CELLULAR AUTOMATA WITH CHAOTIC RULE FORMATION AND ITS CHARACTERISTICS

Christopher Immanuel W* 1 Paul Mary Deborrah S2 Samuel Selvaraj R3

*Research and Development Centre, Bharathiar University, Coimbatore – 641 046.

1Department of Physics, Vel Tech High Tech Engineering College, Chennai, India.

2Department of Physics , TheAmerican College, Madurai, India.

3Department of Physics , Presidency College, Chennai, India.

Christ.phy@gmail.com

+91 09943849838

Abstract

The fractal procedure known as cellular automata are applied to the different field.Cellular

Automata are not only of theoretical interest. One of the main topics in CA research is their

usage as models of phenomena which can be observed in real life, the practical applications of

Cellular Automata can be seen in multiple fields of science and social science. They range from

computer science, technology, physics, biology, and math, to economics, psychology,

philosophy, and even art. Cellular Automata are discrete dynamical systems, of simple

construction but complex and varied behavior.Despite of the simplicity of their construction;

Cellular Automata are capable of very complicated behavior with very little specification. The

present paper introducesa cellular automata and their applicationwith some rules. It is very

useful for future references.

Keywords: Cellular Automaton, Fractal Dimension, Chaotic Rule.

1. Introduction

This paper should give a short overview of what cellular automata are, how they work and what

they can be used for.

The next section starts with the history, how cellular automata were invented and briefly discuss

some of their properties.

Cellular automata are a relatively new development in modern science. It consists of simple

progressions from algorithms or Rules over time.Cellular automata (CA) are mathematical

models for complex natural systems containing large numbers of simple identical

componentswith local interactions. Cellular automata rules can be classified automatically for a

spectrum of ordered, complex and chaotic dynamics, by a measure of the variance of input

entropy overtime. They are discrete dynamical systems exhibiting a wide range of complex

behavior;from on the one hand, point and cyclic behavior yielding structures of relative

periodicity and, on the other hand, utterly chaotic attractors. Chaos and complexity theory

teaches us that simple rules can lead to enormous levels of complexity.

Cellular Automata or Cellular Spaces as they were called then were invented by John von

Neumann half a century ago.

2. The basic idea

What are cellular automata? The word “cellular” here means “consisting of cells” – a cellular

automaton is made up of cells. Each of those cells contains an “automaton”, a finite state

machine. The resulting construct is a space filled with cells, each of those cells containing a

finite state machine. We say “a cell has a state”, meaning the state, the cells finite state machine

is in. Usually the space is divided into a grid or lattice structure. In the cellular space, all the cells

go from their current state to some next state according to the “local rule”, the switching rule of

their state machine.

The input for the local rule is the neighborhood of the cell, which is usually a small number of

cells surrounding the cell. All the cells use the same local rule.The entire cells switch from their

current state to the next state at the same time, the space is synchronous.

A Cellular Automata consists of processors (cells), connected usually in an n-dimensional grid.

Fig.1 One Dimensional CA

Fig.2 Two Dimensional CA

Rules are formed through a definition of the 23= 8 possible progressions of three cells (the cell,

the cells left-hand neighbor, and the cells right-hand neighbor). Each of these progressions gives

a single output, producing a new cell and creating a three to one mapping.

A cell dies or lives according to some transition rule

Fig.3Cell transition from T=0 to T=1

3. Some terms

The following terms will be used in the remainder of this paper.

Cell A single element of a cellular space, the smallest unit of the space.

Cellular Space A lattice space made up of cells, each of which is in one of several predefined

states. This is what most people nowadays call a Cellular Automaton, but I find the distinction

useful.

Cellular Automaton A structure built in a cellular space, an automaton built out of cells. The

term is often used as a synonym for Cellular Space, but to be clearer, I’ll try not use it that way.

“CA” is an abbreviation that is often used for these terms.

Cellular Automata Plural of Cellular Automaton, sometimes used instead of the singular form.

Local Rule The rule governing the transition between states. The definition of a cell’s finite state

machine. It’s called “local”, because it only uses the Neighborhood as its input.

Neighborhood The cells surrounding a cell that influencesits next state. The choice of

neighborhood influences the behavior of the cellular space.

Configuration A snapshot of all cell states, representing a single point in time. When we talk

about a configuration, it’s usually the starting point or a result of running a cellular space.

GenerationOne step in the evolution of a cellular space, an intermediate configuration. Passage

of time in a cellular space is measured in generations.

4. History of CA

In the 1950s, John von Neumann tried to construct self-replicating machines. He wanted to find

the simplest machines that could build copies of themself, but to be useful, they also had to be

universal computers. According to [John von Neumann 1951] started out working with

differential equations until his colleague Stanislaw Ulam suggested a different model, similar to

the lattice spaces Ulam used to study crystals, which lead to a significant simplification.

