Justine Seastres

Saint Mary’s College of CaliforniaDepartment of Mathematics

May 16, 2016

Cellular Automata and the Game ofLife

Supervisors:Professor Porter

Professor Sauerberg

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Contents

1 Introduction 1

2 A Closer Look at Cellular Automata 32.1 Key Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 Local Update Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 One-Dimensional Cellular Automata 53.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Local Transition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Two Dimensional Cellular Automata 6

5 Example: Conway’s Game of Life 75.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.3 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.4 Mathematics of the Game of Life . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.5 Drawing out a Life Board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.6 Pattern Emergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.6.1 Still Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.6.2 Oscillators/Life Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.6.3 Spaceship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.7 Garden of Eden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.7.2 Moore-Myhill Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.8 Applications in the Real World . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.8.1 Physical Reality of the Universe . . . . . . . . . . . . . . . . . . . . . . . 155.8.2 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.8.3 Computers and Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1 Introduction

“Cellular automata may be viewed as computers, in which data represented by initial configura-tions is processed by time evolution”

-Stephen Wolfram

Cellular Automata is an idea formulated in the 1940s by John von Neumann, a Hungarian-American mathematician. He had previously attended the University of Budapest to studychemical engineering, later earning his doctorate in mathematics. Through his interest in bothfields of study, Von Neumann began to develop this idea of self-reproduction. He first exploredthis topic using three dimensional models. His first failed attempt at this concept involved imple-menting replication through robotics. Afterward, studying liquid motion allowed him to further

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observe how different patterns could emerge over certain amounts of time, which influenced VonNeumann even more when studying discrete models. Later on, he was especially interested inbiological phenomenon, and how they could be manipulated and demonstrated through a ma-chine. Over the course of 30 years, he and his partner Stanislaw Ulam researched and developedthis discrete model and new dynamical system.

Their research was primarily focused on the basic ideas of self-reproduction and adaptation,with his work being analogous to the idea of DNA being the instructions for copying preced-ing cells. A main component of Von Neumann’s study was the adaptation of self-reproducingmachines, which have the ability to replicate themselves. When applied to these machines, thisaction can lead to solving self-generated complex and abstract problems.

In particular, there are certain machines which function under conditions set by its neigh-boring environment. We can think of this machine in graphical terms, by considering a modelconsisting of an infinite array of cells in a Euclidean plane, similar to the layout of a checker-board. Each square, or cell, is surrounded by other cells and each has its own previously definedactive or inactive “state”. After some time, this lattice of cells reproduces itself through eachtime step, with the cells either evolving into a new state or maintaining their active status, basedon its surrounding cells. Based off of the size and capability of this computing machine, thesechanges could lead to evolution or self-reproduction. We note that specific cellular automaton isobtained from a cellular space by admitting only a finite, connected set of cells on the checker-board. Therefore, the discrete time steps and evolution of this distinct portion of the grid as awhole is considered local to only those specific cells. More information on the general rules andproperties of neighboring cells will appear later on in this paper.

Therefore, with this knowledge, we can now define a cellular automaton as the collectionof “colored” cells on a grid that evolves through a number of discrete time steps according toa set of rules based on the state of neighboring cells. In other words, it is the coupling of cellsbased on a particular rule set. We refer to the plural of this as cellular automata. With Johnvon Neumann’s revolutionary work towards the automata theory, we are able to gain insight intovarious fields of study, from computational universality, computer processing, cryptography, andeven other sciences related to biology! Von Neumann even created the first virus using cellularautomata, and is now known as the “Father of Computer Virology.” But for now, we will focuson the logistics of cellular automata, and its importance and adaptation towards creating theidea of the Game of Life.

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2 A Closer Look at Cellular Automata

2.1 Key Characteristics

When studying these dynamical systems, we must observe that cellular automata are defined byuniformity, sychronity, and locality.

Definition 2.4.1. For a group of cells to be defined under uniformity , all of the states of thecells are updated by the same rules.

Definition 2.4.2. The synchronity of a cellular automaton is defined as the state of all cellsbeing updated at the same time.

Definition 2.4.3. The locality refers to the set of rules being applied to the specified cells inthe specific, or ”local”, finite lattice of cells.

We can now study the specific properties of cellular automata.

