay bare the dynamics of each and every one of these continuous cellular automata. The results also show that in the class of ACCAs considered, all types of sensitivity can be observed: sensitivity to a change of the number of cells in the grid, sensitivity to slight changes in the parameters of a local rule and sensitivity to the change of a single value in an initial configuration.


Introduction
Continuous cellular automata (CCAs) can be seen as a generalization of cellular automata (CAs), in which time and space are still discrete, but cells can take states from some infinite (often continuous) set. One of the best-known examples of such dynamical systems are coupled map lattices [1][2][3]. A well-known generalization of binary CAs to CCAs with [0, 1] as state set is obtained by "fuzzification" in [4] and was further studied, for example, in [5,6]. This fuzzification process allows to associate with every binary CA some particular CCA (a so-called Fuzzy CA) through an extension of the domain of the local rule. The relationship between Fuzzy CAs and elementary CAs (ECAs) can be found in [7].
A further generalization of the above idea has resulted in the definition of affine CCAs (ACCAs) [8]. This kind of CCAs is considered to be the simplest possible generalization of binary CAs-of course, apart from the Fuzzy CAs-as they have a local rule that is affine in each variable. It appears that ACCAs exhibit a much richer behavior than the binary CAs they stem from. For example, most density-conserving ACCAs can solve the relaxed density classification problem, which is not possible with binary CA, a result obtained through a theoretical study of the dynamics of density-conserving ACCAs [9]. It is known that CCAs, although they include "cellular automata" in their name, usually have a much more complicated dynamics than classical CAs. Together with the introduction of continuity in the set of states and sometimes also in the set of parameters that define the local rule (as in the case of ACCAs), phenomena such as bifurcation or phase transition occur naturally. This translates into a much greater difficulty in studying the dynamics of such dynamical systems. For this reason, the investigation of the dynamics of CCAs is often performed through computer simulation only (see, for example, [3,4,[10][11][12][13]).
However, it should be emphasized that the results of computer simulations should be handled with care, and this for several reasons. First of all, the results may depend on the precision arithmetic used. Secondly, it is not obvious how convergence should be decided upon. Also, in computer simulations, there is no possibility to consider all grids and all corresponding initial configurations and one usually sets a certain "large enough" number of cells and selects some "large enough" subset of initial configurations. However, we must be aware that, for example, for 100 cells, there are as many as 2 100 binary initial configurations, so any reasonable subset suitable for computer simulations will be a tiny fraction of a percent of the total. Additionally, oftentimes CAs and CCAs behave differently depending on the number of cells (for example, the dependence on parity in the case of CAs with radius 1). Furthermore, when the phenomena observed, such as bifurcations or phase transitions, are characterized by numerical parameters, the values of the latter can only be found approximately. Finally, simulations may indicate the existence of a phase transition, but in reality it may not exist (or vice versa). Despite all the reasons described above, computer simulations can be an important tool in studying the dynamics of CCAs as a source of inspiration (see, for example, [5,9,14]).
The situation is quite different for analytical research as its results are fully trustable. Unfortunately, so far there are only a few papers on CCAs and all of them deal with the one-dimensional case only. One of the types of CCAs for which researchers have been able to analyze the dynamics analytically is the class of fuzzy CAs. One can find papers describing the dynamics of individual Fuzzy CAs, like fuzzy rule 90 [6], 110 [15], 30 [16] or 184 [17], as well as papers adopting a much more general approach (see, for instance, [18,19]). The dynamics of Fuzzy CAs called weighted average rules is theoretically explained in [5]. This detailed analytical study confirmed, among other things, the empirical observation that all weighted average rules are periodic in time and space. It is safe to say that all of the 256 elementary Fuzzy CAs have been examined. In contrast, the study of ACCAs has just been initiated. Virtually the only class of ACCAs whose dynamics has been thoroughly examined is the class of one-dimensional density-conserving ACCAs with radius 1 [9].
In this paper, we focus on another class of onedimensional ACCAs with radius 1: ACCAs that are both outer-totalistic and legal. The first property means that such ACCAs update the value of a cell only on the basis of the current value of the cell and the sum of the values of all other cells in the neighborhood of that cell. The second property ensures that nothing can be produced from zeros. A combination of these two properties is very often used when modeling various phenomena (see [20] for some examples). In particular, it is a basic assumption in each "full of common sense" Lifelike CA, i.e., being similar to Conway's Game of Life [21,22]. Thus, such Fuzzy CAs are used to create one-dimensional and two-dimensional CCAs that generalize the Game of Life [23]. For this reason, legal outer-totalistic ACCAs are at the core of interest.
