Monomial Cellular Automata: A number theoretical study on two-dimensional cellular automata in the von Neumann neighbourhood over commutative semigroups

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Degree project

Monomial Cellular Automata

A number theoretical study on two-dimensional cel- lular automata in the von Neumann neighbourhood over commutative semigroups

Author : Linnea Fransson Supervisor : Marcus Nilsson Examiner : Hans Frisk Date: 2015-06-09 Subject : Mathematics Level : Bachelor

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Abstract

In this report, we present some of the results achieved by investigating two-dimensional monomial cellular automata modulo m, where m is a non-zero positive integer. Throughout the experiments, we work with the von Neumann neighbourhood and apply the same local rule based on modular multiplication. The purpose of the study is to examine the behaviour of these cellular automata in three different environments, (i.e. the infinite plane, the finite plane and the torus), by means of elementary number theory. We notice how the distance between each pair of cells with state 0 influences the evolution of the automaton and the convergence of its configurations. Similar impact is perceived when the cells attain the values of Euler’s φ-function or of integers with common divisors with m, when m > 2. Alongside with the states of the cells, the evolution of the automaton, as well as the convergence of its configurations, are also decided by the values attributed to m, whether it is a prime, a prime power or a multiple of primes and/or prime powers.

Keywords: cellular automata, monomial, multiplicative, two-dimensional, von Neumann neighbourhood, number theory

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1 Introduction

As time goes by, it becomes more and more evident that interrelation between subjects is an inevitable fact. Mathematics and linguistics, or mathematics and biology, are examples of subjects that, at a first glance, are completely unrelated to each other. But studies from the last century have shown that they are more connected than people once were capable to believe. A common link that both examples share is the so-called cellular automaton, or cellular automata in plural. These fascinating systems are studied in the present investigation, where we focus on a special case: monomial cellular automata in two dimensions.

Section 1 of this report is designated to the background and definition of cellular automata as well as to present previous studies where monomial cellular automata were introduced. The purpose of the study is also found in the first section. In Section 2, we give an explanation to the neighbourhood, the local rule and the structures used in the investigation. In the same section, the reader will find some basic definitions, along with examples and theorems that are useful for the understanding of the results. Section 3 is dedicated to the results only, leaving the interpretation and analysis of those to Section 4.

1.1 Cellular automata

In the beginning of 1950s, John von Neumann and Stanislaw Ulam introduced to the world a new concept of self-replicating systems that would become known as cellular automata (CA¹).

These were the results of studies regarding biology and robotics. Von Neumann was fascinated by the complexity of self-reproductive biological systems as well as by the design and structure of the computers from his time [4, 5, 6, 9]. The resemblance of the CA to the real world contributed to further studies in the subject and their applications cover different fields of science apart from biology, e.g. hydrodynamics, information theory and coding theory [4, 5].

The underlying topology of the CA is normally addressed by Z^d, where d is the dimension of the CA. Each cell c of the automaton has a state xctaken from a set S, which is usually finite.

A CA configuration is a mapping x ∶ Z^d→ Sthat specifies the states of all cells, and the set of all configurations is denoted by S^Z^d. The states of the cells are changed synchronously at discrete time steps, where the next state of the cell c depends on the current states of the neighbouring cells, or neighbours. The change occurs according to a local rule that is applied to all cells of the system at the same time. This rule is a function f ∶ Sⁿ → S, where n is the size of the neighbourhood. A sequence of a cell and its neighbours forms a neighbourhood, which is denoted by N = (n₁, . . . , n_m), where n_i∈Z^d. When convenient, one can write x[c] = (x_c+n₁, ⋯, x_c+n_m) when referring to the subconfiguration associated with the neighbourhood. Not all cells in the neighbourhood, however, must be included in the local rule, as it is shown in Example 1.1.

Figure 1: Space-time diagram of rule 90. The diagram to the left has an initial configuration containing a single state that is 1 (1=black, 0=white), while the initial configuration in the right diagram was chosen randomly [5].

1the abbreviation CA is used for both the singular and plural forms.

