Lectures on

1
Lectures on Cellular Automata Continued
Modified and upgraded slides of Martijn
Schut schut_at_cs.vu.nl Vrij Universiteit
Amsterdam Lubomir Ivanov Department of Computer
Science Iona College and anonymous from Internet

2
Morphogenetic modeling

• Dynamical Systems
• and Cellular Automata

3
Organisms are dissipative processes in space
and time
matter heat
matter energy
entropy dissipator
4
Dynamical system
Energy
Time
Position, energy, behavior, etc state is
changing in time according to certain dynamics
5
Which space? - which geometry?
geometric space
y
time
x
?
parameter space
state space
velocity
b
a
q
A good source of this kinds of problems may be
robots for robot soccer, robot colony, bee hive
modeling, ants, termites, etc.
6
Which space? - which geometry?
geometric space
ontogeny/phenotype
?
parameter space
state space
genotype environment
phylogeny/ontogeny
Dynamics evolves, we can use evolutionary
algorithms. Concepts of genotype versus phenotype.
7
Discretization has many aspects
t

• discrete time steps T0,1,2,3,
• discrete cells at X,Y,Z0,1,2,3,...
• discretize cell states S0,1,2,3,...

t1
We can mix discrete and continuous values in some
models
t2
8
One Dimensional Cellular Automaton A more
general view
In standard CA the values of cells are discretized
t
neighbours
update rules
t1
Think how can you use this concept to solve SAT?
F can be a complex program, but always
synchronized to macro-pulses example is SAT
solver that will be discussed
9
Components of a maximal problem simplification to
create a Cellular Automaton

• Reduce dimensions D1, i.e. array of cells
• Reduce of cell states binary, i.e. 0 or 1
• Simplify interactions nearest neighbours
• Simplify update rules deterministic, static

Try to think what kind of practical applications
can be found for this kind of models. We will
discuss them later on but may be you will invent
now something new!
10
Effects of simplification
You can model life and physical, chemical,
biological and social phenomena on many levels of
detail.
11
Wolfram's 1D binary CA
We calculate how many neighbors are 1, we dont
care which neighbors

• cell array
• binary states
• 5 nearest neighbours
• sum-of-states update rule

S
Wolfram, S., Physica 10D (1984), 1-35
Observe importance of symmetry of logic function
12
CA rules are now based on Symmetry
We calculate how many neighbors are 1, we dont
care which neighbors
Recall lattices and symmetry indices from ECE 572
class
S ? 1,2,3,4,5
S
S ? 0,1
S
Observe that some kind of symmetry is fundamental
to all science and biology. Symmetry simplifies
rules of Nature.
? 2532 different rules
i.e. placing ones or zeros in five places
13
Code for rule 6 (00110)
We calculate how many neighbors are 1, we dont
care which neighbors
S neighbours
1
2
3
4
5
S(i,T1)
0
0
1
1
0
Rule 6 tells that if sum is 3 or 4 than next
state is alive
This is only a simple example
14
How to encode the rules in a simple and logical
way?
0
0
0
0
0
rule 0
The encoding results from the way of specifying
symmetric functions.
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
The fact that rules are binary string is very
useful, for instance GA can be used.
1
1
1
1
1
rule 32
15
Temporal evolution of a Cellular Automaton
initial configuration (t0) e.g. random 0/1
rule n
t1
rule n
t2
rule n
t3
Please do not mislead the temporal evolution of
the system with given set of rules with the
evolution of the set of rules.
16
Temporal evolution
t0
t10
17
Rule 6 00110
t0
t100
18
Rule 10 01010
t0
t100
Interpretation changing only two bits in rules
(genotype) changes the dynamics of life
(phenotype)
19
CA a metaphor for morphogenesis

• static binary pattern
• translated into
• state transition rule
• rule iteration in configuration space

• genotype
• translation,epigenetic rules
• morphogenesis of the phenotype

20
CA a model for morphogenesis

• update rule
• iterative application of the update rule
• CA state pattern
• state space

• genotype
• epigenetic interpretation of the genotype,
  morphogenesis
• phenotype
• morphospace

21
CA rule mutations
Small change in genetype changes phenotype
1
0
0
1
0
swap
genotype
phenotype
development
negate
0
1
0
1
1

How genotype mutations can change phenotypes?
22
CA rule mutations
0
1
0
1
0
1
0
0
1
0
rule
0
1
1
0
0
1
0
1
0
0
1
0
1
swap
negate
0
1
0
0
1
0
0
0
1
1
23
Predictability?
pattern at t
?
explicit simulation D iterations of rule n
hypothetical "simple algorithm" ?
pattern at tDt
How to predict a next element in sequence? Tough!
24
Predictability
Sum of natural numbers 1 N algorithm 1
12N algorithm2 N(N1)/2 (Gaussian
formula)
Nth prime number algorithm 1 trial and
error algorithm2 ? no general formula
easy
Very tough!
Given a number N predict the sum of numbers from
1 to N.
Given a number N predict the N-th prime number
25
CA there is no effective prediction method for
patterns
pattern at t
pattern at tDt
behavior may be determined only by explicit
simulation
26
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 imply similarity of
  patterns

27
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

28
Meinhardt, H. (1995). The Algorithmic Beauty of
Sea Shells. Berlin Springer
29
Patterns and morphology

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

• Form (Size and Shape)
• Morphogenesis The spatio-temporal processes by
  which an organism changes is size (growth) and
  shape (development)

30
Where is the phene?
phenotype A

• Typification?
• Comparative measures?
• length
• density
• fractal dimension
• Spatio-temporal development?

phenotype B
All active research areas!