4.1. Von Neumann’s automata

Von Neumann realized that a 2-dimensional model would be sufficient. In his 2-dimensional

model, a cellular space was formed of cells that could be in one of several possible states. Each

cell would change its state according to a local rule that calculated the new state as a function of

the four nearest neighboring cells (the orthogonal neighbors, this type of neighborhood is now

called a “von Neumann” neighborhood). Von Neumann called his creation a “cellular space” and

the constructions in that space were the “cellular automata”. The cells in von Neumann’s space

could be in one of 29 states and there was a complicated set of rules for the movement between

these states. For a detailed explanation of these states see e.g. [John von Neumann 1951]. But

self-replication was only half of the problem; there were actually two properties von Neumann

was looking for: universal computation and universal construction. The first was to make sure

that the automata he constructed were not trivial; the second was the basis for self-replication.

Universal computation meant that constructions in the cellular space were capable of computing

any computable function. The simplest way of proving this property was showing that a Turing

machine can be built in the cellular space. Universal construction meant that an automaton could

be built that could create other constructions in the cellular space. A programmable construction

machine was created by coupling a Turing machine with a universal constructor.

Although von Neumann developed the core of his self-replicating machines, he didn’t finish the

work. The complete model was published after his death in 1966 by Arthur W. Burks. Following

this publication, a small number of other works were published, including a monograph by E. F.

Codd.

4.2. Codd’s space

In his Monograph, Codd summarizes his research into the requirements for universal

computation and construction. He shows that there exists no cellular space with 2 states and a

local rule using the von Neumann neighborhood that exhibits universality. He then proceeds to

develop a cellular space that does exhibit universality but has simpler rules than von Neumann’s

space. He shows how both universal computation and universal construction can be realized in

his simpler space. Another first is the fact that he tries out his cellular automata “on-line”, ie. on

a computer terminal as opposed to having simulations run in batch mode. In a section titled

“Why On-Line?” he argues “Rapid interaction between man and machine tends to develop the

intuition and perceptivity of the experimenter. The immediacy of the machine’s response has the

effect of maintaining a high level of cerebral activity in the experimenter.” He may have been the

first to experience the fascination of cellular automata first hand! Codd was not the first to create

a simpler cellular space, though. According to Sarkar, in 1966 Arbib created self-replicating

universal computers in a space with 4 different states and von Neumann neighborhood.

4.3.Konradzuse’s “rechnenderraum”

KonradZuse described how the whole of space might be in reality a cellular space of the kind

von Neumann described. He called this space “Computing Space” (“RechnenderRaum”in

German).

The idea was later picked up by both Edward Fredkin [Hurd 1990] and StephenWolfram.

Compelling as the idea may be, there are no practical applications.

4.4. Conway’s game of life

Short after Codd’s experiments, the British mathematician John Horton Con-way developed

another cellular space that he called “Game of Life”. The Game of Life was a breakthrough

because it is one of the simplest cellular spaces to exhibit universal computation and universal

construction, although that was not known when the Game of Life was first published. Martin

Gardner wrote about the Game of Life in his column in Scientific American and it became very

popular to run simulations of it on the mini computers of the time.

Although Conway constructed the rules for his cellular space in a way that should allow for

complex interaction, the constructions that are possible in this space were discovered after the

rules for the space had been set. In contrast, most of the patterns used by Codd and von Neumann

to construct their automata were developed at the same time as the rules governing their spaces,

the rules are complex and it seems clear that they were crafted specifically to enable certain

constructions. The rules for Conway’s Game of Life are actually very different from the ones

von Neumann and Codd used they are totalistic, which means that only the number of neighbors

that are in a certain state is important, their positions are not taken into account. In Codd’s and

von Neumann’s spaces, most of the rules had directional components only neighbors in certain

positions are considered in each state. This difference means that Conway’s rules are much

simpler than Codd’s and von Neumann’s.

Around this time a trend started to call the cellular spaces “cellular automata”, the former

distinction blurring, possibly because no one thought of the gliders and guns in Conway’s Game

of Life as “automata”, they seemed more like multi celled organisms that sprang to life in this

foreign universe.

4.5. Wolfram’s surveys of cellular space

On a quest to figure out where the complexity in nature comes from, Stephen Wolfram

encountered cellular automata (a colleague told him to have a look and look he did). In contrast

to Codd, who thought that “exhaustive search is out of the question”), he wanted to do an

exhaustive survey of possible cellular automata (or spaces, as Codd would have called them).

But there are too many 2-dimensional cellular spaces – so he simplified the model even

further.Wolfram looked at the simplest possible cellular automata: 1-dimensional automata with

two states and a three cell neighborhood. Since there are eight possible input configurations of

three cells with two possible states, there are 256 possible local rules for these cellular automata.

Of these 256 possible rules, only 32 are legal under the restrictions he used in his initial survey.

As a result of his experiments, he grouped the rules into three classes the first class that evolves

from arbitrary initial conditions into a static pattern, like rules numbers 0 or 4; the second class

that evolves into simple crystalline structures and the third class that evolves into complex self-

similar patterns.

Wolfram refined his classification scheme later into four classes, splitting the class of rules with

complex behavior intocomplex andchaoticrules. The rules with chaotic behavior can be used to

generate randomness, while some of the rules with complex behavior are computation universal.