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Most commonly, CA are constructed over infinite lattices. However, finite structures can also be used depending on the purpose of the study. One way of constructing a CA over a finite structure is through periodic boundary conditions. When working in one-dimension, this is achieved by identifying vertices i and i + n as the same in Z for some n > 1. For higher dimensions, this method is extended to a d-dimensional torus. A second way of creating finite CA structures is by imposing zero boundary conditions. This means that the configuration is bounded by additional cells with fixed stated to zero.

Example 1.1. In this example, we will illustrate the evolution of a finite one-dimensional CA by showing the configurations at each discrete time t in the interval 0 ≤ t ≤ 5. We will apply the zero boundary condition and choose the states of the cells from the set S = Z⁵. The neighbourhood is N = (−1, 0 − 1) and the local rule is f (xi−1, xi, xi+1) =f (x[i]) = xi−1+xi+1

modulo 5.

t = 0 0 0 0 0 1 0 0 0 0

+

t = 1 0 0 0 1 0 1 0 0 0

+

t = 2 0 0 1 0 2 0 1 0 0

t = 3 0 1 0 3 0 3 0 1 0

t = 4 0 0 4 0 1 0 4 0 0

t = 5 0 4 0 0 0 0 0 4 0

The rule applied in this example is known as rule 90, which is the same as the one illustrated in Figure 1. The only difference, however, lies in the number of possible states for each cell, as the CA from the figure work with S = Z² and the CA of the present example takes the states of its cells from S = Z⁵. The space-time diagram in this case presents therefore a similar structure as the left diagram in Figure 1 but with more shades of black:

◇ One-dimensional CA have been studied in detail, specially the so-called Elementary CA. They are the simplest class of one-dimensional CA with only two possible values for each cell, i.e.

S = {0, 1}. Stephen Wolfram is famous for studying them and the behaviour of their 256 local rules. Figure 1 illustrates how two different initial configurations of elementary CA can yield distinct results despite having the same local rule. For further information regarding elementary CA and their rules, the reader is referred to [9].

Two-dimensional CA have also been of great interest for mathematicians. They were orig- inally studied by von Neumann in order to emulate the operations realised by the compo- nents of a computer and other mechanical devices [9]. One of the most common neighbourhood structures used for two-dimensional cases was named after him and is given by N = ((0, 0), (−1, 0), (0, −1), (1, 0), (0, 1)). The Moore neighbourhood is also commonly used and it consists of the cell itself and all its bordering neighbours (see Figure 2).

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Figure 2: The von Neumann neighbourhood (left) and the Moore neighbourhood (right) for two-dimensional CA [5].

Example 1.2. In a zero-bounded plane, let S = Z²and N = ((0, 0), (−1, 0), (0, −1), (1, 0), (0, 1)), the von Neumann neighbourhood. At the discrete time t = 1, the local rule f (x[i, j]) = x_i−1,j+x_i,j−1+x_i,j+x_i+1,j+x_i,j+1modulo 2 gives:

0 0 0 0 0

0 0 0 0 0 1

1 0 0 1

1 1 1 1 1

1 1 0 0 1

0 1 1 1 1

1 1 0 1 0

t = 0

0 0 0 0 0

0 0 0 0 0 1

1 0 0 0

0 1 1 1 0

1 0 1 1 1

1 0 1 0 1

0 1 1 0 0

t = 1

In the first diagram, the coloured areas represent two different cells and their neighbourhoods before the local rule is applied. As it is displayed in the same diagram, the 0s in the frame are included in the neighbourhoods of the bordering cells of the system. They, however, do not have a neighbourhood of their own and neither does the local rule apply to them. In the second diagram, the same colours are used to denote the cells whose states are the results of their respective neighbourhoods and the local rule.

◇ A famous example of a two-dimensional CA is the Game of Life by Conway [3, 4, 9]. It has only two states and uses the Moore neighbourhood. Unlike in the previous example, the states of the cells in the Game of Life are commonly illustrated with black and white cells, where the black cells are referred to as alive and the white cells are dead. The process when a white cell changes to black is called birth and the opposite process is called death. When a black cell does not change its state during a given update, the process is called survival. The conditions for these processes are given by the CA’s outer-totalistic local rules, i.e. the birth and survival of a cell depends only on the number of living cells in its neighbourhood [3]. Birth occurs when there are precisely three live cells in the neighbourhood; the survival condition is that there are exactly two or three living neighbours; and death is caused either when there are less than two live cells in the neighbourhood or when there are more than three living neighbours [3, 4].

