Cellular Automata

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Cellular Automata

This is week 7 of Biologically Inspired
Computing Various credits for these slides, which
have in part been adapted from slides by Ajit
Narayanan, Rod Hunt, Marek Kopicki.
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Cellular Automata

• 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
• So, what are they???

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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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SIMPLE EXAMPLE
Suppose we are interested in understanding how a
forest fire spreads. We can do this with a CA as
follows.
Start by defining a 2D grid of cells, e.g.
This will be a spatial representation of our
forest.
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SIMPLE EXAMPLE continued
Now we define a suitable set of states. In this
case, it makes sense for a cell to be either
empty, tree, or burning_tree meaning
empty no tree here tree there is a tree
here, and its healthy burning_tree there is
a tree here, and its on fire. When we visualise
the CA, we will use colours to represent the
states. In these cases white, green and red seem
the right Choices.
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E.g. here ,might be an initial configuration,
where the density of trees represents a
particular forest we are interested in, or it may
be based directly on an aerial view, and we set a
randomly chosen tree to be burning.
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SIMPLE EXAMPLE continued
Next we define the neighbourhood structure when
we run our CA, cells will change their state
under the influence of their neighbours, so we
have to define what counts as a neighbour.
Youll see example neighbourhoods in a later
slide, but usually you just use a cells 9
immediately surrounding neighbours. Lets do that
in this case. Next we decide what the
neighbourhood will be like at the boundaries of
the grid.
E.g. if this was our grid, the blue are the
neigbours of the green, but what are the
neighbours of the red?
Depending on what makes sense in the application,
it could be this (fixed boundary)
or this (wraparound)
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SIMPLE EXAMPLE continued
Next we define the transition rules here we
will use some simple illustrative rules example
rules for a real forest fire model appear later
in the slides. For now, these are our rules
they are designed (very very roughly) to align
with a timestep being one hour.
R1 If a cell is empty, its state will stay
empty at the next timestep R2 if a cell is
burning, then each of its tree neighbours
becomes burning at the next timestep with
probability 0.2
So, starting with an initially configuration, we
simply use these rules to derive a new state for
every cell at each timestep.
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Lets see that in action. Here is our initial
configuration, at timestep 0.
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Here we are at timestep 1. I have simulated rule
2 with a random number generator somewhere in my
brain.
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Here we are at timestep 2.
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Here we are at timestep 3.
So, what will eventually happen here? What if
the forest were more dense? Can you see a way to
use CAs for planning new forest planting?
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How a CA operates the algorithm
When a CA is set up, we decide beforehand,
according to the application, what the
neighbourhood is, what the wraparound situation
is, what the grid dimensions are, and so on.
Naturally, this includes deciding what states a
cell can be in, what the transition rules are,
and also we need an initial state for each cell.
This is often randomised appropriately according
to background knowledge (e.g. 40 trees, 60 no
trees, then choose a single random tree state
and set it to on_fire).
SYNCHRONOUS CA then operates like this
repeat until finished 1. for each
cell apply the rules to determine what the
cells new state will be at
the next timestep. 2. Update all
cells to be in their new states
An ASYNCHRONOUS CA operates like this
repeat until finished 1. Choose a
cell at random. 2. Apply the rules
for that cell, and then update
it to be in its new state.
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Synchronous CAs
Synchronous CAs can have very interesting
properties, depending on the rules. Conways
Game of Life is a famed very simple 2-state
synchronous CA, which was formative in the
establishment of a new science called Artificial
Life. See the transition rules later, and the
Glider slide (see the animation of that in
Slide Show mode) From remarkably simple rules,
and a very simple computational system, strange,
unpredictable behaviour emerges (it is
unpredictable in the sense that no theory exists
which can relate the observed CA behaviour to
the rules). In particular, with states and rules
designed appropriately, structures which
self-replicate can emerge see the Langtons
loops slides.
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Asynchronous CAs
BOTH synchronous and asynchronous CAs are able to
model natural phenomena the HIV CA model that
comes next is implemented as a synchronous CA
however asynchronous CAs would seem to be a more
realistic way to model certain natural systems.
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Notes on the algorithm
What does apply the rules mean? In the
CA-operation algorithm?
Usually it is straightforward e.g. suppose we
have only these two rules - if the cell is
currently in state 1, and gt5 neighbours are in
state 0, then the cells new state is state
0 - in all other cases, the cells new state
is 1 Then there is never any doubt what a cells
next state will be, whatever the current state
and the states of the neighbours. But, sometimes
the rules might be under-specific, and not
actually provide for a certain case. Or, they may
be in conflict. This is simply something to
watch out for if you design a CA for a specific
purpose, then make sure its operation is fully
specified.
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An example a cellular automaton that models the
HIV infection process in humans
A fairly recent research paper (dos Santos et al)
described a simple CA model of how HIV infection
develops into AIDS. (find it on google
