Cellular Automata

1
Cellular Automata

• BIOL/CMSC 361 Emergence
• 2/12/08

2
The Computational Beauty of Nature

• The topics covered in this book demand varying
  amounts of sophistication from you. Some of the
  ideas are so simple that they have formed the
  basis of lessons for a third grade class. Other
  chapters should give graduate students a
  headache. This is intentional. If you are
  confused by a sentence, section, or chapter,then
  by all means move on. pg. xv

3
A New Kind of Science

• Steven Wolfram (Mathematica)
• The nature of computation must be explored
  experimentally
• Methods relevant to the study of simple programs
  (computation) are relevant to all other fields of
  study
• Non-simple behavior corresponds to a computation
  of equivalent sophistication

Principle of Computational Equivalence
4
Universal Computation

• Turing Machine
• Extremely basic, symbol processing device that
  can be adapted to simulate the logic of any
  computer
• Cellular Automata?

5
Summary

• Chaos simple things ? complex behavior
• Complexity complex collections of simple things
  ? variety of behaviors
• Emergence collection of behaviors ? a whole
• Parts
• Interactions

6
About a Model
?
Input
Output
Top-down formulate overview of system
Bottom-up specify basic elements in great detail
and link together to formulate system
7
What do about a Model?

• Engineers study interesting real-world problems
  but fudge their results. Mathematicians get
  exact results but study only toy problems. But
  computer scientists, being neither engineers nor
  mathematicians, study toy problems and fudge
  their results. pg. xiii
• Engineer ? Experimentalist
• Theorist ? Mathematician
• Simulationist ? Computer Scientist

8
What to do about a Model

• Experimentalist messy real-world problems are
  prone to error
• Theorist must make simplifying assumptions to
  get to the essence of a physical process
• Simulationist attempts to understand the world
  by through computer simulatyions of phenomena
• Makes assumptions
• Simulated results are not perfect match for the
  real world

9
Cellular Automata

• A computational model
• An abstraction of a real-world system
• NOT a type of real-world system
• Other Types of Models
• Mathematical Models
• Differential Equations
• Linear Equations
• Probability Distributions
• Physical Models
• Spatial
• Visual

10
Cellular Automata
Neighbors
Rules
Time
State Space
11
Wolframs Classification

• Class I Always evolve to a homogenous
  arrangement, with every cell in same state

12
Wolframs Classification

• Class II form endlessly cycling periodic
  structures

13
Wolframs Classification

• Class III form aperiodic, or random-like
  patterns

14
Wolframs Classification

• Class IV global pattern is complex due to
  localized structure eventually becomes
  homogenous or settles into a periodic pattern

15
Langtons Scheme

• ? (N nq) / N
• N total number of rules
• nq number of rules that map to a quiescent
  state
• ? 0 ? all rules map to quiescent state
• ? 1 ? all rules map to non-quiescent state

• But
• CA can have high ? and simple behavior if most
  rules map to same state
• Sophisticated programs can produce a variety of
  behaviors
• Cannot account for initial state or long-term
  behavior

I
III
IV
II
16
Bifurcation Diagram
Zero
Steady
Chaos
17
Interactions

• Collections, Multiplicity, Parallelism
• Parallel collections of similar units
• Perform tasks simultaneously
• Multiple problem solutions to be attempted
  simultaneously

18
Interactions

• Iteration, Recursion, Feedback
• Persistence in time (reproduction)
• Self-similarity
• Interaction with environment

19
Interactions

• Adaptation, Learning, Evolution
• Interesting systems change
• Consequence of parallelism and iteration in a
  competitive environment with finite resources
• Multiplicity and iteration ? filter
• Loop in the cause and effect of changes in agents
  and their environments