Transcript Cellular Automata - University of Liverpool

Cellular Automata
COMP308
Unconventional models and paradigms
Failure and Success of Complex Models
New scientific paradigms bring about new
insights:
 Research shows that complexity is a prerequisite for
success of living systems (Jacobs, Langton, Wolfram,
Kaufmann, Fotheringhham, Longley & Batty,
Frankhauser & Sadler)
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Lessons learned:
◦ Complexity  complication
What is a Cellular Automata ?
• Concept introduced by Von Neumann, Ulam and Burk in late 1940-ies
and 1950-ies; (Self-reproducible mechanical automata)
• Conway’s ‘Game of Life’ (Gardner, 1970)
Mathematical object defined as:
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n-dimensional homogeneous and infinite cellular space,
consisting of cells of equal size (2-D CA = torus, not a
plane);
Cells in one of a discrete number of states;
Cells change state as the result of a transition rule;
Transition rule is defined in terms of the states of cells
that are part of a neighbourhood;
Time progresses in discrete steps. All cells change state
simultaneously.
What is a Cellular Automaton?
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A one-dimensional cellular automaton (CA) consists of two
things: a row of "cells" and a set of "rules".
Each of the cells can be in one of several "states". The number
of possible states depends on the automaton. Think of the
states as colors. In a two-state automaton, each of the cells can
be either black or white.
Over time, the cells can change from state to state. The cellular
automaton's rules determine how the states change.
It works like this: When the time comes for the cells to change
state, each cell looks around and gathers information on its
neighbors' states. (Exactly which cells are considered
"neighbors" is also something that depends on the particular
CA.)
Based on its own state, its neighbors' states, and the rules of
the CA, the cell decides what its new state should be. All the
cells change state at the same time.
3 Black = White
 2 Black = Black
 1 Black = Black
 3 White = White
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Now make your
own CA
2-Dimensional Automata
2-dimensional cellular automaton consists
of an infinite (or finite) grid of cells, each
in one of a finite number of states. Time is
discrete and the state of a cell at time t is
a function of the states of its neighbors at
time t-1.
Conway’s Game of Life
The universe of the Game of Life is an infinite twodimensional grid of cells, each of which is either alive or
dead. Cells interact with their eight neighbors.
Example of a Cellular Automata:
Conway’s Life (Gardner, 1970)
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Conway’s Game of Life
At each step in time, the following effects occur:
1.
Any live cell with fewer than two neighbors dies, as if by
loneliness.
2.
Any live cell with more than three neighbors dies, as if by
overcrowding.
3.
Any live cell with two or three neighbors lives, unchanged, to
the next generation.
4.
Any dead cell with exactly three neighbors comes to life.
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The initial pattern constitutes the first generation of the system.
The second generation is created by applying the above rules
simultaneously to every cell in the first generation -- births and
deaths happen simultaneously. The rules continue to be applied
repeatedly to create further generations.
Applications of Cellular Automata
Simulation of Biological Processes
 Simulation of Cancer cells growth
 Predator – Prey Models
 Art
 Simulation of Forest Fires
 Simulations of Social Movement
 …many more.. It’s a very active area of
research.
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Cellular Automata: Life with Simple Rules
Sharks and Fish: Predator/Prey Relationships
Based on the work of Bill Madden, Nancy Ricca and Jonathan Rizzo
(Montclair State University)
Cellular automata can be used to model complex
systems using simple rules.
Key features*
• divide problem space into cells
• each cell can be in one of several finite states
• cells are affected by neighbors according to rules
• all cells are affected simultaneously in a generation
• rules are reapplied over many generations
*Adapted from: Wilkinson,B and M. Allen (1999): Parallel
Programming 2nd Edition, NJ, Pearson Prentice Hall, p189
Main idea
Model predator/prey relationship by CA
 Define set of rules
 Begins with a randomly distributed
population of fish, sharks, and empty cells
in a 1000x2000 cell grid (2 million cells)
 Initially,
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◦ 50% of the cells are occupied by fish
◦ 25% are occupied by sharks
◦ 25% are empty
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Here’s the number 2 million
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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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Initially cells contain fish, sharks or are empty
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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 >= 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)
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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 >=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
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Rules in Detail: 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
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Shark Random Death: Before
I Sure Hope that the
random number
chosen is >.031
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Shark Random Death: After
YES IT IS!!!