The purpose with Game of Life is to simulate artificial life and its evolution, as illustrated in Figure 3.

1.2 Monomial cellular automata

Due to its linear properties², a special class of CA, the additive, has been object of thorough analyses since the beginning [1]. But since this property was extended to CA over commutative

2E.g additivity: f (x + y) = f (x) + f (y) and homogeneity: f (αx) = αf (x), for some scalar α

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Figure 3: The evolution of an object in the Game of Life [4].

semigroups³(e.g. ⟨Z^m, ●⟩,the residue classes of integers modulo m under multiplication), multiplicative versions of CA gained a new focus in the mathematical world. Bartlett and Garzon refer to them as monomial CA and claim that these automata “provide a multiplicative ana- logue of additive cellular automata with novel dynamical features” [1]. Such features consist of the impact on the evolution of the automata caused by the cells with state 0. Many of their observations are similar to those obtained and presented in Section 3, where experiments including the state 0 were performed.

A great deal of Bartlett and Garzon’s article concerns one-dimensional CA over ⟨Z², ●⟩.

Higher-dimensional analogues as well as monomial CA over other semigroups are also introduced but briefly. Their focus lies primarily on the phase portraits⁴ created by two subcases that they distinguish as symmetric and asymmetric monomials. The symmetry in question is related to the neighbourhood and, in the one-dimensional case, the number of neighbours on each side of the cell on focus. The example below is a compilation of some of the rules presented in [1].

Example 1.3. Local rules for symmetric monomials with the neighbourhood N = (−r, 0, r), where r ∈ N, can look as following:

f (x[i]) = xi−rxi+r modulo m;

f (x[i]) = x_i−rxix_i+r modulo m.

Equivalently, local rules for asymmetric monomials with the neighbourhood N = (−r, 0, k), where r, k ∈ N and r > k, can be given by

f (x[i]) = x_i−rx_i+k modulo m;

f (x[i]) = xi−rxixi+k modulo m.

◇ A more theoretical investigation was conducted by Marcus Pivato in [7]. Instead of monomial, Pivato adopts the word multiplicative, taking in consideration the multiplication operation of the group where the CA are studied in. The article explores the algebraical structures of the multiplicative CA and applies probabilistic analysis on the automata. Instead of working in semigroups with an identity element, or monoids, Pivato focus on multiplicative CA over compact groups and studies the structural relations between the automata and the group where it belongs. The results from his study, nevertheless, cannot be extended to all kinds of multiplicative CA [7]. Consequently, Pivato acknowledges that there is still much left to explore where multiplicative, or monomial, CA are concerned.

1.3 Purpose of the study

For the last decades, CA and their pattern evolution have been studied in detail in different settings and from distinct perspectives. Researches have been made in probabilistic settings and through ergodic theory, as well as from a dynamical systems perspective and from a

3In this study, we worked with semigroups with an identity element. Such algebraic structure is known as monoid.

4A phase portrait is a geometric method for describing the sequences of a CA.

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logic, automata and language perspective [5]. Monomial CA, however, have not had the same attention and the number of reports and articles about them is remarkably smaller. Bartlett’s article provides us a study of the phase portraits of different monomials as well the methods to achieve them [1]. Meanwhile, Pivato performed a more abstract and theoretical study on multiplicative CA, focusing on their structure and applying probabilistic analysis [7].

The purpose of this study is to investigate the behaviour of monomial⁵CA over commutative semigroups from a number theoretical perspective. Consequently, most of the results presented in this report are represented by number matrices instead of colour diagrams. In order to make the purpose achievable within the frames of the project, the subject of interest was limited to two-dimensional CA and restricted to one neighbourhood, in which the local rule was the same in all experiments. More detailed information concerning the method in which the study was conducted with is given in the following section.

5The word monomial was chosen over multiplicative as the local rule, as well as its appearance, used in this study was inspired by Bartlett and Garzon’s experiments [1].