2.2 State

We write S for the finite set of states. A state, denoted as s, where s ∈ S is one of the theone or more finite numbers that determines the cell’s condition. It was mentioned earlier that acellular automaton is a collection of colored cells, which can indicate the state of the cell. Thesimplest, most common case is when there are two different states. There are various ways todefine these states: active/quiescent, alive/dead, on/off, 0/1, black/white.

2.3 Dimension

We can denote a d−dimensional cellular space as Zd, where the cells it consists of are theelements of this space. We observe a d-dimensional figure; we restrict our observation regardingthis area of study to d ≥ 2 dimensions. A one-dimensional figure is presented as a row, whereasa two-dimensional figure gives us a wider range of rows resulting in various shapes, the mostcommon of which is a square.

2.4 Neighborhoods

A neighborhood consists of all the cells surrounding the observed, central cell in Zd. Theneighborhood vector with size k is

N = (n1, n2, ..., nk)

for all nk ∈ Zd. In other words, each element nk identifies the relative locations of theneighboring cells, where n ∈ Zd has k neighbors n+ np for p = 1, 2, ...k.

2.5 Local Update Function

The state of a cell is affected at some discrete time step, t. A cellular automaton evolves throughthis transition function. This local update function governs how each cell changes its statefrom the present state to the next. It can be denoted as

f : Sm → S

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The change of each cell’s state is based on a set of rules that utilizes the present state of thecell and its neighbors. At the end of each time step, we see a resulting pattern arise, known asthe configuration, c. These instantaneous patterns are functions written out as

c : Zd → S

Thus, the state of each cell n ∈ Zd is c(n).

2.6 Conclusion

In the end, to be able to fully define a cellular automaton, one must be able to identify theseproperties:

1. Dimension of d ∈ Z+

2. Finite state set S

3. Neighborhood vector N = (n1, n2, ..., nm)

4. Local Update Function c : Zd → S

Under these properties are we able to contain and observe the behavior of these self-replicatingmachines.

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3 One-Dimensional Cellular Automata

3.1 Definition

A one-dimensional cellular automaton refers to a configuration having d = 1 dimension.[5]Therefore, the cells are displayed side-by-side in a row, with two neighboring cells, and areindexed by Z.

ci−2 ci−1 ci ci+1 ci+2

3.2 Local Transition Function

We now consider a neighborhood surrounding the cell c with a range r that represents the cellsto the left and right of c. Thus, the cell has a neighborhood with 2r + 1 cells.

Let the ith cell’s state at time t be ci(t), so the subsequent time step is denoted as ci(t+ 1).Then, we can define ρ, the local transition function, as a function where the next statedepends on the current state and the state of the two neighbors. It can be denoted as

ρ[ci−1(t), ci(t), ci+1(t)] ≡ ci(t+ 1) mod 2

In other words, we see the mathematical representation of the cells’ state at each time step.It is in mod 2 since there are two different states.

Example 3.1. Let us consider the set of rules in which ci(t + 1) = ci−1(t) + ci(t) + ci+1(t).Thus, by modular arithmetic, we are given

as the states of each cell at each time step, using the local transition function.

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4 Two Dimensional Cellular Automata

Two-dimensional cellular automata have most of the same properties as the one-dimensionalcellular automata. However, instead of being displayed in a row of cells, the cells forming thiscellular automata are indexed by Z2. This is presented to be a larger shape consisting of multiplerows of cells. Many are displayed, however not limited, in a square formation.

There are two types of neighborhoods that are considered with two-dimensional cellular au-tomata. Moore neighborhood consists of all eight cells that surround the central cell.VonNeumann neighborhoods are composed of the four cells which are directly adjacent to the cen-tral cell’s edges. A Moore neighborhood consists of all eight cells that surround the central cell.[5]

Figure 1: Figure 2: Moore Neighborhood(left), von Neumann neighborhood(right)

We can then denote the central cell as (i, j) as a way to identify the location of the the cellsrelative to it.

