Cellular Automata

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Cellular Automata

Various credits for these slides Ajit
Narayanan, Rod Hunt, several more.
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Cellular Automata

• A 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 t1 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. Hence, k and r are the parameters
  of the CA.
• CAs are often described as discrete dynamical
  systems with the capability to model various
  kinds of natural discrete or continuous dynamical
  systems

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Example of 1-D cellular automaton

• For a binary input N long, are there more 1s than
  0s?
• Set k2 and r1 with the following rule

000 001 010 011 100 101 110 111
0 0 0 1 0 1 1 1
Cell 2 neighbours
Result
That is, the value of a cell at time t1 will
depend on its value and the values of its two
immediate neighbours at time t. This is a form of
majority voting between all three cells.
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Density classification

• In the above example, we have assumed
  wrap-around, and r1.
• In this case, the CA has reached a limit point
  from which no escape is possible.
• CAs have been used for simulating fluid dynamics,
  chemical oscillations, crystal growth, galaxy
  formation, stellar accretion disks, fractal
  patterns on mollusc shells, parallel formal
  language recognition, plant growth, traffic flow,
  urban segregation, image processing tasks, etc

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Types of neighbourhood
Many more neighbourhood techniques exist - see
http//cell-auto.com and follow the link to
neighbourhood survey
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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)
probably equivalent to a regular grammar/finite
state automaton 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
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John Conways Game of Life

• 2D cellular automata system.
• Each cell has 8 neighbors - 4 adjacent
  orthogonally, 4 adjacent diagonally. This is
  called the Moore Neighborhood.

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Simple rules, executed at each time step

• A live cell with 2 or 3 live neighbors survives
  to the next round.
• A live cell with 4 or more neighbors dies of
  overpopulation.
• A live cell with 1 or 0 neighbors dies of
  isolation.
• An empty cell with exactly 3 neighbors becomes a
  live cell in the next round.

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Is it alive?

• http//www.bitstorm.org/gameoflife/
• Compare it to the definitions

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Glider
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Sequences
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More
Sequence leading to Blinkers Clock Barbers pole
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A Glider Gun
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Loops

• Assumptions
• Computation universality not required
• Characteristics
• 8 states, 2D Cellular automata
• Needed CA grid of 100 cells
• Self Reproduction into identical copy
• Input tape with data and instructions
• Concept of Death
• Significance Could be modeled through computer
  programs

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Langtons Loop
0 Background cell state 3, 5, 6 Phases of
reproduction 1 Core cell state 4 Turning arm
left by 90 degrees 2 Sheath cell state
state 7 Arm extending forward cell state
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Loop Reproduction
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Loop Death
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Langtons Loops
Chris Langton formulated a much simpler form of
self-rep structure - Langton's loops - with only
a few different states, and only small starting
structures.
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Example

• Modelling Sharks and Fish
• Predator/Prey Relationships
• Bill Madden, Nancy Ricca and Jonathan Rizzo
• Graduate Students, Computer Science Department
• Research Project using Departments 20-CPU
  Cluster

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• This project modeled a predator/prey relationship
• Begins with a randomly distributed population of
  fish, sharks, and empty cells in a 1000x2000 cell
  grid (2 million cells)
• Initially,
• 50 of the cells are occupied by fish
• 25 are occupied by sharks
• 25 are empty

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Heres the number 2 million

• Fish red sharks yellow empty black

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Rules

• A dozen or so rules describe life in each cell
• birth, longevity and death of a fish or shark
• breeding of fish and sharks
• over- and under-population
• fish/shark interaction
• Important what happens in each cell is
  determined only by rules that apply locally, yet
  which often yield long-term large-scale patterns.

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Do a LOT of computation!

• Apply a dozen rules to each cell
• Do this for 2 million cells in the grid
• Do this for 20,000 generations
• Well over a trillion calculations per run!
• Do this as quickly as you can

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Do a LOT of computation!

• We used a 20-CPU cluster in the Computer Science
  Department (Galaxy)
• Gal is the smaller of two clusters run by the
  Department (larger one has 64 CPUs)
• 15x faster than a single PC
• Longest runs still took about 45 minutes
• GO PARALLEL !!!

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Rules in detail Initial Conditions

• Initially cells contain fish, sharks or are empty
• Empty cells 0 (black pixel)
• Fish 1 (red pixel)
• Sharks 1 (yellow pixel)

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Rules in detail Breeding Rule

• Breeding rule if the current cell is empty
• If there are gt 4 neighbors of one species, and
  gt 3 of them are of breeding age,
• Fish breeding age gt 2,
• Shark breeding age gt3,
• and there are lt4 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)

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Breeding Rule Before

EMPTY

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Breeding Rule After



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Rules in Detail Fish Rules

• If the current cell contains a fish
• Fish live for 10 generations
• If gt5 neighbors are sharks, fish dies (shark
  food)
• If all 8 neighbors are fish, fish dies
  (overpopulation)
• If a fish does not die, increment age

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Rules in Detail Shark Rules

• If the current cell contains a shark
• Sharks live for 20 generations
• If gt6 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

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Shark Random Death Before
I Sure Hope that the random number chosen is
gt.031



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Shark Random Death After
YES IT IS!!! I LIVE ?



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Sample Code (C) Breeding
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Results

• Next several screens show behavior over a span of
  10,000 generations

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Generation 0
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Generation 100
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Generation 500
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Generation 1,000
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Generation 2,000
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Generation 4,000
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Generation 8,000
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Generation 10,500
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Long-term trends

• Borders tended to harden along vertical,
  horizontal and diagonal lines
• Borders of empty cells form between like species
• Clumps of fish tend to coalesce and form convex
  shapes or communities

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Variations of Initial Conditions

• Still using randomly distributed populations
• Medium-sized population. Fish/sharks
  occupy 1/16th of total grid Fish 62,703
  Sharks 31,301
• Very small population. Fish/sharks
  occupy 1/800th of total grid Initial
  population Fish 1,298 Sharks 609

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Medium-sized population (1/16 of grid)
Generation 100
2000
1000
4000
8000
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Very Small Populations

• Random placement of very small populations can
  favor one species over another
• Fish favored sharks die out
• Sharks favored sharks predominate, but fish
  survive in stable small numbers

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Very Small Populations
Gen. 100
4000
6000
1500
8000
10,000
12,000
14,000
Ultimate welfare of sharks depends on initial
random placement of fish and sharks
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Very small populations

• Fish can live in stable isolated communities as
  small as 20-30
• A community of less than 200 sharks tends not to
  be viable

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Community image
This is what a community of virtual plants looks
like
Contrasting tones show patches of resource
depletion
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CSR type, frame 1
This is a single propagule of a virtual plant
It is about to grow in a resource-rich above- and
below-ground environment
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ditto f. 2
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ditto f. 3
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ditto f. 4
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ditto f. 5
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ditto f. 6
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ditto f. 7
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ditto f. 8
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ditto f. 9
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ditto f. 10
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ditto f. 11
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ditto f. 12
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ditto f. 13
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ditto f. 14
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ditto f. 15
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ditto f. 16
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ditto f. 17
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ditto f. 18
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ditto f. 19
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ditto f. 20
The plant has produced abundant growth above- and
below-ground
and zones of resource depletion have appeared
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See Rod Hunt at http//www.ex.ac.uk/rh203/for
lots more about the plant life CA and its uses