Lectures on Cellular Automata Continued

1. Lectures on Cellular Automata Continued Modified and upgraded slides of Martijn Schut schut@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 energy matter heat entropy dissipator

4. Model animals as dynamical systems Tissues Groups Space Dynamical system Energy Time

5. morphometric space S A Q=S/A Which space? - which geometry? geometric space y time x ? state space parameter space velocity b a q

6. morphometric space 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.

7. 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

8. Cellular Automaton In standard CA the values of cells are discretized t neighbours update rules t+1

9. 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

10. Effects of simplification CA mo d e l Mor ph oge n e si s S pa ti a l d is c r e ti za ti on ~ l e ve l o f d e t ai l ( o r g a ni sm , ce l l, mo le c ul e ) T e m pora l d i sc r e ti za ti on ~ l e ve l o f d e t ai l ( cel l c y cl e , r eac t i o n r a t e) ® pro f ound e ff ec ts R e duc ti on o f d im en s ion ( 2 D “ G a m e of Li f e” ) B i na ri za ti on co m pu t a ti ona l conven i ence ® ® ® ® ( 0 1 A , 1 0 T, 0 0 G , 1 1 C ) nea r e st -ne i ghbou r ~ spa ti a l re st r i c ti on s i n t e r ac ti ons s im p l e upda t e ru l e s; ~ non- li nea rit y ® p r ofound e f fe ct s s im u lt ane it y , ub i qu it y

11. 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

12. 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

13. Code for rule 6 (00110) S neighbours 1 2 3 4 5 S(i,T+1) 0 0 1 1 0 Number of “alive” neighbors

14. 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

15. Temporal evolution initial configuration (t=0) e.g. random 0/1 rule n t=1 rule n t=2 rule n t=3

16. Temporal evolution t=0 t=10

17. Rule 6: 00110 t=0 t=100

18. Rule 10: 01010 t=0 t=100

19. static binary pattern 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 morphogenesis

20. update rule iterative application of the update rule CA state pattern state space genotype epigenetic interpretation of the genotype, morphogenesis phenotype morphospace CA: a model for morphogenesis

21. 0 1 0 1 0 CA rule mutations 1 0 0 1 0 swap genotype negate development phenotype 0 1 0 1 1 How genotype mutations can change phenotypes?

22. 0 1 1 1 0 1 1 0 1 0 1 0 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 t+Dt How to predict a next element in sequence? Tough!

24. 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!

25. CA: no effective pattern prediction pattern at t pattern at t+Dt behaviour 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 implysimilarity 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. 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 morphology

30. Where is the phene? phenotype A • Typification? • Comparative measures? • length • density • fractal dimension • Spatio-temporal development? phenotype B