Complex Systems Engineering
SwE 488
Artificial Complex Systems
- Cellular Automata -
Cellular Automata
2
Purpose
• In Theory:
• Computation of all computable functions
• Construction of (also non-homogenous) automata by
other automata, the offspring being at least as
powerful (in some well-defined sense) as the parent
• In Practice:
• Exploring how complex systems with emergent
patterns seem to evolve from purely local interactions
of agents. I.e. Without a “master plan!”
3
Cellular Automata
• A cellular automata is a family of simple, finitestate machines that exhibit interesting,
emergent behaviors through their interactions
in a population
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Emergent Behavior
The famous BOIDS model
shows how flocking
behavior can emerge from
a collection of agents
following a few simple
rules.
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•
Original concept of CA is most strongly associated with
John von Neumann who was interested in the connections
between biology and the new study of automata theory
•
Stanislaw Ulam suggested to von Neumann the use a
cellular automata as a framework for researching these
connections.
•
The original concept of CA can be credited to Ulam, while
the early development of the concept is credited to von
Neumann.
•
Ironically, although von Neumann made many
contributions and developments in CA, they are commonly
referred to as “non-von Neumann style”, while the standard
model of computation (CPU, globally addressable memory,
serial processing) is know as “von Neumann style”.
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Cellular Automata (CAs)
•
Have been used as:
•
•
•
•
•
VLSI Testing
Data Encryption
Error Correcting Code Correction
Testable Synthesis
Generation of hashing Function
•
Currently being investigated as model of quantum computation
(QCAs)
• massively parallel computer architecture
• model of physical phenomena (Fredkin, Wolfram)
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• Grid
• Mesh of cells.
• Simplest mesh is one dimensional.
• Cell
• Basic element of a CA.
• Cells can be thought of as memory
elements that store state
information.
• All cells are updated synchronously
according to the transition rules.
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• Local interaction leads to global
dynamics.
• One can think of the behaviour of a
cellular automata like that of a
“wave” at a sports event.
• Each person reacts to the state of
his neighbours (if they stand, he
stands).
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• Rule Application
• Next state of the core cell is related to the states of the
neighbouring cells and its current state.
• An example rule for a one dimensional CA: 011->x0x
• All possible states must be described.
• Next state of the core cell is only dependent upon the sum
of the states of the neighbouring cells.
• For example, if the sum of the adjacent cells is 4 the
state of the core cell is 1, in all other cases the state of
the core cell is 0.
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Structure
• Discrete space (lattice) of regular cells
• 1D, 2D, 3D, …
• rectangular, hexagonal, …
• At each unit of time a cell changes state in
response to (Time advances in discrete steps):
• its own previous state
• states of neighbors (within some “radius”)
• All cells obey same state update rule, depending
only on local relations
• Synchronous updating (parallel processing)
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Structure: Neighborhoods
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1-DIMENSIONAL AUTOMATA
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One-Dimensional CA’s
• Game of Life is 2-D. Many simpler 1-D CAs have
been studied
• For a given rule-set, and a given starting setup, the
deterministic evolution of a CA with one state (on/off)
can be pictured as successive lines of colored squares,
successive lines under each other
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Neighborhoods
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•
•
•
•
3 Black = White
2 Black
= Black
1 Black = Black
3 White = White
Now make your own CA
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“A New Kind of Science”
ISBN 1-57955-008-8
Stephen Wolfram
www.wolframscience.com
1-D CA Example
cell#
1
2
3
4
5
6
7
8
9
10 11 12 13
Rules
time = 1