Our motivation to consider one-dimensional ACCAs only is rather simple: We have set ourselves the goal to explore the dynamics theoretically and not by analyzing computer simulations only. As will become clear, even in the one-dimensional case this is a challenging task. Up to now, there are no such theoretical results on the dynamics of two-dimensional cellular automata, such as the mentioned Game of Life, despite the ample interest of researchers (see, for instance, [24][25][26]). Moreover, even in the case of one-dimensional cellular automata, virtually all existing research describes the dynamics on the basis of computer simulations (see, for instance, [27,28]).
Among the ECAs, there are 64 outer-totalistic CAs. Only four of them are still outer-totalistic after the "fuzzification" process and all show a very simple dynamics. Restricting our attention to the legal ones, we are left with only two ECAs, i.e., only two ECAs are still legal and outer-totalistic after fuzzification: ECA 0 (the zero rule) and ECA 204 (the identity rule). The situation looks quite different for ACCAs. There are infinitely many legal outer-totalistic ACCAs and their dynamics are very diverse, which will be extensively demonstrated in this paper. To be more precise, we will uncover the dynamics of all such ACCAs. Preliminary hypotheses will be generated on the basis of computer simulations, but will then be confirmed analytically. More specifically, we will use the results of preliminary computer simulations (sometimes quite advanced) to identify the proper mathematical tools for our theoretical analysis. This approach will allow to lay bare the dynamics of each and every outer-totalistic legal ACCA. The results also imply that in this class of continuous cellular automata one can observe all types of sensitivity: sensitivity to a change of the number of cells in the grid, sensitivity to slight changes in the parameters of a local rule and sensitivity to the change of a single value in an initial configuration. All the results presented in this paper are new (except when the rule under consideration is density-conserving, as this case was considered in [9]). This paper is organized as follows. Definitions and theoretical facts are given in Sect. 2. Computational experiments are described in Sect. 3, while the ACCA dynamics are discussed in Sect. 4. The paper is concluded with Sect. 5.

Outer-totalistic ACCAs
A CCA is a dynamical system consisting of a grid of cells, where each cell is assigned one of infinitely many states. We consider the one-dimensional case, radius 1 and periodic boundary conditions, so that the cells are arranged in a circular linear array and each cell has two adjacent cells, which together with the cell itself constitute its neighborhood.
As state set we consider the interval where n ∈ {0, 1, . . . , N − 1} is the index of the nth cell of the grid. (All operations on the indices are performed modulo N .) Further, F t (x) denotes the result of the tth application of F to the configuration x, corre-sponding to the tth time step. Moreover, we will write x t n to denote the value of the nth cell in F t (x). For a given configuration x ∈ [0, 1] N , we define the sum of its states and its density, respectively, by Furthermore, for x ∈ [0, 1] N and t ∈ N, we denote the minimal and maximal value among by min(F t (x)) and max(F t (x)), respectively. The Euclidean norm in R N is denoted by the symbol · . For the sake of brevity, we often express configurations in a more compact form and write, for instance, 1 2 0 4 1(01) 2 as a self-explanatory shorthand for 11000010101. In this paper, we restrict our attention to the interesting class of affine CCAs (ACCAs), i.e., CCAs whose local rule is affine in each variable (see [9] The general form of a LUT is given in Table 1.
Next, we extend the function to the entire domain [0, 1] 3 . Obviously, the extension that is affine in each variable is unique and can be expressed as the following polynomial: ACCAs are a generalization of Fuzzy CAs that are constructed from ECAs by "fuzzification" of their disjunctive normal form [29]. Thus, there are exactly 256 Fuzzy CAs, because each ECA corresponds to a unique Fuzzy CA, whose local rule is also defined by Eq. (1), where l 0 , . . . , l 7 are the LUT entries of the given ECA.
To understand the difference between ECAs, Fuzzy CAs and ACCAs, it is worth emphasizing that the local rules of all these CA families are defined by Eq. (1), but only in the case of ECAs both (l 0 , l 1 , . . . , l 7 ) ∈ {0, 1} 8 Table 1 General form of the LUT of the local rule of an ACCA and (x, y, z) ∈ {0, 1} 3 . In the case of Fuzzy CAs, still (l 0 , l 1 , . . . , l 7 ) ∈ {0, 1} 8 but (x, y, z) ∈ [0, 1] 3 , while in the case of ACCAs both (l 0 , l 1 , . . . , l 7 ) ∈ [0, 1] 8 and In the investigations presented in this paper, we will further restrict our attention to legal outer-totalistic ACCAs. An outer-totalistic CA is a CA whose local rule depends only on the state of the focal cell and the sum of the states of the adjacent cells. The following theorem provides a necessary and sufficient condition for an ACCA to be outer-totalistic.
