Cellular AutomataCellular AutomataBIOL/CMSC 361: Emergence2/12/08

The Computational Beauty of The Computational Beauty of NatureNature“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

A New Kind of ScienceA New Kind of ScienceSteven 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 EquivalencePrinciple of Computational Equivalence

Universal ComputationUniversal Computation“Turing Machine”

Extremely basic, symbol processing device that can be adapted to simulate the logic of any computer

Cellular Automata?

SummarySummaryChaos: simple things complex

behavior

Complexity: complex collections of simple things variety of behaviors

Emergence: collection of behaviors a whole◦Parts◦Interactions

About a ModelAbout a Model

Input Output

))(1)(()1( xnxrnxn

Top-down: formulate overview of systemTop-down: formulate overview of systemBottom-up: specify basic elements in great detail and link together to formulate system

Bottom-up: specify basic elements in great detail and link together to formulate system

What do about a Model?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 ExperimentalistTheorist MathematicianSimulationist Computer Scientist

What to do about a ModelWhat to do about a ModelExperimentalist: 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

Cellular AutomataCellular AutomataA computational modelAn abstraction of a real-world systemNOT a type of real-world system

Other Types of Models:◦ Mathematical Models

Differential Equations Linear Equations Probability Distributions

◦ Physical Models

SpatialVisual

Cellular AutomataCellular Automata

Time

Neighbors

Rules

State Space

Wolfram’s ClassificationWolfram’s ClassificationClass I: Always evolve to a

homogenous arrangement, with every cell in same state

Wolfram’s ClassificationWolfram’s ClassificationClass II: form endlessly cycling

periodic structures

Wolfram’s ClassificationWolfram’s ClassificationClass III: form aperiodic, or

“random”-like patterns

Wolfram’s ClassificationWolfram’s ClassificationClass IV: global pattern is

complex due to localized structure; eventually becomes homogenous or settles into a periodic pattern

Langton’s SchemeLangton’s Schemeλ = (N – nq) / NN = total number of rulesnq = number of rules that map to a quiescent stateλ = 0 all rules map to quiescent stateλ = 1 all rules map to non-quiescent state

But… CA can have high λ 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

But… CA can have high λ 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

III IV III

Bifurcation DiagramBifurcation Diagram

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x0 =0.1

Rate

Lon

g-r

un

Va

lue

(Att

ract

or)

Zero Steady Chaos

InteractionsInteractionsCollections, Multiplicity,

Parallelism◦Parallel collections of similar units◦Perform tasks simultaneously◦Multiple problem solutions to be

attempted simultaneously

InteractionsInteractionsIteration, Recursion, Feedback

◦Persistence in time (reproduction)◦Self-similarity◦Interaction with environment

InteractionsInteractionsAdaptation, 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