time = 2

time = 3

time = 4

0
0
1
1
0
1
1
Rule# = 2 + 4 + 16 + 32 = 54
time = 5

time = 6

time = 7
18
0
Wolfram Model
1
1
1
1
1
1
1
0
Rule #126 = 64 + 32 + 16 + 8 + 4 + 2 + 0 = 126
1
1
1
1
1
1
0
0
Rule #124 = 64 + 32 + 16 + 8 + 4 + 0 + 0 = 124
Most of the rules are degenerate, meaning they create
repetitive patterns of no interest.
However there are a few rules which produce surprisingly
complex patterns that do not repeat themselves.
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Wolfram Model
we can view the state of the model at any time in the future as
long as we step through all the previous states.
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Hundred generations of Rule 30
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CA Example: Rule 30
111 110 101 100 011 010 001 000
0
0
0
1
1
1
1
0
Rule (0 + 0 + 0 + 16 + 8 + 4 + 2 +0 ) = 30
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Conus Textile pattern
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The pattern is neither regular nor completely random.
It appears to have some order, but is never predictable.
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Wolfram Model
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.
See Demo - NetLogo
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Wolfram Rule 90
0 1
0 1 1 0
1 0
Rule (0 + 2 + 0 + 8 + 16 + 0 + 64) = 90
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Wolfram Rule 110
Proven to be Turing Complete - Rich enough for universal computation
interesting result because Rule 110 is an extremely simple system, simple enough to
suggest that naturally occurring physical systems may also be capable of universality
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Wolfram Rule 99
Rule# 99 = 0 27 + 1 26 + 1 25 + 0 24 + 0 23 + 0 22 + 1 21 + 1 20
0 + 64 + 32 + 0 + 0 + 0 + 2 + 1 28
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Mollusc Pigmentation Patterns
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Wolfram’s CA classes 1,2
From observation, initially of 1-D CA spacetime
patterns, Wolfram noticed 4 different classes of rulesets. Any particular rule-set falls into one of these:-:
CLASS 1: From any starting setup, pattern converges to
all blank -- fixed attractor
CLASS 2: From any start, goes to a limit cycle, repeats
same sequence of patterns for ever. -- cyclic attractors
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Wolfram’s CA classes 3,4
CLASS 3: Turbulent mess, chaos, no patterns to be seen.
CLASS 4: From any start, patterns emerge and
continue without repetition for a very long
time (could only be 'forever' in infinite grid)
Classes 1 and 2 are boring, Class 3 is messy,
Class 4 is 'At the Edge of Chaos' - at the transition
between order and chaos -- where Game of Life is!.
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2-DIMENSIONAL
AUTOMATA
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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.
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Neighborhoods
Von Neumann
Moore
margolus
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Snowflakes
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Example:
Conway’s Game of Life
• Invented by Conway in late 1960s
• A simple CA capable of universal
computation
• Structure:
•
•
•
•
•
2D space
rectangular lattice of cells
binary states (alive/dead)
neighborhood of 8 surrounding cells (& self)
simple population-oriented rule
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Example:
Conway’s Game of Life
Cell
State = dead/off/0
State = alive/on/1
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Example:
Conway’s Game of Life
• A cell dies or lives according to some transition rule
transition
rules
T=0
T=1
• The world is round (flips over edges)
• How many rules for Life? 20, 40, 100, 1000?
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State Transition Rule
• Live cell has 2 or 3 live neighbors
 stays as is (stasis)
• Live cell has < 2 live neighbors
 dies (loneliness)
• Live cell has > 3 live neighbors
 dies (overcrowding)
• Empty cell has 3 live neighbors
 comes to life (reproduction)
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State Transition Rule
• Survive with 2 or 3 live neighbors
Live cell  stays as is (stasis)
otherwise dies from loneliness or
overcrowding
• Generate with 3 live neighbors
Empty cell  comes to life (reproduction)
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State Transition Rule
Three simple rules:
• dies if number of alive neighbor cells =< 1
(loneliness)
• dies if number of alive neighbor cells >= 4
(overcrowding)
• generate alive cell if number of alive neighbor cells = 3
(procreation)
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State Transition Rule
Examples of the rules
• loneliness
(dies if #alive =< 1)
• overcrowding (dies if #alive >= 4)
• procreation
(lives if #alive = 3)
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CA: Discrete Time,
Discrete Space
Initial Setup
After Pass 1
Number of Neighbors
After Pass 2
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Game of Life: 2D Cellular
Automata using simple rules
neighboring values
T=0
Emergent pattern:
Blinker
T=1
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Emergent patterns
Conway automaton can simulate a number of different effects that
can be found in the evolution of a living population.
Equilibria
Oscillation
Movement
square
2 steps
beehive
2 steps
diagonal
horizontal
instability
boat
3 steps
ship
15 steps
toast
(all the space
is filled up by
horizontal lines)
instability
chaos?
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Game of Life:
emergent patterns
Gosper’s glider gun : emits glider stream
Conway’s Rules: Game of Life
Survive with 2 – 3 living neighbors
Generate with 3 living neighbors
gliders: patterns that moves constantly across the grid
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Emergent Patterns
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Emergent Patterns
Oscillators-objects that change from step to step, but
eventually repeat themselves. These include, but are not
limited to, period 2 oscillators, including the blinker and the
toad.
Blinker
Toad
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Emergent Patterns:
A Clock
See Demo: Game of Life
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Emergent Patterns:
Oscillator
Pulsar
SpaceShip
Beacon
Glider
See Demo: Game of Life
Barber’s Pole
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Emergent Patterns:
Gosper’s Glider Gun
See Demo: Game of Life
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Emergent Patterns:
Puffer train
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Emergent Patterns:
Double-Barreled Gun
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Emergent Patterns:
Edge Shooter
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Emergent Patterns:
Evolution of a breeder ...
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Conclusions
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Conclusions
• This topic is very hot and has widespread implications
• Biology
• Chemistry
• Computer science
• Complexity
• We’ve seen the basic concepts …
• But we’ve only scratched the surface!
 From now on, Think Biology, Emergence, Complex
Systems …
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References
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References
• Jay Xiong, New Software Engineering Paradigm Based on Complexity Science, Springer
2011.
• Claudios Gros : Complex and Adaptive Dynamical Systems. Second Edition, Springer,
2011 .
• Blanchard, B. S., Fabrycky, W. J., Systems Engineering and Analysis, Fourth Edition,
Pearson Education, Inc., 2006.
• Braha D., Minai A. A., Bar-Yam, Y. (Editors), Complex Engineered Systems, Springer,
2006
• Gibson, J. E., Scherer, W. T., How to Do Systems Analysis, John Wiley & Sons, Inc.,
2007.
• International Council on Systems Engineering (INCOSE) website (www.incose.org).
• New England Complex Systems Institute (NECSI) website (www.necsi.org).
• Rouse, W. B., Complex Engineered, Organizational and Natural Systems, Issues
Underlying the Complexity of Systems and Fundamental Research Needed To Address
These Issues, Systems Engineering, Vol. 10, No. 3, 2007.
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References
• Wilner, M., Bio-inspired and nanoscale integrated
computing, Wiley, 2009.
• Yoshida, Z., Nonlinear Science: the Challenge of
Complex Systems, Springer 2010.
• Gardner M., The Fantastic Combinations of John
Conway’s New Solitaire Game “Life”, Scientific
American 223 120–123 (1970).
• Nielsen, M. A. & Chuang, I. L. ,Quantum
Computation and Quantum Information, 3rd ed.,
Cambridge Press, UK, 2000.
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