Let us note that for Fuzzy CAs the necessary and sufficient condition for being outer-totalistic is exactly the same as for ACCAs in Theorem 1 with the additional assumption that a, b, c, d ∈ {0, 1}. As a consequence, we obtain that among the 64 outer-totalistic ECAs, only four of them are still outer-totalistic when we extend the domain of their local rule from {0, 1} 3 to [0, 1] 3 . These are: ECA 0, ECA 51 (the negation), ECA 204 (the identity) and ECA 255. Thus, there are only four outer-totalistic Fuzzy CAs with the following local rules: f 1 (x, y, z) = 0, f 2 (x, y, z) = 1 − y, f 3 (x, y, z) = y and f 4 (x, y, z) = 1. All of these rules give rise to a very simple dynamics.
It is easy to see that if some ACCA is outertotalistic and its local rule f is parameterized by a point (a, b, c, d), then the conjugate CA, whose local rule f C is given by Note that the behavior of F C is strongly related to the behavior of F. Indeed, if x is any configuration from [0, 1] N and y is the configuration given by For binary CAs the conjugation F → F C is often called the white-to-black transformation.
The following theorem shows a basic property of the local rule of ACCAs.  where the set V is defined as We omit the proof, as it is an immediate consequence of the monotonicity of f w.r.t. each variable.
In this paper, we focus on a subclass of outertotalistic ACCAs, known as legal ACCAs: A CA is called legal if it cannot produce anything from zeros, i.e., its local rule f satisfies l 0 = f (0, 0, 0) = 0. According to Eqs. (2) and (3), the local rule of a legal outer-totalistic ACCA is given by where 0 ≤ b ≤ 1 2 , 0 ≤ c ≤ 1 and c ≤ 2d ≤ c + 1. These restrictions on the parameters b, c, d generate a parallelepiped P in the Cartesian coordinate system, which is shown in Fig. 1.
As one can see, the parallelepiped P has eight vertices; thus, there are exactly eight extreme points in the set of all legal outer-totalistic ACCAs (see Table 2). The local rule of any legal outer-totalistic ACCA can be written as a convex combination of the local rules listed in the last column of Table 2. Table 2 Analytical expressions of the local rules corresponding to the extreme points in the set of legal outer-totalistic ACCAs

Vertex
Corresponding local rule

Computer simulations
Our goal is to examine the dynamics of all legal outertotalistic ACCAs; more specifically, we want to determine how these dynamics evolve when starting from binary configurations. We concentrate on such initial configurations for two reasons. Firstly, many problems concerning cellular automata are centered on binary configurations (as, for example, the density classification problem). Secondly, we want to deal with a finite set of initial configurations to be able to examine all of them using computer simulations. Our investigation started with very detailed simulations, described below, whose results facilitated to formulate preliminary hypotheses (depending on the range of the parameters b, c and d), some of which could be easily confirmed theoretically, while other ones required often very sophisticated additional simulations, revealing the underlying mechanism of a given ACCA in detail. This was essential in order to be able to choose the proper analytical tools and mathematical proof methods. As a result, it became possible to describe and theoretically confirm the dynamics of all legal outer-totalistic ACCAs. Below, we present the experimental setup for the first part of the simulations.
If the number of cells N is given, then X N denotes the set of all binary configurations of length N , i.e., X N = {0, 1} N . Additionally, we use the notations X 0 N , X 1 N and X 0,1 N to refer to the set X N without 0 N , 1 N or both of these elements, respectively, i.e., In the first part of our experiments, we sampled the set of all legal outer-totalistic ACCAs parameterized by (b, c, d) by varying the parameter c from 0 to 1 with a step size of 0.1 and-taking into account Theorem 1-the parameters b, d from 0 to 0.5 and from 0.5c to 0.5c +0.5, respectively, with a step size of 0.05. For each such set of parameters, we defined the local rule f by the expression in Eq. (4), resulting in a set R containing 1331 rules.
Because we did not want to be dependent on randomly selected configurations, we chose small N and examined all binary configurations of this length. As our investigations conducted in a previous work [9] showed that the behavior of an ACCA may depend on whether N is odd or even, we considered two specific sizes of the grid: N = 10 and N = 11. Note that |X 10 | = 2 10 and |X 11 | = 2 11 . For every local rule f ∈ R, we carried out the following simulations. For each x ∈ X N , we ran the corresponding global rule F for = 10000 iterations and classified x, by checking the output configurations with a matching condition: : all cell states are close to 0 and configurations F (x) and F −1 (x) are almost the same, i.e., max(F (x)) < ε and : all cell states are close to 1 and configurations F (x) and F −1 (x) are almost the same, i.e., min( : all cell states are close to the same value and configurations F (x) and : all cell states are close to the same value and configurations F (x) and F −2 (x) are almost the same, i.e., max(F (x)) − min(F (x)) < ε and F (x) − F −2 (x) < η. In this case, we conjecture that F 2t (x) tends to c N 1 and F 2t+1 (x) tends to c N 2 , for some c 1 , c 2 ∈ [0, 1] (a convergence with period 2).
