Lectures on 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

Morphogenetic modeling Dynamical Systems and Cellular Automata

matter heat matter heat matter energy matter energy entropy dissipator Organisms are dissipative processes in space and time

Dynamical system Time Space Energy Position, energy, behavior, etc – state is changing in time according to certain dynamics

Which space? - which geometry? parameter space a b geometric space x y state space morphometric space Q=S/A S A ? time velocity  A good source of this kinds of problems may be robots for robot soccer, robot colony, bee hive modeling, ants, termites, etc.

Which space? - which geometry? ? parameter space genotype environment state space phylogeny/ontogeny morphometric space morphology geometric space ontogeny/phenotype Dynamics evolves, we can use evolutionary algorithms. Concepts of genotype versus phenotype.

discrete time steps T=0,1,2,3,… discrete cells at X,Y,Z=0,1,2,3,... discretize cell states S=0,1,2,3,... Discretization has many aspects t t+1 t+2 We can mix discrete and continuous values in some models

One Dimensional Cellular Automaton: A more general view t t+1 update rules neighbours In standard CA the values of cells are discretized F can be a complex program, but always synchronized to macro- pulses – example is SAT solver that will be discussed Think how can you use this concept to solve SAT?

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 Components of a maximal problem simplification to create a Cellular Automaton 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!

Effects of simplification You can model life and physical, chemical, biological and social phenomena on many levels of detail.

  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: Wolfram, S., Physica 10D (1984), 1-35 Observe importance of symmetry of logic function We calculate how many neighbors are “1”, we don’t care which neighbors

  CA rules are now based on Symmetry S  { 0,1 }   { 1,2,3,4,5 } S  2 5 =32 different rules We calculate how many neighbors are “1”, we don’t care which neighbors Recall lattices and symmetry indices from ECE 572 class Observe that some kind of symmetry is fundamental to all science and biology. Symmetry simplifies rules of Nature. i.e. placing ones or zeros in five places

Code for rule 6 (00110) 0 S (i,T+1)  neighbours We calculate how many neighbors are “1”, we don’t care which neighbors Rule 6 tells that if sum is 3 or 4 than next state is alive This is only a simple example

How to encode the rules in a simple and logical way? rule 0 rule 1 rule 2 rule 3 rule 4 rule 5 rule 32 The fact that rules are binary string is very useful, for instance GA can be used. The encoding results from the way of specifying symmetric functions.

Temporal evolution of a Cellular Automaton initial configuration (t=0) e.g. random 0/1 t=1 t=2 t=3 rule n Please do not mislead the temporal evolution of the system with given set of rules with the evolution of the set of rules.

Temporal evolution t=0 t=10

Rule 6: t=100 t=0

Rule 10: t=100 t=0 Interpretation: changing only two bits in rules (genotype) changes the dynamics of life (phenotype)

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   1   0   1   0  

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

CA rule mutations swap genotypephenotypedevelopment negate How genotype mutations can change phenotypes? Small change in genetype changes phenotype

CA rule mutations negate swap rule

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!

Sum of natural numbers 1… N algorithm 1: 1+2+…+N algorithm2: N(N+1)/2 (Gaussian formula) Sum of natural numbers 1… N algorithm 1: 1+2+…+N algorithm2: N(N+1)/2 (Gaussian formula) Predictability Nth prime number algorithm 1: trial and error algorithm2: ? no general formula Nth prime number algorithm 1: trial and error algorithm2: ? no general formula easyVery tough! Given a number N predict the sum of numbers from 1 to N. Given a number N predict the N-th prime number

CA: there is no effective prediction method for patterns pattern at t+  t pattern at t behavior 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 imply similarity of patterns 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

Aspects of CA morphogenesis complex relationship between „genotype“ and „phenotype“ pheneseffects of „genes“ are not localizable in specific phenes (pleiotropy) epistasisphenes 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

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)

Where is the phene? Typification? Comparative measures? –length –density –fractal dimension Spatio-temporal development? phenotype A phenotype B All active research areas!