Figure 2: The coordinates of each cell displayed

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5 Example: Conway’s Game of Life

5.1 History

“The game made Conway instantly famous, but it also opened up a whole new field of mathe-matical research, the field of cellular automata ... Because of Life’s analogies with the rise, falland alterations of a society of living organisms, it belongs to a growing class of what are called“simulation games” (games that resemble real life processes)”

-Martin Gardner

John Conway, an English mathematician, was a prominent figure in studying number theory,coding theory, knot theory, and finite groups. He is most widely known for his work on computersimulations that run by a set of simple rules to exemplify simple cellular life. In the 1970s, Con-way invented this idea of the Game of Life, utilizing Von Neumann’s idea of cellular automata.He wanted to create a method in which one can convert biological reproductions into a game-likesimulation. Through this, he closely studied realistic patterns of life, and intertwined it withthe ideas of the self-replicating machines. In turn, he produced a system which closely imitatedreal-life processes to simulate the rise and fall of a society’s population.

5.2 Description

This game requires no players, implementing the predetermined rules based on its neighbor-ing cells as a means of “advancing” through each time step. The dimensions and states aredetermined using the ideas and structure of the Moore neighborhood, with states of all the sur-rounding eight cells being of interest. We consider this chessboard-like set up of a 3x3 square ofcells, each occupying a state of “alive” or “dead”. These states can be denoted mathematicallyas “1” or “0”, or geometrically on the board as “black” or “white.” We now consider the ruleswhich determine the subsequent state of each cell. [1]

5.3 Rules

Conway realized that the process of maintaining or transforming the numbers in a populationis governed by various key elements: Deaths, Survivals, and Births. These elements are imple-mented into the rules below.

• Survivals: A cell alive at a certain time t will remain alive at the next time step, t+ 1, ifand only if it is surrounded by 2 or 3 living cells. Each of these cells are surrounded by 3cells, and therefore all will remain in their current states.

t=0 t=1

Figure 3: Survival

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• Deaths: Overcrowding occurs when a living cell at time t is surrounded by 4 or moreliving cells at time t+ 1.

t = 0 t = 1 t = 2

Figure 4: Death by overcrowding

We see in the Figure r that there are some cells which do not die according to the idea ofunderpopulation. Instead, there is another phenomena which is occurring called under-population. This occurs when a living cell at time t is surrounded by less than 2 living cellsat time t + 1. As we iterate through each time step, cells continually die until we are leftwith no more at t =3.

t=0 t=1 t=2

Figure 5: Death by overpopulation

• Births: A dead cell at time t can be “revived” if surrounded by exactly 3 living cells.

t=0 t=1

Figure 6: An example of birth, or ’revival’

These rules resemble real-life patterns. Each time there is a state of overpopulation or under-population, then the center cell will “die” or switch to a state of 0. If the center cell is surroundedby 2 or 3 cells, it is in survival mode and stays in a state of “alive”. A “dead” cell can be revivedto an “alive” state when it is surrounded by exactly 3 “alive” cells.

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5.4 Mathematics of the Game of Life

We want to observe the mathematics behind Game of Life. First we must revisit the mathematicsof a general Cellular Automatan[8]. The configuration is denoted as

c : Zd → S

The Game of Life deals with a two-dimensional cellular automata. We can then similarly definea life board set as the set G of all life board functions, g, where g ∈ G such that

G = {g | g : Z× Z→ {1, 0}}

However, to observe the set of functions within a given area, we must look at a finite life boardset, FXY ⊆ G. The finite life board set is denoted as

FXY = {g | g : Z× Z→ {1, 0}}

where

g(x, y) = 0 if (x, y) 6∈ [i, j]× [k, l]

for some closed, bounded integral intervals [i, j] [k, l] particular to g, where X = j−i and Y = l−k

Now, we can define the game of life function as the function φ : G→ G′,

φ(g(i, j)) =

{1 ( if z = 2, 3) OR if (g(i, j) = 0 and z = 3)

0 otherwise

where for each g(i, j)

z =

i+1∑a=i−1

j+1∑b=j−1

g(a, b)

− g(i, j)

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5.5 Drawing out a Life Board

We can draw out a our Game of Life configurations by graphing the coordinates of each stateonto the life board. Let’s generate our own life board g under the these local rules:

φ(g(i, j)) =

{1 (i = −2,−1, or 0) and (j = 1)

0 otherwise

Figure 7: Coordinates Graphed

The states of these cells would be graphed as

Figure 8: Representing the State of Each Cell through Binary States

We can then incorporate this into colored cells on a grid:

Figure 9: Representing the State of Each Cell through Color

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5.6 Pattern Emergence

We can now observe the idea of patterns. But first we must define two key elements in observingpattern emergence. Note that we are still using Von Neumann Neighborhoods.