In this case, we assume that F t (x) tends to some fixed configuration. G : all other configurations.
We will drop the upper index (N ) unless confusion is possible.
For the reported experiments, we chose ε = 0.001 and η = 0.01. As a result, for every local rule f ∈ R, we obtained 12 subsets of binary configurations: G (10) 0 , . . . , G (10) 5 and G (11) 0 , . . . , G 11 . On this basis, we formulated the preliminary hypothesis on the dynamics of f . = X 11 , (while the other subsets are empty). Thus, it is obvious that we conjecture that for each binary configuration x ∈ X N , it holds that F t (x) tends to 0 N .
As mentioned above, the results of these first experiments allowed to detect the nature of many legal outertotalistic ACCAs. Moreover, it is not hard to confirm these observations theoretically (see, for example, Sect. 4.3). However, for some local rules we were forced to perform additional simulations. In particular, we varied N , ε, η, and used various data exploration techniques that allow for tracking both the positions and values of min(F t (x)) and max(F t (x)). In some cases, we relied on a visualization of the dynamics.
To visualize the dynamics of an ACCA, we use a slight modification of the radial view of CCAs-a method proposed in [30]. If the number of cells N is given, we consider a circle with radius 1.1 and center C divided in N equal sectors by points A 0 , A 1 , . . ., The value x n of the nth cell is represented by a dot lying on the line segment C A n at a distance 0.1+x n from C. For the sake of convenience, we connect subsequent dots using a polyline. Four examples of the radial view of some configurations from [0, 1] 10 are shown in Fig. 2.
The radial representations at subsequent time steps allow to observe the evolution of a given CCA and gain some intuition about the rules that govern it.  The simulation suggests a convergence with period 2: F 2t (x) → (α0) 5 and F 2t+1 (x) → (0α) 5 , for some α ∈ (0, 1). As we will see further on, Theorem 7 will confirm this observation.

Dynamical properties of legal outer-totalistic ACCAs
In this extensive section, we provide a detailed description of the dynamics of all legal outer-totalistic ACCAs.

General convergence results
We start with the following simple, yet crucial observation concerning local rules parameterized by points of P not lying on the left face (P 0 P 1 P 3 P 2 ), back face (P 0 P 5 P 7 P 2 ) and upper face (P 2 P 3 P 6 P 7 ), i.e., points (b, c, d) ∈ P satisfying b, c > 0 and 2d − c < 1.
For any x ∈ X 0,1 N and any t ≥ 2, the configuration F t (x) does not contain any 1s.
Among the values different from f (0, 0, 0), only contains 1. If so, then according to (P1), F t−1 (x) has to contain a string 01 and, consequently, F t−2 (x) contains a string x yzu such that f (x, y, z) = 0 and f (y, z, u) = 1. However, the first equality implies y = z, while the second one is possible only when y = z. This contradiction shows that  Since we consider legal ACCAs, it holds that F t (0 N ) = 0 N for any t > 0. Moreover, as we will see, for many (b, c, d) ∈ P and many x ∈ X N , it holds that F t (x) → 0 N . The following lemma characterizes the homogeneous configurations s N ∈ X N that may be the limit of some sequence F t (x) t∈N when b + c − d = 0.
Note that although 2b+c−1 2(b+c−d) is bounded from above by 1, it still might be negative, in which case the only possible homogeneous limit of F t (x) is 0 N .  are parameterized by points lying on the upper face P 2 P 3 P 6 P 7 of the parallelepiped P (see Fig. 4a).
In this case, the expression of the local rule f reduces to  points (b, c, d) representing five distinct dynamics observed in the simulation of double-legal outertotalistic ACCAs lying on the upper face P 2 P 3 P 6 P 7 of the parallelepiped P Thus, the set of double-legal outer-totalistic ACCAs can also be parameterized by the points of the square P 2 P 3 P 6 P 7 shown in Fig. 4b and denoted in the remainder by S. Note that if f is double-legal, then also f C is double-legal. Moreover, the points that parameterize f and f C are symmetric w.r.t. the diagonal P 3 P 7 . As the behavior of F C is strongly related to the behavior of F, it is sufficient to consider the case b + d ≤ 1.
In Table 3, we list a few points (b, c, d) corresponding to double-legal outer-totalistic ACCAs with distinct dynamics and indicate the corresponding groups of initial configurations identified through simulation. These distinct dynamics will be discussed analytically further on.
Theorem 3 Let f be the local rule parameterized by a point (b, d) ∈ S and let F be the corresponding global rule.