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2 Method and preliminaries

2.1 Neighbourhood and local rule

For this study, we choose to work with only one neighbourhood as a way to deepen the understanding of the behavioural patterns created by the CA. The neighbourhood that we work with is N = ((0, 0), (−1, 0), (0, −1), (1, 0), (0, 1)), i.e. the von Neumann neighbourhood. The main reason behind the choice of this specific neighbourhood lies in the studies performed by Bartlett and Garzon [1]. When presenting monomial CA in higher dimensions, they introduced the von Neumann neighbourhood for two-dimensional cases as the most obvious analogy to one-dimensional neighbourhoods. In their article, they use the cardinals to designate the neighbours of the cell i, as depicted in Figure 4.

i S N

W E

Figure 4: The von Neumann neighbourhood described in [1], where W , S, E and N correspond to West, South, East and North on the neighbourhood centered at cell i. Compare the notation with the one used in [7] and in Figure 2.

The local rule that we apply throughout the study is also inspired by Bartlett and Garzon’s article [1]. Following the thread of analogies to one-dimensional CA, the rule in question can be considered as an extension and adaptation of rule 90 (see Example 1.1 and Figure 1) to two-dimensional CA over commutative semigroups with multiplication modulo m as their operation. More precisely, the rule is given by:

f (xi−1,j, xi,j−1, xi,j, xi+1,j, xi,j+1) =x^p_i−1,j¹ ⋅x^p_i,j−1² ⋅x^p_i+1,j³ ⋅x^p_i,j+1⁴ mod m. (1) The exponents pk, where k ∈ {1, 2, 3, 4}, are positive integers and they are not necessarily different from each other. For practical reasons, the cardinal indexes suggested by [1] have been chosen to be used in this work instead of the ones used in (1). Besides, we also propose a new notation that will give a better representation of the evolution of the state of the cell xi

at the discrete time t + 1, where t ∈ N. Combining the indexes from Figure 4 with the newly proposed notation, the rule is now given by:

x_i(t + 1) = x^p_W^W(t) ⋅ x^p_S^S(t) ⋅ x^p_E^E(t) ⋅ x^p_N^N(t) mod m. (2) For xi(0), no rule has been applied since it stands for the state of the cell xi in the initial configuration. The rule is therefore only applicable when t ≥ 0, i.e. for xi(1). Furthermore, it is implicitly understood in (2) that the indexes correspond to the respective cardinal neighbours adjacent to the cell xiwith their respective exponents. In this study, however, all experiments were performed with pk = 1, giving us the final mapping in (3). An illustration of the local rule is given by Figure 5.

xi(t + 1) = xW(t) ⋅ xS(t) ⋅ xE(t) ⋅ xN(t) mod m. (3) Some of the experiments presented in this report obtained the same configurations that can be achieved by traditional two-dimensional CA. In Sections 3.1, 3.1.1 and 3.2.1, the conditions for birth, survival and death of the cells are discussed and compared to traditional CA, whose rule numbers⁶are also given in the text.

6During his researches about traditional CA, Wolfram gave them rule numbers that represent the evolution of the cells in the configurations [9].

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xW(t) x_S(t)

xE(t) x_N(t)

xi(t)

time t

⋅

⋅ ⋅

mod m

xi(t + 1)

time t + 1

Figure 5: An illustrative representation of equation (2) in the von Neumann neighbourhood.

2.2 Structures of cellular automata

In the first section of this report, we named three different types of structures and conditions that are commonly used in the studies of CA: the infinite lattice, the periodic and the zero boundary conditions. For the present investigation of two-dimensional CA, we have taken all of them into consideration in order to fulfil the purpose of the study. Some of the first experiments are possible to realise by hand but the need of a computational device is obvious very early in the process. The program used to simulate the monomial CA was Mathematica v.10 and the main code can be found in the Appendix⁷.

Working with no boundary conditions would be ideal when it comes to dealing with CA.

However, since no program can recreate a structure such as the infinite plane, we simulate it artificially by setting high values to the borders and following the development of the interior of the system for a reasonable amount of time. By “reasonable”, we mean the number of steps before the states of the cells in the middle are influenced by the states of the boundaries. In this report, some of the CA created in the infinite plane are illustrated as the matrix below:

⋮ ⋮ ⋮ ⋮

⋯ α1,1 α1,2 ⋯ α1,n ⋯

⋯ α_2,1 α_2,2 ⋯ α_2,n ⋯

⋯ ⋮ ⋮ ⋱ ⋮ ⋯

⋯ α_m,1 α_m,2 ⋯ α_m,n ⋯

⋮ ⋮ ⋮ ⋮

No boundary condition: infinite plane

When establishing the boundary conditions to our finite plane, we had to make a necessary adaptation of the frame. Instead of fixing the states of the borders to 0, we fix them to 1, preserving the frame’s fundamental purpose. In this work, we propose the name unit boundary condition for this kind of finite structure. Throughout this report, the following matrix is used to represent some of the examples created in a bounded m × n-plane:

7For almost all of the experiments, it is possible to run the predefined function CellularAutomaton in Mathematica. The only exception is experiment 7.

