Cellular Automata BIOL/CMSC 361: Emergence 2/12/08

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

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

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

Summary Chaos: simple things  complex behavior Complexity: complex collections of simple things  variety of behaviors Emergence: collection of behaviors  a whole ◦ Parts ◦ Interactions

About a Model InputOutput Top-down: formulate overview of system Bottom-up: specify basic elements in great detail and link together to formulate system

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

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

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

Cellular Automata TimeTime Neighbors Rules State Space

Wolfram’s Classification Class I: Always evolve to a homogenous arrangement, with every cell in same state

Wolfram’s Classification Class II: form endlessly cycling periodic structures

Wolfram’s Classification Class III: form aperiodic, or “random”-like patterns

Wolfram’s Classification Class IV: global pattern is complex due to localized structure; eventually becomes homogenous or settles into a periodic pattern

Langton’s Scheme λ = (N – n q ) / N N = total number of rules n q = number of rules that map to a quiescent state λ = 0  all rules map to quiescent state λ = 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 II I IV III

Bifurcation Diagram ZeroSteady Chaos

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

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

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