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Lectures on Cellular Automata ContinuedPowerPoint Presentation

Lectures on Cellular Automata Continued

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Cellular Automata Continued

Modified and upgraded slides of

Martijn Schut

Vrij Universiteit Amsterdam

Lubomir Ivanov

Department of Computer Science

Iona College

and anonymous from Internet

S

A

Q=S/A

Which space? - which geometry?geometric space

y

time

x

?

state space

parameter space

velocity

b

a

q

morphology

Which space? - which geometry?geometric space

ontogeny/phenotype

?

parameter space

state space

genotype

environment

GA evolution environment

phylogeny/ontogeny

Dynamics of species in environments.

Discretization has many aspects

t

- discrete time stepsT=0,1,2,3,…
- discrete cells at X,Y,Z=0,1,2,3,...
- discretize cell statesS=0,1,2,3,...

t+1

We can mix discrete and continuous values in some models

t+2

A simple Cellular Automaton

- Reduce dimensions: D=1, i.e. array of cells
- Reduce # of cell states: binary, i.e. 0 or 1
- Simplify interactions: nearest neighbours
- Simplify update rules: deterministic, static

Effects of simplification

CA

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Wolfram's 1D binary CA

- cell array
- binary states
- 5 nearest neighbours
- „sum-of-states“ update rule:

S

Wolfram, S., Physica10D (1984), 1-35

Observe importance of symmetry of logic function

Number of “alive” neighbors

Wolfram‘s 32 „Symmetry-based“ CA rules in 5-neighborhood

S{1,2,3,4,5}

S

S

S{0,1}

25=32 different rules

Number of “alive” neighbors

Rule codes

0

0

0

0

0

rule 0

0

0

0

0

1

rule 1

0

0

0

1

0

rule 2

0

0

0

1

1

rule 3

0

0

1

0

0

rule 4

0

0

1

0

1

rule 5

1

1

1

1

1

rule 32

Number of “alive” neighbors

translated into

state transition rule

rule iteration in configuration space

genotype

translation,epigenetic rules

morphogenesis of the phenotype

1

2

3

4

5

0

1

0

1

0

0

1

0

1

0

CA: a metaphor for morphogenesisiterative application of the update rule

CA state pattern

state space

genotype

epigenetic interpretation of the genotype, morphogenesis

phenotype

morphospace

CA: a model for morphogenesis1

0

1

0

CA rule mutations1

0

0

1

0

swap

genotype

negate

development

phenotype

0

1

0

1

1

How genotype mutations can change phenotypes?

1

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0

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CA rule mutations0

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1

0

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Predictability?

pattern at t

?

explicit simulation:

D iterations of rule n

hypothetical

"simple algorithm" ?

pattern at t+Dt

How to predict a next element in sequence? Tough!

Predictability

Sum of natural numbers 1… N

algorithm 1:

1+2+…+N

algorithm2:

N(N+1)/2

(Gaussian formula)

Nth prime number

algorithm 1:

trial and error

algorithm2: ?

no general formula

easy

Very tough!

CA: no effective pattern prediction

pattern at t

pattern at t+Dt

behaviour may be determined only by explicit simulation

CA: deterministic, unpredictable, irreversible

- Simple rules generate complex spatio-temporal behavior
- For non-trivial rules, the spatio-temporal behaviour is computable but not predictable
- The behavior of the system is irreversible
- Similarity of rules does not implysimilarity of patterns

Aspects of CA morphogenesis

- complex relationship between „genotype“ and „phenotype
- effects of „genes“ are not localizable in specific phenes (pleiotropy)
- phenes cannot be traced back to specific single genes (epistasis)
- phenetic effects of "mutations" are not predictable

Meinhardt, H. (1995). The Algorithmic Beauty of Sea Shells. Berlin: Springer

Pattern: a spatially and/or temporally ordered distribution of a physical or chemical parameter

Pattern formation

Form (Size and Shape)

Morphogenesis: The spatiotemporal processes by which an organism changes is size (growth) and shape (development)

Patterns and morphologyWhere is the phene? of a physical or chemical parameter

phenotype A

- Typification?
- Comparative measures?
- length
- density
- fractal dimension

- Spatio-temporal development?

phenotype B

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