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1 1 ⋯ 1 1 α_1,1 α_1,2 ⋯ α_1,n 1 1 α2,1 α2,2 ⋯ α2,n 1

⋮ ⋮ ⋮ ⋮ ⋱ ⋮

1 αm,1 αm,2 ⋯ αm,n 1

1 1 ⋯ 1

Unit boundary condition: m × n-plane

The third structure used in this study, the one with the periodic boundary condition, is given by a flat torus. It is finite, since the configuration is set in a limited number of rows and columns but the last row and the last column are neighbours to the first row and the first column respectively. Consequently, monomial CA over this peculiar structure present more unique behaviours than over any of the previous structures. An example of a torus is displayed below:

α1,1 α1,2 ⋯ α1,n

α_2,1 α_2,2 ⋯ α_2,n

⋮ ⋮ ⋱ ⋮

α_m,1 α_m,2 ⋯ α_m,n

≃

αm,n αm,1 ⋯ α_m,n−1 α_1,n α_1,1 ⋯ α_1,n−1

⋮ ⋮ ⋱ ⋮

α_m−1,n α_m−1,1 ⋯ α_m−1,n−1

Periodic boundary condition: m × n-torus

The matrix to the right side of the equivalence sign is a representation of how the torus looks like after being shifted once to the right and once down, showing that the order of the structure is maintained. For convenience, however, only the left matrix is used to illustrate CA over a torus. Furthermore, the examples used in Sections 3.2 and 3.3 are taken directly from Mathematica and look therefore somehow differently from the matrices used in this section. To minimise any possible risk of confusion, the choice of structure will be explicit in the context.

2.3 Basic definitions

For the presentation of the experiments in this report, as well as for the analysis and interpretation of their results, we use some basic definitions from number theory, dynamical systems and the theory of CA. The proofs for the theorems are excluded as they are irrelevant for the present work. The reader, however, is welcome to read about them in [2] (dynamical systems), [4] (theory of CA) and [8] (number theory). Most of the definitions, examples and theorems are extracted from their respective literature with some slight modifications.

Later in this report, we talk about CA converging to ^c_c¹₃^c_c²₄, where c₁, c₂, c₃ and c₄ are elements of S. By this we mean that the cells of the configurations of the CA will gradually be taken over by the states c₁, c₂, c₃, c₄until the whole CA is covered by them. In the environments with boundary conditions, this will happen at a certain discrete time t > 0, while, in the infinite plane, cells with these states will only keep on spreading themselves throughout the CA according to the neighbourhood and the local rule applied. We also talk about CA converging to (^cc¹3^cc²4,^cc²4^cc¹3), where we mean that the states c1and c3switch places with c2and c4respectively at time t + 1, switching back at t + 2.

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n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

φ(n) 1 1 2 2 4 2 6 4 6 4 10 4 12 4 8 8 16

Table 1: The values of Euler’s phi function for 1 ≤ n ≤ 17.

Terms such as period and periodic, commonly used in the theory of dynamical systems, are present in the report as well. In this study, a configuration is said to be periodic if xi(t + n) = x_i(t), for all i and for n ∈ N. Then we say that the period of the CA is n, where n is the smallest possible value. A configuration can also be eventually periodic of period n (in [6], the author prefers the usage of the term ultimately periodic), if it is not periodic but there exists m > 0 such that x_i(t + n + k) = x(t + k) for all k ≥ m. Examples of the latter are given in the experiments described in this report.

