Cellular Automata

Introduction

Cellular Automata originally devised in the late 1940s by Stan Ulam (a mathematician) and John von Neumann.

Originally devised as a method of representing a stylized universe, with rules (e.g. laws of thermodynamics) acting over the entire universe.

Have subsequently been used for a wide variety of purposes in simulating systems from chemistry and physics

CAs have started to be used in bioinformatics

Some Facts

Cellular Automata but … one Automaton !! Cellular Automata can be defined in multiple

dimensions: one, two, three… or more

Cellular Automata have applications in many scientific domains:

1. Physics2. Material Science3. Biology4. Epidemiology5. etc

Cellular Automata

CA is a dynamical system in which

space and time are discrete and the

space of Cellular automata is changed

with time

Cellular Automata

CA is a spatial lattice of N cells, each of which is one of k states at time t.

•Each cell follows the same simple rule for updating its state.

•The cell's state s at time t+1 depends on its own state and the states of some number of neighbouring cells at t.

• For one-dimensional CAs, the neighbourhood of a cell consists of the cell itself and r neighbours on either side.

Cellular Automata An automaton consists of a

grid/lattice of cells each of which can be in a number of states

The figure shows a 5x5 automaton where each cell can be in a filled or empty state.

Cell

State = empty/off/0

State = filled/on/1

1-D Cellular Automata

It is a string of cells (one dimensional array of

cells). Each cell in one state of State Set and

change its state with time depend on the

neighbor cells.

1D Cellular Automata

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

T=0

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

T=1

Application

1-D cellular Automata example is the Traffic

road (system)

Let us consider the cell with state 0 is empty

of car and cell with state 1 has a car

Traffic Road

1 1 1 1 1 1

T=0

1 1 1 1 1 1

T=1

Traffic road

The idea is to consider a set of adjacent cells

representing a street along which a car can move.

The car jumps to its nearest neighbor cell unless

this cell is already occupied by another car.

Traffic System

The rule of motion can be expressed by

X (t+1) = Xin (t) (1-X(t)) + Xout(t) X(t)

X is the cell, Xin is the cell from which the car

come, Xout the destination cell.

Traffic Simulation

Decelerate, if tailing distance to the next car is less than strength of pheromone suggests

Accelerate, if there is no pheromone or tailing distance is greater than suggested by pheromone strength

Traffic Simulation

Driving, changing lanes, stopping

State Set

Each cell of the cellular automata is a

finite automaton and then it has a

state, the set of all possible states is

called state set

State Transition Rules The states of an automaton change over time

in discrete timesteps The state of each cell is modified in parallel at

each timestep according to the state transition rules

These determine the new states of each of the cells in the next timestep from the states of that cells neighbours

For (int i=0 to CellCount){

Cell[i].State[t+1] = STR(Cell[i].Neighbour.State[t]

}

Boundary conditionsthere are two kinds of boundaries; periodic

and fixed value boundaries.

1. Periodic boundary is obtained by periodically extending the array or lattice

2. Fixed value boundary is obtained by simply prescribing a fixed value for the cells on the boundary

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

Neighborhood size

The set of cells that neighbor a given cell X

in traffic example, the neighborhood size is 3 ( cell before and

cell after plus the cell X itself)

1 0 1

2-D Cellular Automata

Is an array nxn of cells. Each cell has a different state from state set.

1 0 1 0 1

1 1 0 1 1

1 0 1 1 0

0 1 1 1 0

0 1 0 0 1

2 D Cellular Automata

The neighborhood size of 2D cellular Automata is 2 types: Von Neuman and Moore

R=1 R=2Von Neuman

2 D Cellular Automata

The neighborhood size of 2D cellular Automata is 2 types: Von Neuman and Moore

R=1 R=2Moore

2-D Traffic System

Using 2 D cellular Automata we can construct road networks with crossing where cars can move

Game of life

Consider 2D cellular automata with neighbors size 8 and states 0 and 1

The transition rule is:– If cell has 3 of its neighbors live, then it

live next time– Live cell and has 2 neighbors live, it will

be live next time– Otherwise, it will die next time step

Game of life

Game of life

Game of life

Game of life

Game of life

Is it alive?

Sequences

Loop ReproductionLoop Reproduction

Loop DeathLoop Death

Classes of cellular automata (Wolfram)

Class 1: after a finite number of time steps, the CA tends to achieve a unique state from nearly all possible starting conditions (limit points)

Class 2: the CA creates patterns that repeat periodically or are stable (limit cycles)

Class 3: from nearly all starting conditions, the CA leads to aperiodic-chaotic patterns, where the statistical properties of these patterns are almost identical (after a sufficient period of time) to the starting patterns (self-similar fractal curves) – computes ‘irregular problems’

Class 4: after a finite number of steps, the CA usually dies, but there are a few stable (periodic) patterns possible (e.g. Game of Life) - Class 4 CA are believed to be capable of universal computation

The 2 million cell

Fish: red; sharks: yellow; empty: black

Initial Conditions

Initially cells contain fish, sharks or are empty

Empty cells = 0 (black pixel) Fish = 1 (red pixel) Sharks = –1 (yellow pixel)

Breeding Rule

Breeding rule: if the current cell is empty If there are >= 4 neighbors of one species,

and >= 3 of them are of breeding age,• Fish breeding age >= 2,• Shark breeding age >=3,

and there are <4 of the other species:

then create a species of that type • +1= baby fish (age = 1 at birth)• -1 = baby shark (age = |-1| at birth)

Breeding Rule: Before

EMPTY

Breeding Rule: After

Fish Rules

If the current cell contains a fish: Fish live for 10 generations If >=5 neighbors are sharks, fish dies

(shark food) If all 8 neighbors are fish, fish dies

(overpopulation) If a fish does not die, increment age

Shark Rules

If the current cell contains a shark: Sharks live for 20 generations If >=6 neighbors are sharks and fish

neighbors =0, the shark dies (starvation) A shark has a 1/32 (.031) chance of

dying due to random causes If a shark does not die, increment age

Shark Random Death: Before

I Sure Hope that the random number chosen is >.031

Shark Random Death: After

YES IT IS!!! I LIVE

Generation: 0

Generation: 100

Generation: 500

Generation: 1,000

Generation: 2,000

Generation: 4,000

Generation: 8,000

Generation: 10,500

Application Guidelines

To apply a cellular automaton to a problem:– A representation of a cell must be determined

– Cell states must be defined

– A grid of cells must be constructed

– A set of rules must be created to modify states

– A neighbourhood should be defined

Reaction/Diffusion with Cellular Automata

CA Methods in Games

SimCity 2000

The SIMS