(s1) If b + d < 1, then for any initial configuration x ∈ X 1 N , it holds that 2 , then for any initial configuration x ∈ X N , it holds that Moreover, from the above calculations, it is obvious that if at least one variable x, y, z is strictly smaller than 1, then also f (x, y, z) < 1. This means that if for some t 0 ∈ N the configuration F t 0 (x) does not contain any 1s, i.e., M = max(F t 0 (x)) < 1, then for each t ≥ t 0 and 0 ≤ n ≤ N − 1, it holds that x t n ≤ M. Hence, This means that σ (F t (x)) tends to zero (since 1− , which concludes the proof of statement (s1). As mentioned before, statement (s2) is a direct consequence of (s1), since local rules param- )y and f is density-conserving, i.e., for any x ∈ X N , it holds that The dynamics of such ACCAs is described in [9], including the proof of statements (s3) and (s5). Statement (s4) is obvious.
The dynamics of ACCAs whose local rules are parameterized by these points is very simple. The state of each cell tends to zero, irrespective of the initial configuration. The following theorem formally characterizes the behavior of such CAs. a point (b, c, d) ∈ P and let F be the corresponding global rule. If 2b + c < 1 and 2d − c < 1, then for any initial configuration x ∈ X N , it holds that lim x→∞ F t (x) = 0 N .

Theorem 4 Let f be the local rule parameterized by
Since 2b + c < 1 and 2d − c < 1, we see that, in both cases, σ (F t (x)) tends to zero, which concludes the proof.

4.4
The case 2b + c ≥ 1 and 2d − c < 1 To conclude our investigation, we explore the dynamics of the local rules parameterized by points belonging to the polyhedron P 1 P 4 P 5 P 3 P 6 P 7 without the upper face P 3 P 6 P 7 , denoted by P 0 and shown in Fig. 6. This set is explicitly given as In this subsection, the expression 2b+c−1 2(b+c−d) will be denoted by λ and will turn out to be a key parameter. Note that if (b, c, d) ∈ P 0 , then it holds in particular that b + c − d > 0, hence λ is well defined and lies in [0, 1].
This section is organized as follows. We start with some technical lemmata. Then we describe the dynam-ics of the rules parameterized by points belonging to P 0 , except for the three edges P 1 P 3 , P 4 P 6 and P 5 P 7 . Next, we consider the edges P 4 P 6 , P 1 P 3 and P 5 P 7 one by one. The latter case is the most complicated one, since the rules parameterized by points from P 5 P 7 exhibit the most surprising behavior.

Some technical lemmata
In order to prove the main theorems in this section, we introduce a few lemmata. a point (b, c, d)

Lemma 3 Let f be the local rule parameterized by
Proof Assume first that 2b + c > 1, then the graph of the function g is shown in Fig. 7a (under the proviso that s v may lie to the right of 1 or s 2 may be equal to 1). Indeed, the function g is a quadratic function with roots s 1 = 0 and s 2 = 2b+c 2(b+c−d) , restricted to the domain [0, 1]. It has two fixed points, namely 0 and λ. Since c ≤ 2d ≤ c + 1, it holds that λ ≤ 1 ≤ s 2 . Directly from the graph of g, we can conclude that (g1) holds, since for s ≥ λ we have g(s) ≤ s. Moreover, the fixed point s 1 = 0 is a repeller, while λ is an attractor with basin including (0, 1). This implies (g2).
Next, assume that 2b + c = 1. In that case, it holds that λ = 0 and the line y = s is tangent to the graph of g at s 1 = 0 (see Fig. 8) and one can see that 0 is an attractor with basin [0, 1].
As a simple consequence of Lemma 3, we get the following description of the orbits for the initial configuration 1 N . point (b, c, d) ∈ P 0 and let F be the corresponding global rule. If c = 2d (i. e., (b, c, d) belongs to the lower face P 1 P 4 P 5 ), then F t (1 N ) = 0 N , for each t ≥  Proof Note that f (1, 1, 1) = 2d − c. If 2d = c, then it holds that F(1 N ) = 0 N , whence F t (1 N ) = 0 N , for any t ≥ 1. If 2d = c, then it holds that 2d − c ∈ (0, 1) (since (b, c, d) ∈ P 0 ). Taking into account that g(s) = f (s, s, s), it then follows that Since the behavior of F on 1 N is now understood, in the remainder we will only consider initial configurations x from X 0,1 N . point (b, c, d) ∈ P 0 and let F be the corresponding global rule. For each ε > 0, for each t 0 ∈ N and for any initial configuration x ∈ X N , there exists t ε ≥ t 0 such that

Lemma 5 Let f be the local rule parameterized by a
Proof We give a proof by contradiction. Let ε > 0 and t 0 ∈ N be given and assume that for some initial configuration x it holds for each t ≥ t 0 that For every t ≥ t 0 and any n ∈ {0, 1, . . . , N −1}, it holds that Note that 1−2(b+c−d)ε < 1, so the above inequality implies σ (F t (x)) t→∞ − −− → 0, contrary to the assumption in Eq. (7).