I LIVE 
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Use 2-dimensional array to
represent grid
 At any one (x, y) position, value
is:
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Programming
logic
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Positive integer (fish present)
Negative integer (shark present)
Zero (empty cell)
Absolute value of cell is age
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Parallelism
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A single CPU has to do it all:
◦ Applies rules to first cell in array
◦ Repeats rules for each successive cell in array
◦ After 2 millionth cell is processed, array is
updated
◦ One generation has passed
◦ Repeat this process for many generations
◦ Every 100 generations or so, convert array to
red, yellow and black pixels and send results
to screen
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Parallelism
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How to split the work among 20 CPUs
◦ 1 CPU acts as Master (has copy of whole
array)
◦ 18 CPUs act as Slaves (handle parts of the
array)
◦ 1 CPU takes care of screen updates
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Problem: communication issue concerning
cells along array boundaries among slaves
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Send Right Boundary Values
26
Receive Left Boundary Values
27
Send Left Boundary Values
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Receive Right Boundary Values
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Send Right Boundary Values
30
Receive Left Boundary Values
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Send Left Boundary Values
32
Receive Right Boundary Values
33
At intervals, update the master CPU
has copy of entire array
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Illustration
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Next several screens show behavior over
a span of 10,000+ generations (about 25
minutes on a cluster of 20 processors )
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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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Variations of Initial Conditions
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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
1000
4000
8000
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2000
Random placement of very small populations can favor one
species over another
Fish favored: sharks die out
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Sharks favored: sharks predominate,
but fish survive in stable
small numbers
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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Elementary Cellular Automata
As before think of every cell as having a left and right neighbor
and so every cell and its two neighbors will be one of the
following types
Replace a black cell with 1 and a white cell with 0
111
111
110
110
101
101
100
100
011
011
010
010
001
001
000
000
Every yellow cell above can be filled out with a 0 or a 1 giving a
total of 2 8=256 possible update rules.
This allows any string of eight 0s and 1’s to represent a distinct update
rule
01101010
Example: Consider the string
it can be
taken to represent the update rule
111
0
110
1
101
1
100
0
011
1
010
0
001
1
Or equivalently
Now think of 01101010 as the binary expansion of the number
0 27+ 1 26+ 1 25+ 0 24+1 23+ 0 22+ 1 21+ 0 20 = 64+32+8+2=106.
So the update rule is rule # 106
000
0
Rule #45=32+8+4+1
= 0 27+ 0 26+ 1 25+ 0 24+1 23+ 1 22+ 0 21+ 1 20 =0 0 1 0 1 1 0 1
Rule #30=16+8+4+2
= 0 27+ 0 26+ 0 25+ 1 24+1 23+ 1 22+ 1 21+ 0 20 =0 0 0 1 1 1 1 0
This naming convention of the 256 distinct update rules is due to
Stephen Wolfram. He is one of the pioneers of Cellular Automata and
author of the book a New Kind of Science, which argues that discoveries
about cellular automata are not isolated facts but have significance for all
disciplines of science.
Let’s visualize some of these rules using a Mathematica notebook
Automata generated using Rule 30
appear in nature, on some shells.
current
pattern
111
new state
for center 0
cell
110
101
100
011
010
001
000
0
0
1
1
1
1
0
Rule 184
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Rule 184 is a one-dimensional binary cellular automaton rule,
notable for solving the majority problem as well as for its ability to
simultaneously describe several, seemingly quite different, particle
systems:
Rule 184 can be used as a simple model for traffic flow in a single
lane of a highway, and forms the basis for many cellular automaton
models of traffic flow with greater sophistication. In this model,
particles (representing vehicles) move in a single direction, stopping
and starting depending on the cars in front of them. The number of
particles remains unchanged throughout the simulation. Because of
this application, Rule 184 is sometimes called the "traffic rule.”
Rule 184 also models a form of deposition of particles onto an
irregular surface, in which each local minimum of the surface is
filled with a particle in each step. At each step of the simulation, the
number of particles increases. Once placed, a particle never moves.
Rule 184
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In each step of its evolution, the Rule 184
automaton applies the following rule to
determine the new state of each cell, in a
one-dimensional array of cells:
current
pattern
111
110
101
100
011
010
001
000
new state
1
0
1
1
1
0
0
0
The rule set for Rule 184 may also be described intuitively, in several ways:
At each step, whenever there exists in the current state a 1 immediately
followed by a 0, these two symbols swap places.