Definition. A pattern represents the different formations or configurations.

Definition. The period represents the number of repeating formations triggered from the pat-tern through each time step.

Knowing these two terms allows us to observe the discrete nature of these patterns, and howthey evolve at each time step. We can now observe the three most common categories of Gameof Life patterns. Note that we are not limited to these three, however they are the ones whichcome up the most often when visually representing these configurations.

5.6.1 Still Life

• Definition. A still life is a stable, finite, and nonempty pattern. It has a period of 1,since it does not oscillate past one pattern.

• Example: Block Through each time step, the configuration stays the same.

t = 0, 1, 2...∞

5.6.2 Oscillators/Life Forms

• Definition. An oscillator is a pattern which oscillate periodically. It has a period of atleast two, to oscillate between a set of variations of patterns.

• Example: Oscillator

t = 0 t = 1 t = 2 t = 3

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5.6.3 Spaceship

• Definition. A spaceship is a pattern that moves across the game board. It can have aninfinite period, which can end or be terminated according to the finite life board that it isdisplayed on.

• Example: An example of this is the glider pattern. This specific pattern is one whichconsists of 5-cells that repeats itself every four generations. It appears to move across agrid at a right, diagonal, and downward motion, as shown below.

t = 0 t = 1 t = 2 t = 3 t = 4

One of the most common spaceships is called the gosper glider gun. Through each iteration,we see two guns shooting at each other to produce a glider, which moves in a diagonal, downwardmotion. It is of the form below

The discovery of this pattern led to more findings in the relationship between cellular au-tomata and the Turing machine, and how both can be applied to the idea of universal computa-tion.[2]

5.7 Garden of Eden

When observing patterns, it is easier to discover patterns which do no have a predecessor, orthe configuration at the immediate, previous time step. Observe both patterns below at timestep t = 0 and t = 1

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t = 0 t = 1

t = 0 t = 1

We see two patterns (out of other various patterns) which are predecessors of the block con-figuration on the left hand side. Though most patterns commonly have various predecessors, itis interesting to closely study the idea of configurations which do not have any.

John W. Tukey came up with a cellular automaton configuration that can have no priorconfiguration generating it and have no inverse function.[5] In other words, at one time, thereare patterns of cell states that cannot arise in a given cellular space except at time zero. Anothername for this configuration is called an orphan. By connecting the ideas of injectivity and sur-jectivity for the global function of cellular automata in the Euclidean plane, it has been proventhat this configuration exists. There is no past, but only a present and future. Essentially, theseconfigurations can only arise as the initial state, since they have no ancestors.

How did this idea come about? If we consider a finite n number of cells, it is evident thatthere are at most 2n possible outcomes from the transition function. We also note that, from ourobservation above, that having two different configurations with the same predecessor leaves uswith 2n − 1 configurations, where one configuration does not have a predecessor. Both EdwardMoore and John Myhill further studied this phenonemon, but in terms of an infinite number ofcells.

5.7.1 Definitions

Definition 5.7.1. Finite configurations G are ones such that all but a finite number of cellsin the lattice are in quiescent state.

Definition 5.7.2. A distinct f is injective, or one-to-one, if x 6= y, implies f(x) 6= f(y). Inother words, all points x and y must have distinct images f(x) and f(y).

Definition 5.7.3. For a F : G → G, Surjectivity (onto) means that for every g ∈ G, thereexists g′ ∈ G such that f(g) = g′.

5.7.2 Moore-Myhill Theorem

Moore-Myhill Theorem. If C is the collection of all finite configurations and F : C → C isthe global transition function, then F is not injective if and only if it is not surjective.[5]

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Theorem Explained: If F is not injective (with more than one predecessor), then converselyF is not surjective (no predecessor). Therefore, F relates to how a cellular automaton is reversible(or invertible). Being reversible, we are able to go back to a specific predecessor at each time step.

We also note that the Garden of Eden has no pre-images, which means that the set G−1 isempty. To reiterate, if the cellular automata is not surjective, then there are Garden of Edenconfigurations.

This illustration shows the display of one Garden of Eden configuration, discovered by RogerBanks. This configuration is one of a few that have been found to exist. This cannot be derivedfrom another configuration, therefore it is in its initial state.