5. Cellular automata and the mechanisms of nature

Cellular automata are examples of simple programs, that work by having the color of each cell in

successive rows be determined by the same simple rule that certain simple rules like the one on

the right can produce patterns and behavior of immense complexity.

Rule 30 cellular automaton

Rule 30 is a one-dimensional binary cellular automaton.

current pattern 111 110 101 100 011 010 001 000new state for center cell 0 0 0 1 1 1 1 0

Fig.4One-dimensional binary cellular automaton Rule 30

Rule 30 exhibits class 3 behaviors, meaning even simple input patterns such as that shown lead

to chaotic, seemingly random histories.

With the rule above, a simple pattern is produced.

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step 8

Fig.5Successive lines give configurations obtained on successive time steps, with white and black squares representing sites with values 0 and 1 respectively.

The above figure showed patterns produced by evolution according to various Cellular

automaton rules, starting from typical disordered initial conditions, in which the value of

each site is randomly chosen to be zero or one.

Such cellular automata can be used as models of randomness in nature.

some class 3 cellular automata yield rather simple patterns, which are typically periodic or

at least self-similar (almost periodic). There are never the less class 3 cellular automatawhich

yield complex patterns, even from simple initial states. Their evolution can intrinsically produce

apparent randomness, withoutexternal input of random initial conditions.

Many class 3 cellular automata seem to perform very complicated trans-formations on their

initial conditions. Their evolution thus corresponds to a complicated computation. But any

predictions of the cellular automaton behavior must also be obtained through computations. Most

class 3cellular automata are expected to be computationally irreducible. A few rules however

have special simplifying features which make predictions and analysis possible. Most class 3

cellular automata are however nonlinear.

But the two rules that seem best as random sequence generators are nonlinear , and are

given by

sin+1 =si-1

nXOR (sinOR s i+1

n) (1)

or, equivalently,

ai’ = (ai-1 + ai +ai+1 + aiai+1) mod 2 (2)

The cellular automaton rule 30 is essentially nonlinear.

Any class 3 (chaotic) cellular automaton rule can be considered a candidate random

sequence generator.

Here, we shall restrict ourselves to the case of one-dimensional binary CA ofNcells with

neighborhood ofsize equal to 3, defined as elementaryby Wolfram. In these models, Lis a linear

array where each celladopts one of two possible states inS={0,1}and thedynamical rule is the

form sin+1 =f (s i-1

n, sin,s i+1

n) means that,N(i) → sin+1 =f (s i-1

n, sin,s i+1

n) In this case, there

are 256 possible rules f. If {bi}is a binary string, corresponding to the result of applying f to

each different set of 3-tuples (s i-1n, si

n , s i+1n) sorted in descending order,it is possible to use the

Wolfram′s naming scheme to assign, to each rule, a label given by the decimal representation of

the binary sequence. Rules of elementary automata were classified qualitatively by [Wolfram

1984] according to its asymptotic behavioras Class I to IV, if the automata evolution

asymptoticallygoes to: a constant state, an isolated periodic segments,chaotic regime or isolated

chaotic segments, respectively.In contrast with the continuous case, the elements ofthe Class III

can be linear or nonlinear CA. The linear arethe rules that can be written as

In the case of elementary CA this class contains 16 elementsand the rest are nonlinear.

6. Main characteristics of cellular automata

a. Space is discrete, and there is a regular array of sites (cells). Each cell has a finite set of values.

b. Time is discrete, and the value of each site is up-dated in a sequence of discrete time steps.

c. The “Rule” for the new value of a site depends only on the values of a local neighborhood of

sites near it.

d. The variables at each site are up-dated simultaneously (“synchronously”) based on the values

of the variables at the previous time step.

Conclusion

The model studied in this paper was developed in an attempt to increase our understanding of

complex behavior in dynamical systems. It is well known that dynamical systems differ

dramatically in several important aspects of their behavior, such as periodicity, predictability,

stability, dependence on initial conditions, etc. Systems that are “nasty” in these respects are

called “chaotic”. There are many important, and sometimes deep, facts known about chaotic

dynamical systems.

The cellular automata are the new style, high performance simulation tool. We hope researchers

can bring into play well the modeling power of the CA approach in future in variety complex

systems.

Cellular automata have been successfully used to model many variety complex systems, such as

biology, chemistry, mathematics, physics field.

A cellular automaton model allows the formulation of a dynamic complex system application in

simple rules. Based on standard CA, there are of course many corrective and extended

computational models for different applied objectives. According to simple local transition

functions, CA are intuitively regarded as a set of interacting elements are updated during a

discrete time interval.

References

1.John von Neumann,The general and logical theory of automata, Cerebral Mechanisms in Behavior.[ The Hixon Symposium, John Wiley & Sons Inc., New York, N. Y., 3] 1951, pp. 1–31.

2. K. C., Hurd, L. P., Yu, S.: Computation Theoretic Aspects of Cellular Automata, Physica D, 45, 1990,. 357–378.

3.Wolfram, S. "Universality and Complexity in Cellular Automata." Physica D 10, 1-

35,1984. Wolfram