The following lemma lists some dependencies between the elements of the set V introduced in Theorem 2. Note that if f is a local rule defined by Eq. (4), then the set V contains at most six different numbers: Moreover, V 2 always lies between V 1 and V 4 , while V 5 always lies between V 3 and V 6 (since f is affine also w.r.t. x+z). The following lemma presents some dependencies between V 1 , V 3 , V 4 and V 6 .
Lemma 6 Let f be the local rule parameterized by a point (b, c, d) ∈ P 0 and let 0 ≤ m < M ≤ 1. Then it holds that Proof Statements (d1)-(d5), respectively, follow from which concludes the proof.
Equipped with the above facts, we study the behavior of ACCAs parameterized by points in P 0 . We divide P 0 into several parts, because for each part we use different proof methods.
In this subsection, we uncover the dynamics of local rules parameterized by points (b, c, d) that belong to P 0 but do not lie on any of the edges P 1 P 3 , P 4 P 6 and P 5 P 7 , i.e., 0 < b < 1 2 or 0 < c < 1. In this case, the dynamics turns out to be very simple: Each initial configuration x ∈ X 0,1 N tends to the same homogeneous one, as stated in the following theorem.
If m 0 = M 0 , i.e., F t ε (x) is a homogeneous configuration (m 0 ) N , then according to Lemma 3, it holds that F t (x) tends to λ N . Therefore, we can assume that m 0 < M 0 .
Consider the following two sequences (m t ) t≥0 and (M t ) t≥0 : According to Theorem 2, it holds that We will show that the sequence of N -dimensional cubes [m t , M t ] N collapses to a single point λ N . We consider two cases.
Case one. Assume that for some t 0 ≥ 0, it holds that Then obviously m t 0 ≤ b b+c−d and from Lemma 6 it follows that and By simple induction, we get M t ≤ b b+c−d for t ≥ t 0 , and consequently, M t+1 = g(M t ) and m t+1 = g(m t ). Therefore, according to Lemma 3(g2), we have Case two. Assume that for all t ≥ 0, it holds that M t > b b+c−d . Let us note that for all t ≥ 0 it holds that m t < λ + ε. Indeed, from Lemma 3(g1), it follows that .
It follows from Lemma 6(d2), (d3) and (d4) that for all t ≥ 0 it holds that which is only possible for c = 1, as otherwise the sequence (M t ) ∞ t=0 would decrease exponentially to zero, contradicting the assumption that M t > b b+c−d > 0, for all t ≥ 0. Note that if c = 1, then b b+c−d = λ < M t and the sequence (M t ) ∞ t=0 is decreasing. Again, we consider two cases.
(a) Assume that for all t ≥ 0, it holds that also m t ≥ λ.
It then holds that λ ≤ m t ≤ M t . Using Eq. (8), it follows that Since 0 < b < 1 2 , the above upper bound shows that the difference between M t and λ decreases exponentially to 0, i.e., lim t→∞ M t = λ, which implies that also lim t→∞ m t = λ. (b) Assume that for some τ ≥ 0, it holds that m τ < λ.
Of course, then due to Lemma 3(g1), it holds that m t ≤ λ for all t ≥ τ . It follows from Lemma 6(d1) and (d2) that for all t ≥ τ , it holds that Let δ = min(m τ , g(M τ )). On the one hand, it holds that 0 < m τ ≤ λ. On the other hand, λ < M τ < 1 implies g(M τ ) > 0. This yields 0 < δ ≤ λ. We will show that m t ≥ δ for all t ≥ τ (it holds for t = τ ). The following reasoning is based on Fig. 7a. Since λ < M t ≤ M τ , it holds that g(M t ) ≥ min(λ, g(M τ )) ≥ δ. Moreover, since m t ≤ λ, it holds that g(m t ) ≥ m t ≥ δ. Together, according to Eq. (9), we obtain m t+1 ≥ δ. Now, let us compare the differences M t+1 − m t+1 and M t − m t for t ≥ τ . If m t+1 = g(M t ), then it follows from Eq. (8) and Lemma 3 that Similarly, if m t+1 = g(m t ), then In both cases, we have Note that c = 1 implies that 1 2 ≤ d < 1. Together with δ > 0 and b < 1 2 , it then follows that α < 1. This means hat In this subsection, we discuss the dynamics of local rules parameterized by points belonging to the edge P 4 P 6 . In this case, λ = 1 3−2d and the local rule is given by An interesting observation is the following bifurcation: For any even N ∈ N, it holds that Theorem 6 Let f be the local rule parameterized by a point 1 2 , 1, d ∈ P 0 and let F be the corresponding global rule. If 1 2 < d < 1, then for any initial configuration x ∈ X 0,1 N , it holds that -if N is even, then for any x ∈ X 0,1 Proof Let x ∈ X 0,1 N . For any t ≥ 0, let μ t denote the maximal absolute difference between x t n and λ, i.e., μ t = max 0≤n≤N −1 |x t n − λ|. As 1 2 ≤ λ < 1, it holds that μ t ≤ λ. Additionally, which means that In particular, since μ t ≤ λ = 1 3−2d , it holds that μ t+1 ≤ μ t .