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5.8 Applications in the Real World

We now come to see the specfic applications of Game of Life and Cellular Automata.

5.8.1 Physical Reality of the Universe

An interesting study is one that refers to the idea of Cellular Automata, particularly the Gameof Life, being a physical representation of the universe. Andrew Ilachinski studies this idea morein depth in his book Cellular Automata: A Discrete Universe [6]. He states that “if withoutin any way disturbing a system, we can know with certainty the value of a physical quantityof that system, then there exists an element of physical reality corresponding to that physicalquantity” (56). With that, the patterns of automata can mimic the behavior of the universe.We can also view the observations that Stephen Wolfram has made in his book A New Kindof Science. There, he connects cellular automata with other fields of sciences, and names themall “rules”. He links mathematics, empirical studies, and engineering together to emphasize theidea of computing. With this computing, he is able to prove in Rule 110 that the idea of TuringComplete, another dynamical system, is the basis for universal cellular automata. Both arethen proven to create this system can compute any mathematical computation. With this ideaof computational universality, we can relate this idea to being able to represent the universe.Albert Eisntein once asked “why is it possible that mathematics, a product of human thoughtthat is independent of experience, fits so excellently the objects of physical reality?” Though notheory has been proven in this area, it is interesting to note the conjectures that many scientistsand mathematicians have made in speculating this seemingly outlandish, yet somewhat tangibleidea.

5.8.2 Biology

The behavior of life or cells can be simulated through cellular automata. These biological modelscan display bacterial growth, seashell patterns, and predator-prey behavior. But one of the mostintriguing displays of cellular automata in biology are the genetic algorithms. We can display thepatterns of natural selection in this way. When observing each chromosome, we want to apply afitness test in order to assess their capabilities. Applying these rules, which are similar to thoseof the Game of Life, will result in a representation of the evolution. This is a perfect exampleof emergent computation, or incorporating natural selection and analyzing pattern structures.Below is an image showing this evolution at various, increasing time steps.

We can also utilize this method to observe water and fish/shark populations.

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And even further, we can also display the time evolution of a single avalanche.

This method of incorporating cellular automata is extremely useful for biologists, and anyother scientists, wanting to visually observe patterns in nature and in life.[5]

5.8.3 Computers and Cryptography

• Computer VirusesJohn von Neumann is known as the “Father of Computer Virology” because of his inno-vations with cellular automata 1949. He created a computer program which implementedthe idea of cellular automata, incorporating patterns which were untraceable. Thus, theidea of a virus was created. [4]

• CryptographyStephen Wolfram also studied the idea of cellular automata being able to calculate random-ness in physical systems.[7] With this, it is a method which can be utilized in cryptography.Utilizing stream ciphers, it is easy to apply the local time function (as introduced in Section3) in order to represent a random sequence. Reiterating

ρ[ci−1(t), ci(t), ci+1(t)] = ci(t+ 1) mod 2

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This represents all iterations of the function. As stated earlier, the behavior of cellularautomata makes it is easier to acquire future iterations as opposed to past iterations, sinceit is possible to have numerous predecessors for each configuration. Because of its nature,this idea has also been a concept in public key cryptography.

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5.9 Bibliography

1. Berlekamp, Elwyn R., John H. Conway, and Richard K. Guy. Winning Ways for YourMathematical Plays. Wellesley, MA: A.K. Peters, 2004. Print.

2. ”Cellular Automaton.” from Wolfram MathWorld. N.p., n.d. Web. 15 Apr. 2016.

3. Codd, E. F. Cellular Automata. New York: Academic, 1968. Print.

4. Kari, Jarkko. ”Properties of Limit Sets of Cellular Automata.” Cellular Automata andCooperative Systems (1993): 311-21.

5. Schiff, Joel L. Cellular Automata: A Discrete View of the World. Hoboken, NJ: Wiley-Interscience, 2008.

6. Ilachinski, Andrew. Cellular Automata: A Discrete Universe. Singapore: World Scientific,2001. Print.

7. Wolfram, Stephen. ”Cryptography with Cellular Automata.” Lecture Notes in ComputerScience Advances in Cryptology — CRYPTO ’85 Proceedings (n.d.): 429-32. Web.

8. J.J.Behrens, The Game of Life, Senior Mathematics Major Essay, St. Mary’s College ofCA, 1999.

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