At each step, if a cell with value 1 has a cell with value 0 immediately to its
right, the 1 moves rightwards leaving a 0 behind. A 1 with another 1 to its
right remains in place, while a 0 that does not have a 1 to its left stays a 0
(traffic flow modeling)
If a cell has state 0, its new state is taken from the cell to its left. Otherwise,
its new state is taken from the cell to its right.
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Rule 184, run for 128
steps from random
configurations with
each of three different
starting densities: top
25%, middle 50%,
bottom 75%.
The view shown is a
300-pixel crop from a
wider simulation.
Rule 184 dynamics and
majority problem
111
110
101
100
011
010
001
000
1
0
1
1
1
0
0
0
Two important properties of its dynamics may
immediately be seen:
 First, in Rule 184, for any finite set of cells with
periodic boundary conditions the number of 1s and
the number of 0s in a pattern remains invariant
throughout the pattern's evolution.
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◦ Similarly, if the density of 1s is well-defined for an
infinite array of cells, it remains invariant as the
automaton carries out its steps.
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And second, although Rule 184 is not symmetric
under left-right reversal, it does have a different
symmetry: reversing left and right and at the same
time swapping the roles of the 0 and 1 symbols
produces a cellular automaton with the same
update rule.
Patterns in Rule 184
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Patterns in Rule 184 typically quickly stabilize, either to a pattern in
which the cell states move in lockstep one position leftwards at
each step, or to a pattern that moves one position rightwards at
each step.
Specifically, if the initial density of cells with state 1 is less than 50%,
the pattern stabilizes into clusters of cells in state 1, spaced two
units apart, with the clusters separated by blocks of cells in state 0.
Patterns of this type move rightwards.
If, on the other hand, the initial density is greater than 50%, the
pattern stabilizes into clusters of cells in state 0, spaced two units
apart, with the clusters separated by blocks of cells in state 1, and
patterns of this type move leftwards.
If the density is exactly 50%, the initial pattern stabilizes (more
slowly) to a pattern that can equivalently be viewed as moving
either leftwards or rightwards at each step: an alternating sequence
of 0s and 1s.
Majority Problem
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One can view Rule 184 as solving the majority problem, of
constructing a cellular automaton that can determine whether an
initial configuration has a majority of its cells active:
if Rule 184 is run on a finite set of cells with periodic boundary
conditions, and the number of active cells is less than half of all
cells, then
◦ each cell will eventually see two consecutive zero states infinitely often,
and two consecutive one states only finitely often,
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while if the number of active cells forms a majority of the cells then
◦ each cell will eventually see two consecutive ones infinitely often and
two consecutive zeros only finitely often.
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The majority problem cannot be solved perfectly if it is required
that all cells eventually stabilize to the majority state but the Rule
184 solution avoids this impossibility result by relaxing the criterion
by which the automaton recognizes a majority.
Traffic flow
Rule 184 interpreted as a simulation of
traffic flow. Each 1 cell corresponds to a
vehicle, and each vehicle moves
forwards only if it has open space in
front of it.
Although very primitive, the Rule 184 model of traffic flow already predicts
some of the familiar emergent features of real traffic: clusters of freely moving
cars separated by stretches of open road when traffic is light, and waves of
stop-and-go traffic when it is heavy.
Surface deposition
111 110 101 100 011 010 001 000
184
1
0
1
1
1
0
0
0
 Rule 184 as a model of surface deposition. In a layer of
particles forming a diagonally-oriented square lattice, new
particles stick in each time step to the local minima of the
surface.
 We model a segment with slope +1 by an automaton cell with state
0, and a segment with slope -1 by an automaton cell with state 1.
The local minima of the surface are the points where a segment of
slope -1 lies to the left of a segment of slope +1; that is, in the
automaton, a position where a cell with state 1 lies to the left of a cell
with state 0.
 Adding a particle to that position corresponds to changing the states
of these two adjacent cells from 1,0 to 0,1, which is exactly the
behavior of Rule 184.
Computer processors
CA processors are a physical, not software only,
implementation of CA concepts, which can process
information computationally.
 Processing elements are arranged in a regular grid of
identical cells. The grid is usually a square tiling, or
tessellation, of two or three dimensions; other tilings are
possible, but not yet used.
 Cell states are determined only by interactions with the
small number of adjoining cells. Cells interact, communicate,
directly only with adjoining, adjacent, neighbor cells. No
means exists to communicate directly with cells farther away.
 Cell interaction can be via electric charge, magnetism,
vibration (phonons at quantum scales), or any other
physically useful means. This can be done in several ways so
no wires are needed between any elements.
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