One easily verifies that in this case In this subsection, we discuss the dynamics of local rules parameterized by points belonging to the edge P 1 P 3 . In this case, the local rule is given by which yields Moreover, the alternating sum of states, i.e., is an invariant of this dynamical system. Indeed, Due to the periodic boundary conditions, we have The following theorem describes the dynamics of such CAs.
Theorem 7 Let f be the local rule parameterized by a point ( 1 2 , 0, d) ∈ P 0 , where 0 ≤ d < 1 2 , and let F be the corresponding global rule.
(i) If N is even, then for any x ∈ X N , it holds that Proof Let x ∈ X N . First, we consider even N . For each t ≥ 0, we consider the following two sets Note that Let α t = max j∈E t x t j and β t = max j∈O t x t j . According to Eq. (12), for any t ≥ 0 it holds that α t+1 ≤ α t and β t+1 ≤ β t . This means that both sequences (α t ) t≥0 and (β t ) t≥0 are convergent. Hence, α = lim t→∞ α t and β = lim t→∞ β t . Now consider δ > 0, then there exists τ ≥ 0 such that for any t ≥ τ , it holds that α ≤ α t < α + δ. We will show that for any t ≥ τ and any j ∈ E t , it holds that Indeed, suppose that this is not true, i.e., there exist t 0 ≥ τ and j 0 ∈ E t 0 such that x t 0 j 0 < α − ε, where ε = 2 N 2 −1 − 1 δ. According to Eq. (12) we have and so on. Thus, after N 2 − 1 steps, we get that for each which contradicts the fact that α t 0 +k−1 ≥ α, which According to Eqs. (11) and (13), the numbers α and β satisfy the following equations: The first two imply that if α = 0, then β = 0 and vice versa. Therefore, if . This concludes the proof for even N .
Second, we consider odd N . Consider the following concatenation: Due to the periodic boundary conditions, it holds for each t ≥ 0 that However, for the initial configuration xx, the alternating sum equals 0 and the first part of the theorem implies that This concludes the proof for odd N .

The case
In this subsection, we discuss the dynamics of local rules parameterized by points belonging to the edge P 5 P 7 . In this case, λ = 0 and the local rule is given by In particular, f (x, 0, z) = 0 and f (0, y, 0) = y. This means that any binary configuration x ∈ X N not containing two neighboring 1s is a fixed point of f . Furthermore, each substring of x ∈ X N starting and ending with 0 and not containing two neighboring 1s remains unchanged in F t (x) for every t ≥ 0. Additionally, according to Lemma 4, we have F t (1 N ) t→∞ − −− → 0 N . For these reasons, to uncover the behavior of f , it suffices to consider strings of the type 011 . . . 110, where the number of 1s equals K with 2 ≤ K < N . W.l.o.g., let us assume that . . 110 is then described by the system of recurrence relations: t=0 is decreasing for every n ∈ {1, 2, . . . , K } and thus converges. We denote this limit as q n = lim t→∞ x t n ∈ [0, 1]. According to Eq. (15), it holds that for any n ∈ {1, 2, . . . , K }, where q 0 = q K +1 = 0. Since f is outer-totalistic, it holds that for any n ∈ {1, 2, . . . , K } and any t ≥ 0, and thus, for any n ∈ {1, 2, . . . , K }.
We start with some technical lemmata.
Proof Since, 1 2 ≤ d < 1, it follows from Eq. (16) that q n−1 q n + q n q n+1 = 0, for any n ∈ {1, 2, . . . , K − 1}. Since q 0 = 0, we get q 1 q 2 = 0. A simple induction yields the first part of the claim. If K is even, then, according to Eq. (18), it holds for n = K 2 that q K Proof According to Eq. (15), for any t > 0, it holds that Hence, also using x 0 1 = x 0 2 = 1, we have Since x 0 2 x 0 3 = 1 and x l 2 x l 3 ≥ 0, for any l ≥ 0, it then follows that Further, if d = 1 2 , then the local rule is given by f (x, y, z) = y(1 − 1 2 (x + z)), and the second part of the lemma readily follows.