In Experiment 5, when investigating the behaviour of CA over ⟨Z⁺m, ●⟩, where m is not a prime and Z⁺ = {1, 2, 3, ...}, we work with the Euler phi-function φ(m). As m is a positive integer, φ(m) is defined to be the number of positive integers not exceeding m that are relatively prime to m. Table 1 shows the values of Euler’s phi function for 1 ≤ n ≤ 17. For the analysis of the results, a couple of theorems related to Euler’s phi function are necessary and, therefore, they are included in this section. After each theorem, an example is given to illustrate them.

Theorem 2.1. Let p be a prime and α a positive integer. Then φ(p^α) =p^α−p^α−1.

Example 2.1. From Theorem 2.1, we have that

φ(5³) =5³−5²=100 and φ(2¹⁰) =2¹⁰−2⁹=512.

◇ Theorem 2.2. Let m and n be relatively prime positive integers. Then φ(mn) = φ(m)φ(n).

Example 2.2. Combining Theorems 2.1 and 2.2, we have that φ(36) = φ(4)φ(9) = (2²−2)(3²−3) = 2 ⋅ 6.

◇ Finally, for the interpretation and analysis of the results gathered from Experiments 3 and 7, we talk about primitive roots. If r and n are relatively prime integers with n > 0, then r is called a primitive root modulo n, or a primitive root of n, if r^φ⁽ⁿ⁾≡1 modulo n, and φ(n) is the least positive integer x such that r^x≡1 modulo n.

Theorem 2.3. The positive integer n, where n > 1, possesses a primitive root if and only if n = 2, 4, p^t, or 2p^t,

where p is an odd prime and t is a positive integer.

Example 2.3. The integer 3 is a primitive root of 7, as

3¹≡3 (mod 7), 3²≡2 (mod 7), 3³≡6 (mod 7), 3⁴≡4 (mod 7), 3⁵≡5 (mod 7), 3⁶≡1 (mod 7).

And the elements of the set {1, 2, 3, 4, 5, 6} = Z^∗7 are all relatively prime to 7.

◇

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Example 2.4. The integer 65 = 5 ⋅ 13 does not have a primitive root as it does not fulfil any of the criteria in Theorem 2.3.

◇

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3 Results

The experiments conducted in this study are performed in three phases. Firstly, we look at configurations where the states of the cells are taken from the set S = Z². The simulations realised at this stage are analogous to the one-dimensional symmetric cases in [1]. Secondly, we examine the behaviour of CA over ⟨Z^p, ●⟩ as well as over ⟨Z⁺p, ●⟩, where p is prime and p ≥ 3.

Lastly, we consider the configurations generated by S = Zm, for m not a prime. Different behaviours are observed for cases when m = p^α, where α ∈ N and α ≥ 2 (i.e. prime powers), and when m attains other values (i.e. products of primes and/or prime powers). The examples present very low values of p and m, but the reason is partially aesthetic and partially practical.

Big numbers would take a lot of space in the configurations and the diagrams would be more difficult to read. The results documented in this section are analysed and interpreted in subsections with similar names under Section 4.1.

3.1 Binary configurations

We begin the first phase of the study by choosing the set of the states of the cells in the configurations to be S = Z2 and the local rule to be given by:

xi(t + 1) = xW(t) ⋅ xS(t) ⋅ xE(t) ⋅ xN(t) mod 2.

Thereafter, we investigate the evolution of a CA whose initial configuration has only one cell with the state 0. The first n steps of the automaton are illustrated below:

⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋰

⋯ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⋯

⋯ 1 1 ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋰ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 1 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 0 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 1 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋰ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ 1 1 ⋯

⋯ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⋯

⋰ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱

t = 0

⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋰

⋯ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⋯

⋯ 1 1 ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋰ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 1 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 0 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 0 1 0 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 0 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 1 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋰ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ 1 1 ⋯

⋯ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⋯

⋰ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱

t = 1

⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋰

⋯ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⋯

⋯ 1 1 ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋰ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 1 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 0 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 0 1 0 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 0 1 0 1 0 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 0 1 0 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 0 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 1 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋰ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ 1 1 ⋯

⋯ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⋯

⋰ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱

t = 2

⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋰

⋯ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⋯

⋯ 1 1 ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋰ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 1 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 0 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 0 1 0 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 0 1 0 1 0 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 0 1 0 1 0 1 0 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 0 1 0 1 0 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 0 1 0 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 0 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 1 1 1 1 1 1 1 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋰ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ 1 1 ⋯

⋯ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⋯

⋰ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱

t = 3

(15)

⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋰

⋯ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⋯

⋯ 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 ⋯

⋯ 1 1 ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋰ 1 1 ⋯

⋯ 1 1 ⋯ 0 1 0 1 0 1 0 1 0 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 0 1 0 1 0 1 0 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 0 1 0 1 0 1 0 1 0 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 0 1 0 1 0 1 0 1 ⋯ 1 1 ⋯

⋯ 1 0 ⋯ 0 1 0 1 0 1 0 1 0 ⋯ 0 1 ⋯

⋯ 1 1 ⋯ 1 0 1 0 1 0 1 0 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 0 1 0 1 0 1 0 1 0 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 0 1 0 1 0 1 0 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 0 1 0 1 0 1 0 1 0 ⋯ 1 1 ⋯

⋯ 1 1 ⋰ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ 1 1 ⋯

⋯ 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 ⋯

⋯ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⋯

⋰ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱

t = n − 1

⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋰

⋯ 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 ⋯

⋯ 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 ⋯

⋯ 1 1 ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋰ 1 1 ⋯

⋯ 1 1 ⋯ 1 0 1 0 1 0 1 0 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 0 1 0 1 0 1 0 1 0 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 0 1 0 1 0 1 0 1 ⋯ 1 1 ⋯

⋯ 1 0 ⋯ 0 1 0 1 0 1 0 1 0 ⋯ 0 1 ⋯

⋯ 0 1 ⋯ 1 0 1 0 1 0 1 0 1 ⋯ 1 0 ⋯

⋯ 1 0 ⋯ 0 1 0 1 0 1 0 1 0 ⋯ 0 1 ⋯

⋯ 1 1 ⋯ 1 0 1 0 1 0 1 0 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 0 1 0 1 0 1 0 1 0 ⋯ 1 1 ⋯

⋯ 1 1 ⋯ 1 0 1 0 1 0 1 0 1 ⋯ 1 1 ⋯

⋯ 1 1 ⋰ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ 1 1 ⋯

⋯ 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 ⋯

⋯ 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 ⋯

⋰ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱

t = n

As a result of both the neighbourhood and the local rule being symmetric, the cells with state 0 spread themselves throughout the automaton in a diamond-shaped pattern. Observe that, as the figure spreads itself by every step of the evolution of the CA, the cells inside the outer frame of 0s have their states switching from 0 to 1 (and vice-versa) at time t + 1, and then switching back from 1 to 0 (and vice-versa) at time t + 2. This is consequence of the local rule:

whenever a cell xi has all its neighbours with state 1 at time t, then xi(t + 1) = 1. Likewise, if at least one of its neighbours has state 0 at time t, then xi(t + 1) = 0. In other words, a cell with state 1 is alive and a cell with state 0 is dead. For both birth and survival of the cell, it needs four living neighbours, while death occurs whenever there are three or less live cells in the neighbourhood. The CA is obviously outer-totalistic and behaves exactly as a traditional two-dimensional CA with rule number 768.

As a consequence of the conditions for birth, survival and death of the cells given by the update rule, the diamond-shaped figure consists of diagonals formed by cells with state 0 (dead) between diagonals of cells with state 1 (alive). Thereupon, as long as the diagonals of 0s are surrounded by diagonals of 1s, the pattern will continue to grow undisturbed. For future notice, we claim this as a necessary condition for the automaton to converge to (^{0 1}_{1 0},^{1 0}_{0 1}).

The diagrams on the previous page demonstrate a CA in the infinite plane. Because of the lack of boundaries, the figure will keep on growing endlessly. Meanwhile, in a finite plane and in a torus, the cells with state 0 will eventually reach the boundaries and the figure will no longer be able to spread itself. For the finite plane, it means that the configuration converges to (^{0 1}_{1 0},^{1 0}_{0 1}). For the torus, however, the configuration can either converge to (^{0 1}_{1 0},^{1 0}_{0 1})or to

0 0

0 0, depending on the size of its sides. Examples of both environments are presented here:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1

→

1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1

1 1 1 1 1 1 1 1 1

↔

1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1

1 1 1 1 1 1 1 1 1 9 × 9-plane

Monomial Cellular Automata: A number theoretical study on two-dimensional cellular automata in the von Neumann neighbourhood over commutative semigroups

Degree project