scholar) There are many alternative and
so-called classical techniques that try to
model the same process usually this involves
differential equations. However, this CA model
seems to capture the quantitative dynamics of the
infection more convincingly than mathematical
models. This is something to do with the fact
that CAs effortless combine space and time
dynamics. Other techniques tend to have
trouble with spatial issues, but clearly CAs are
excellent at modelling processes in which the key
element is neighbour interactions.
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This graph is from real patient data notice the
two very different timescales on the x axis.
Black is HIV-infected cells, white is healthy
cells after initial infection, HIV spreads
rapidly but then subsides rapidly within weeks,
after the immune system launches an initial
assault which kills most of the virus. However,
the virus stays around and gradually wears down
the immune system in a process that can take
several years. At a particular point, the density
of infected cells overtakes that of healthy cells
(here it is at the 8 year point) this is the
onset of AIDS. Mathematical models either get the
first 10 weeks correct and make a mess of the
rest, or vice versa. A properly configured CA,
however, can reproduce both.
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dos Santos HIV CA details
A cell can be in one of 4 states Healthy (H),
Infected1 (I1), Infected2 (I2), Dead (D)
There are just four rules, about what happens to
a cell in the Next timestep.
Rule 1 - If an H cell has at least one I1
neighbour, or if has at least 2 I2 neighbours,
then it becomes I1. Otherwise, it stays
healthy.   Rule 2 An I1 cell becomes I2 after 4
time steps (simulated weeks). (to operate this
the CA maintains a counter associated with each
I1 cell).   Rule 3 - An I2 cell becomes D.   Rule
4 A D cell becomes H, with probability
I1, with probability
otherwise, it
remains D
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Notes on the HIV CA
A 2D grid is used, which seems to need to be at
least 400 by 400 in order to model the dynamics
appropriately. Initially, a certain very small
proportion of cells are randomly set to be
I1. Each rule is designed based on simple but
sensible notions of the biological mechanism.
The probability parameters in the rules are
guessed within reasonable and plausible limits
based on known data. Note the counter in rule 2
operating such a counter goes outside
the pure CA operation algorithms presented
earlier. However, this can be simulated within
the pure algorithm simply by having more states
and new and different rules. (how?). But it is
common practice to simplify things by using such
mechanisms where appropriate. Put another way, a
pure CA is computationally complete it can do
anything but in many modelling tasks it is
simpler to make it impure.
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Now give it a try
A previous student implemented the dos Santos HIV
CA in java Find that linked from my teaching
page compile with javac .java and the run with
java Main Then choose Run, obviously, from the
menu. Blue cells are healthy, yellow are I1 and
red are I2 each update is one week. Notice
craziness (but qualitatively correct dynamics) in
the first Few weeks then, notice the onset of
AIDS at around the 8 year point. If it doesnt
happen, then just try it again. Naturally you
can view the code and see how the rules are
Implemented. (I dont suggest that this is
exemplary java though)
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Here are some well-known 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)
You recall we said that SYNCHRONOUS CAs had
rather interesting properties, and were central
to the research field of Artificial Life
these classes of CAs, defined by Steve Wolfram,
are about such CAs basically, depending on the
transition rules, you tend to get one of these
four types of behaviour. Class 4 is where the
special things are. 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 (animated see this in slideshow mode)
if you happen to have the initial configuration
right, this is what happens
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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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Langtons loops a synchronous CA that leads to
self-replicating structures
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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There remains debate and interest about the
essentials of life issue with CAs, but their
main BIC value is as modelling techniques.
Weve seen HIV here are some more examples.

• 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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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
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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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Forest Fire Model (FFM)
Forest Fire Model is a stochastic 3-state
cellular automaton defined on a d-dimensional
lattice with Ld sites. Each site is occupied by
a tree, a burning tree, or is empty.
During each time step the system is updated
according to the rules

• empty site ? tree with the growth rate
  probability p
• tree ? burning tree with the lightning rate
  probability f, if no nearest neighbour is burning
  
• tree ? burning tree with the probability 1-g, if
  at least one nearest neighbour is burning, where
  g defines immunity.
• burning tree ? empty site

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The application
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Simulation
The average cluster size is small in comparison
to lattice size L.
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Simulation
forest density 60 first signs of fire
Forest density reaches the critical value 59 -
the percolation threshold for square lattice.
The average cluster size goes to infinity for
infinite lattice size.
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Simulation
Fire spreads quickly burning down all connected
tree clusters. A variety of global structures
emerges.
The whole process repeats and after some time
forest reaches the steady state in which the
mean number of growing trees equals the mean
number of burning trees.
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Next time
L systems