Finally, we consider the case 1 2 < d < 1 and K ≥ 3. The following theorem describes the dynamics of F on strings 011 . . . 110, where the number of 1s is even.
The above theorem can be illustrated as follows. Suppose that in a string 011 . . . 110 the number of 1s is even, then For an odd number K ≥ 3, we introduce the following notations: s = K +1 2 , 2n o − 1 is the largest odd number not greater than s and 2n e is the largest even number not greater than s.

Lemma 9
Let K ≥ 3 be odd. For any t ≥ 0, it holds that Additionally, Moreover, if t > s, then the inequalities in Eqs. (21) and (22) are strict.
Proof One can prove Eq. (21) using induction in a similar way as in the proof of Theorem 8. We can also prove Eq. (22) using induction. Note that for t = 0 the inequality is trivially fulfilled and assume that it is satisfied for some t ≥ 0. If s is odd, then 2n o − 1 = s and 2n e = s−1; thus, Eq. (21) and the induction hypothesis imply Similarly, if s is even, then 2n e = s and 2n o −1 = s−1, and it follows using Eq. (21), the induction hypothesis and the fact that x t s−1 = x t s+1 according to Eq. (17), that Observe that if 2 j + 1 < K 2 and x t 2 j−1 > x t 2 j+1 , or x t 2 j−2 < x t 2 j , or x t 2 j < x t 2 j+2 , then x t+1 2 j−1 > x t+1 2 j+1 and, consequently, x t+k 2 j−1 > x t+k 2 j+1 for k ≥ 1. Analogously, if 2 j + 2 < K 2 and x t 2 j < x t 2 j+2 , or x t 2 j−1 > x t 2 j+1 , or x t 2 j+1 > x t 2 j+3 , then x t+1 2 j < x t+1 2 j+2 and, consequently, x t+k 2 j < x t+k 2 j+2 for k ≥ 1. Since 1 3 , by simple induction one can obtain strict inequalities in Eq. (21) for t ≥ s. This also implies that the inequalities in Eq. (22) are strict.
Lemma 10 Let K ≥ 3 be odd. For any t ≥ 0, it holds that Proof For odd s, we have x t 2n e = x t 2n e +2 ; thus, For even s, we have x t 2n o −1 = x t 2n o +1 ; thus, Theorem 9 Let K ≥ 3 be odd. For any 2 j − 1 ≤ s, it holds that q 2 j−1 > 0, while for any 2 j ≤ s, it holds that q 2 j = 0. Moreover, q 1 + q 3 + . . . + q K = 1.
The above theorem can be illustrated as follows. For even s, we have

Conclusions
In this paper, we have provided a complete description of the dynamics of legal outer-totalistic ACCAs. On the one hand, such CAs are the simplest generalization of elementary cellular automata, while on the other hand, they are dynamical systems that exhibit some properties that do not occur in the binary case. Thanks to massive numerical simulations, we have been able to partition the rule space in a number of classes with a distinct behavior. Through the use of a panoply of proof techniques and oftentimes tedious proofs, we have been able to provide undeniable analytical evidence for the dynamics observed. Table 4 summarizes the results obtained. In order to keep the presentation manageable, we focus on initial configurations from X 0,1 N only and add a short explanation for the other configurations separately below the table.
The results obtained show that in the set of ACCAs considered, one can observe various types of sensitivity: -Sensitivity to the change of a single value in an initial configuration. This kind of sensitivity may concern all initial configurations, as, for example, for f (x, y, z) = 0.4(x + z) + 0.2y (see (s3) in Theorem 3) or only some initial configurations, as for example, for f (x, y, z) = 0.2(x +z)y+0.4(x + z) − 0.2y (see (s1) in Theorem 3). -Sensitivity to the change of the number of cells in the grid. Note that if we add one cell to the considered grid, then the parity of N will change. For example, the local rule f (x, y, z) = −0.1(x + z)y + 0.5(x + z) is not immune to such interference in the grid structure (see Theorem 7). -Sensitivity to slight changes in the parameters of a local rule. Perhaps the best example of this sensitivity are local rules parameterized by points lying on the edge P 1 P 3 , i.e., local rules given by the following expression: f (x, y, z) = −( where 0 ≤ d ≤ 1 2 . The dynamics of such ACCAs for 0 ≤ d < 1 2 (see Theorem 7) is completely different than for d = 1 2 (see (s3) in Theorem 3).
Although we limited our investigation to binary initial configuration only, nearly all results can be readily extended to all configurations from [0, 1] N . However, the case described in Sect. 4.4.5 would require a more substantial effort.
Data availability Data sharing is not applicable to this article as any datasets were generated by presented formulas and analyzed during the current study.