1. Complex Systems Engineering SwE 488 Artificial Complex Systems
- Cellular Automata -

2. Cellular Automata
Each of a set of units in a mathematical model which have simple rules governing their replication and destruction, used to model complex systems composed of simple units such as living things or parallel processors.

3. 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!”

4. Cellular Automata
A cellular automata is a family of simple, finite-state machines that exhibit interesting, emergent behaviors through their interactions in a population
A cellular automaton is a collection of "colored" cells on a grid of specified shape that evolves through a number of discrete time steps according to a set of rules based on the states of neighboring cells.

5. Emergent Behavior
The famous BOIDS model shows how flocking behavior can emerge from a collection of agents following a few simple rules.

6. history
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”.

7. Cellular Automata (CAs)
Part 2A: Cellular Automata
9/12/2018
Cellular Automata (CAs)
Have been used as:
massively parallel computer architecture
model of physical phenomena (Fredkin, Wolfram)
VLSI Testing
Data Encryption
Error Correcting Code Correction
Testable Synthesis
Generation of hashing Function
Currently being investigated as model of quantum computation (QCAs)
Edward Fredkin, MIT
CS 420/594

8. components Grid Cell Mesh of cells. Simplest mesh is one dimensional.
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.

9. Behavior 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).

10. Behavior 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.

11. Part 2A: Cellular Automata
9/12/2018
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)
CS 420/594

12. Structure: Neighborhoods

13. 1-Dimensional Automata

14. 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

15. Neighborhoods

16. 3 Black = White 2 Black = Black 1 Black = Black 3 White = White
Now make your own CA

17. “A New Kind of Science” www.wolframscience.com Stephen Wolfram
Dr. Wolfram is founder of Wolfram Research, the creators of Mathematica.
Stephen Wolfram over the past twenty years contemplated the workings of the universe and he has come to the realization that equations fail to explain the complexity of the universe.
In this provocative book, “A New Kind of Science” ( ), Wolfram seeks to convince us to change our world view of science, and to re-interpret the universe through computer programs which use cellular automata.
Wolfram proposes that we adopt simple rules to explain the workings of the universe
These rules can be easily modelled with a few lines of computer code.
He observed that these simple rules can lead to complex behaviour
ISBN
Stephen Wolfram

18. 1-D CA Example
Rules
Rule# = = 54

19. Wolfram Model
Rule #126 = = 126
Rule #124 = = 124
Thus in a three cell arrangement there are eight possible input situations ( 23 = 8 ).
Each rule maps all combinations of these eight input situations to an output (black or white). So there are 28 = 256 possible rules for such a one-dimensional, two-color, adjacent-cell automaton.
Half of the 256 possible rules map onto the other half because of left-right symmetry.
We can map half of them again because of black-white equivalence, so we are left with 64 rule types.
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.

20. 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.

21. Hundred generations of Rule 30

22. CA Example: Rule 30
Rule ( ) = 30

23. Conus Textile pattern

24. The pattern is neither regular nor completely random.
It appears to have some order, but is never predictable.

25. Wolfram Model Rule #45=32+8+4+1= =
Rule #30= = =
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

26. Wolfram Rule 90
Rule ( ) = 90

27. 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

28. Wolfram Rule 99
A pyramid !!
Rule# 99 =

30. Mollusc Pigmentation Patterns

31. Wolfram’s CA classes 1,2
From observation, initially of 1-D CA spacetime patterns, Wolfram noticed 4 different classes of rule-sets. 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

32. 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!.

33. 2-Dimensional Automata

34. 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.

35. Von Neumann Moore margolus
Neighborhoods
Von Neumann Moore margolus

36. Snowflakes

37. Example: Conway’s Game of Life
Part 2A: Cellular Automata
9/12/2018
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
CS 420/594

38. Example: Conway’s Game of Life
Part 2A: Cellular Automata
9/12/2018
Example: Conway’s Game of Life
Cell
State = dead/off/0
State = alive/on/1
CS 420/594

39. Example: Conway’s Game of Life
A cell dies or lives according to some transition rule
The world is round (flips over edges)
How many rules for Life? 20, 40, 100, 1000?
transition
rules
T = 0
T = 1

40. Part 2A: Cellular Automata
9/12/2018
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)
Alternate formulation:
turned on if 3 of neighbors turned on [3 are needed for birth]
remains on if 2 or 3 of neighbors on [2 or 3 are needed for survival]
otherwise is turned off
(Emmeche, p. 6)
CS 420/594

41. Part 2A: Cellular Automata
9/12/2018
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)
Alternate formulation:
turned on if 3 of neighbors turned on [3 are needed for birth]
remains on if 2 or 3 of neighbors on [2 or 3 are needed for survival]
otherwise is turned off
(Emmeche, p. 6)
CS 420/594

42. Part 2A: Cellular Automata
9/12/2018
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)
Alternate formulation:
turned on if 3 of neighbors turned on [3 are needed for birth]
remains on if 2 or 3 of neighbors on [2 or 3 are needed for survival]
otherwise is turned off
(Emmeche, p. 6)
CS 420/594

43. State Transition Rule Examples of the rules
loneliness (dies if #alive =< 1)
overcrowding (dies if #alive >= 4)
procreation (lives if #alive = 3)

44. CA: Discrete Time, Discrete Space
Initial Setup
After Pass 1
Number of Neighbors
After Pass 2

45. Emergent pattern: Blinker
Game of Life: 2D Cellular Automata using simple rules
neighboring values
T = 0
Emergent pattern: Blinker
T = 1

46. 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
diagonal
beehive
horizontal
boat
3 steps
instability
(all the space
is filled up by
horizontal lines)
ship
15 steps
toast
chaos?

47. 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

48. Emergent Patterns

49. 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

50. Emergent Patterns: A Clock
See Demo: Game of Life

51. Emergent Patterns: Oscillator SpaceShip Pulsar Beacon Glider
Barber’s Pole
See Demo: Game of Life

52. Emergent Patterns: Gosper’s Glider Gun
See Demo: Game of Life

53. Emergent Patterns: Puffer train

54. Emergent Patterns: Double-Barreled Gun

55. Emergent Patterns: Edge Shooter

56. Emergent Patterns: Evolution of a breeder ...

57. Conclusions

58. 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 …

59. References

60. References
Jay Xiong, New Software Engineering Paradigm Based on Complexity Science, Springer 2011.
Claudios Gros : Complex and Adaptive Dynamical Systems. Second Edition, Springer,
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 (
New England Complex Systems Institute (NECSI) website (
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.

61. 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 –123 (1970).
Nielsen, M. A. & Chuang, I. L. ,Quantum Computation and Quantum Information, 3rd ed., Cambridge Press, UK, 2000.

62. Q & A