Cellular Automataweb.cecs.pdx.edu/~mperkows/temp/May22/B0028... · Lectures on Cellular Automata...
Transcript of Cellular Automataweb.cecs.pdx.edu/~mperkows/temp/May22/B0028... · Lectures on Cellular Automata...
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Lectures onLectures onLectures onLectures onCellular Automata Cellular Automata Cellular Automata Cellular Automata ContinuedContinuedContinuedContinued
Modified and upgraded slides of
Martijn [email protected] Universiteit Amsterdam
Lubomir IvanovDepartment of Computer ScienceIona College
and anonymous from Internet
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MorphogeneticMorphogeneticmodelingmodeling
Dynamical SystemsDynamical Systemsand Cellular Automataand Cellular Automata
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matterheat
matterheat
matterenergymatterenergy
entropy dissipator
Organisms aredissipative processes
in space and time
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Dynamical system
Time
Space
8
S(�8
x , t)
8
state S at�8
x at time t ?
Energy
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Which space? - which geometry?
parameter space
a
b
geometric space
x
y
x2
a2 +y2
b2 =1
state space
x =a costy = b sin t
morphometric space
Q=S/A
S
A
?
time
velocity
θ
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Which space? - which geometry?
? parameter space
genotypeenvironment
state space
phylogeny/ontogeny
morphometric space
morphology
geometric space
ontogeny/phenotype
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• discrete time stepsT=0,1,2,3,…
• discrete cells atX,Y,Z=0,1,2,3,...
• discretize cell statesS=0,1,2,3,...
Discretization has many aspects
t
t+1
t+2We can mix discrete andcontinuous values in somemodels
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Cellular AutomatonCellular Automaton
t
t+1
update rules
neighbours
S(i, t +1) = f (S(neighbours,t), update rules)
S(i,t)
S(i,t +1)
In standard CA the values of cells are discretized
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• 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
A simple Cellular Automaton
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Effects of simplification
CA mo d el Mor ph oge n esi sS pa ti al d is creti za ti on ~ le ve l o f d e tai l
(organi sm , ce ll, mo lecule)T em pora l d isc reti za ti on ~ le ve l o f d e tai l
(cel l cycle , reac tion rate)R e duc ti on o f d im en sion → pro found e ff ec ts
(2D “G am e of Li fe” )B ina ri za ti on co m pu tati ona l conven ience
(01→ A , 10→ T, 00→ G , 11→ C)nea rest -ne ighbou rin terac ti ons
~ spa ti a l re st ric ti on s
sim p le upda te ru les;sim u lt ane it y , ub iqu it y
~ non- li nea rit y→ profound effe ct s
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ΣΣ
Wolfram's 1D binary CA
• cell array• binary states• 5 nearest neighbours• „sum-of-states“ update
rule:
• cell array• binary states• 5 nearest neighbours• „sum-of-states“ update
rule:
S(i, t +1) = f S(i + j, t)j= −2
j =2
∑
Wolfram, S., Physica 10D (1984), 1-35
S(i, t)S(i −1, t) S(i +1, t)
S(i, t +1)
S(i − 2, t) S(i + 2, t)
Observe importance ofsymmetry of logic function
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ΣΣ
CA rules
S ∈ {0,1}S ∈ {0,1}
Σ ∈ {1,2,3,4,5}Σ ∈ {1,2,3,4,5}
S
→ 25=32 different rules→ 25=32 different rules
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Code for rule 6 (00110)
0S(i,T+1)
Σ neighbours 11
0
22
1
33
1
44
0
55
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Rule codes0 0 0 0 0
0 0 0 0 1
0 0 0 1 0
0 0 0 1 1
0 0 1 0 0
0 0 1 0 1
1 1 1 1 1
rule 0rule 1rule 2rule 3rule 4rule 5
rule 32
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Temporal evolutioninitial configuration (�t=0)
e.g. random 0/1
t=1
t=2
t=3
rule n
rule n
rule n
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Temporal evolution
t=0
t=10
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Rule 6: 00110
t=100
t=0
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Rule 10: 01010
t=100
t=0
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CA: a metaphor for morphogenesis static binary
pattern
translated into state transition rule
rule iteration inconfigurationspace
genotype
translation,epigenetic rules
morphogenesisof the phenotype
00 11 00 11 00
0
11
1
22
0
33
1
44
0
55
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CA: a model for morphogenesis
update rule
iterative application of theupdate rule
CA state pattern
state space
genotype
epigenetic interpretation of thegenotype, morphogenesis
phenotype
morphospace
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CA rule mutations
11 00 0 1 0
0 1 0 1 11
swap
�
0 1 0 1 0
genotype phenotypedevelopmentnegate
How genotype mutations can change phenotypes?
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CA rule mutations0 1 0 1 0 1 0 0 1 0
0 1 1 0 0 1 0 1 0 0 1 0 1
0 1 0 0 1 0 0 0 1
0 1 1 1 0
1 1
0 1 0 1 0 1
negate
swaprule
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Predictability?
pattern at t+∆t
pattern at t
explicit simulation:∆ iterations of rule n
?hypothetical"simple algorithm" ?
How to predict a next element in sequence? Tough!
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Sum of naturalnumbers 1… N
algorithm 1:1+2+…+N
algorithm2:N(N+1)/2
(Gaussian formula)
Sum of naturalnumbers 1… N
algorithm 1:1+2+…+N
algorithm2:N(N+1)/2
(Gaussian formula)
Predictability
n1
N
∑
Nth prime number
algorithm 1:trial and error
algorithm2: ?no general formula
Nth prime number
algorithm 1:trial and error
algorithm2: ?no general formula
pN ∈ pn{ }= p | p / i ∉ N;i ∈ 1, p] [{ }
easy Very tough!
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CA: no effective patternprediction
pattern at t+∆t
pattern at t
behaviour may be determined only by explicit simulation
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CA: deterministic, unpredictable,irreversible
• Simple rules generate complex spatio-temporal behavior
• For non-trivial rules, the spatio-temporalbehaviour is computable but not predictable
• The behavior of the system is irreversible• Similarity of rules does not imply similarity
of patterns
• Simple rules generate complex spatio-temporal behavior
• For non-trivial rules, the spatio-temporalbehaviour is computable but not predictable
• The behavior of the system is irreversible• Similarity of rules does not imply similarity
of patterns
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Aspects of CA morphogenesisAspects of CA morphogenesis
• complex relationship between „genotype“and „phenotype
• effects of „genes“ are not localizable inspecific phenes (pleiotropy)
• phenes cannot be traced back to specificsingle genes (epistasisepistasis)
• phenetic effects of "mutations" are notpredictable
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Meinhardt, H. (1995). TheAlgorithmic Beauty of Sea Shells.Berlin: Springer
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Patterns and morphology• Pattern: a spatially
and/or temporallyordered distribution ofa physical or chemicalparameter
• Pattern formation
• Form (Size and Shape)
• Morphogenesis: Thespatiotemporalprocesses by which anorganism changes issize (growth) andshape (development)
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Where is the phene?
• Typification?• Comparative
measures?– length– density– fractal dimension
• Spatio-temporaldevelopment?
phenotype A
phenotype B