one dimensional cellular...

ONE DIMENSIONAL CELLULAR

AUTOMATA(CA).

bertrand LUT, 21May2010

By Bertrand Rurangwa

Cellula automata(CA)

OUTLINE

- Introduction.

-Short history.

-Complex system.

-Why to study CA.

-One dimensional CA.

bertrand LUT, 14Mary2010

Complex Systems

- From the turbulence in fluids, to global

weather patterns, to beautifully intricate

galactic structures, to the complexity of living

organisms.

Historical examples of ornamental art.

bertrand LUT, 14Mary2010

Five generic characteristics(CA) :

• Discrete lattice of cells: the system substrate

consists of a one, two or three-dimensional

lattice of cells.

• Homogeneity: all cells are equivalent.

• Discrete states: each cell takes on one of a

finite number of possible discrete states.

• Local interactions: each cell interacts only

with cells that are in its local neighborhood.

• Discrete dynamics: at each discrete unit time,

each cell updates its current state according to

a transition rule taking into account the states

of cells in its neighborhood.

Why Study CA?

Four partially overlapping motivations for

studying CA :

• As powerful computation engines.

• As discrete dynamical system simulators.

• As conceptual vehicles for studying pattern

formation and complexity.

• As original models of fundamental.

As powerful computation engines.

- С A allow very efficient parallel computational

implementations to be made of lattice models

in physics and thus for a detailed analysis of

many concurrent dynamical processes in

nature.

As discrete dynamical system

simulators

- CA allow systematic investigation of complex

phenomena by embodying any number of

desirable physical properties. CA can be used

as laboratories for studying the relationship

between microscopic rules and macroscopic

behavior- exact computability ensuring that the

memory of the initial state is retained exactly

for arbitrarily long periods of time.

As conceptual vehicles for studying

pattern formation and complexity

- CA can be treated as abstract discrete

dynamical systems embodying intrinsically

interesting, and potentially novel, behavioral

features.

As original models of fundamental

- CA allow studies of radically new discrete

dynamical approaches to microscopic physics,

exploring the possibility that nature locally and

digitally processes its own future states.

One-dimensional cellular automata

- One-dimensional cellular automata consist

of a number of uniform cells arranged like

beads on a string. If not stated otherwise

arrays with finite number of cells and

periodic boundary conditions will be

investigated, i.e. the beads form a

necklace.

-The state of cell i at time t is referred to as . The finite number of possible states are

labelled by non-negative integers from 0 to

k -1.

The state of each cell develops in time by iteration of the map

F is called the automata rule.

( ) ( 1) ( 1) ( 1) ( 1)

( ) ( 1) ( ) ( ), ,... ,...t t t t t

i i r i r i i ra F a a a a

( )t

i ka

The state of the ith cell at the new time

level t depends only on the state of the ith

cell and the r (range) neighbors to the left

and right at the previous time level t- 1.

( ) ( 1)

( )

j rt t

i j i j

j r

a f a

where the are integer constants and thus f the function has a single integer as argument.

Number of automata rulesConsider a CA with K possible states

per cell and a range r the different

combinations are .

j

2 1rK

Cellular automata as a discretization of

partial differential equations

Lattice-gas cellular automata - a special

type of cellular automata are relatively new

numerical schemes to solve physical

problems ruled by partial differential

equations.

2

2

C Ck

t x

The discretization forward in time and

symmetric in space reads

( ) ( 1) ( 1) ( 1) ( 1)

1 12

.2

( )

t t t t t

i i i i i

t kC C C C C

x

1( 1)

1

jt

j i j

j

C

1( 1)

1

jt

j i j

j

f C

Fundamental differences:

-The coefficients in general are real

numbers and not integers.

-The number of states of is infinite.

j

jC

Footer

2

2

1

4

C C

t x

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