Evolving Transition Rules for Multi Dimensional Cellular Automata

1. Evolving Transition Rules for Multi Dimensional Cellular Automata University of Leiden (LIACS) Ron Breukelaarrbreukel@liacs.nl

2. Evolving Transition Rules for Multi DimensionalCellular Automata

3. Evolving Transition Rules for Multi DimensionalCellular Automata Cellular Automata

4. Evolving Transition Rules for Multi DimensionalCellular Automata Cellular Automata Transition Rules for Cellular Automata

5. Evolving Transition Rules for Multi DimensionalCellular Automata Cellular Automata Transition Rules for Cellular Automata Evolving Transition Rules for Cellular Automata

6. Evolving Transition Rules for Multi DimensionalCellular Automata Cellular Automata Transition Rules for Cellular Automata Evolving Transition Rules for Cellular Automata Multi Dimensional Cellular Automata

7. Evolving Transition Rules for Multi DimensionalCellular Automata Cellular Automata Transition Rules for Cellular Automata Evolving Transition Rules for Cellular Automata Multi Dimensional Cellular Automata Transition Rules for Multi Dimensional Cellular Automata

8. Evolving Transition Rules for Multi DimensionalCellular Automata Cellular Automata Transition Rules for Cellular Automata Evolving Transition Rules for Cellular Automata Multi Dimensional Cellular Automata Transition Rules forMulti Dimensional Cellular Automata Evolving Transition Rules forMulti Dimensional Cellular Automata

9. Evolving Transition Rules for Multi Dimensional Cellular Automata Cellular Automata C = {a1, a2, …, an} a1 a2 a3 a4 a5 a6 a7 a8 an an is linked to a1 ai {0,1} 2n different states of C

10. Evolving Transition Rules for Multi Dimensional Cellular Automata r r ai Cellular Automata si si is the neighborhood of ai with r as radius (here r = 3)

11. Evolving Transition Rules for Multi Dimensional Cellular Automata 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 1 1 0 0 1 0 Cellular Automata Ct = state of CA at time t C0 C1 C2 C3

12. Evolving Transition Rules for Multi Dimensional Cellular Automata 0 0 1 0 1 0 1 1 0 Cellular Automata Ct = state of CA at time t C0 : {0,1}2r+1 {0,1}(si)  ai C1 1 1 0 0 0 1 0 1 1

13. Evolving Transition Rules for Multi Dimensional Cellular Automata 0 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 Transition Rules 0 1 0 1 0 1 0 0 1 1 0 …10010111011010…22r+1 bits in rule

14. Evolving Transition Rules for Multi Dimensional Cellular Automata Majority Problem • = relative number of ones in C0 Task: > 0.5  iterate to ‘all ones’ state  < 0.5  iterate to ‘all zeros’ state within maximum I iterations.

15. Evolving Transition Rules for Multi Dimensional Cellular Automata Majority Problem (GKL) (81,6% correct classifications)

16. Evolving Transition Rules for Multi Dimensional Cellular Automata M. Mitchell, J.P. Crutchfield, P.T. Hraber • A Genetic Algorithm to evolve the rules: • A ‘pool’ of 100 transition rules • Evaluation by iterating CA on 100 random initial states uniformly dist. over nr. of ones • Selecting 10% to survive every generation • Generating the other 90% using crossover on the selected 10% and then mutation • r = 3, therefore 27 = 128 bits in rule and2128 possible rules

17. Evolving Transition Rules for Multi Dimensional Cellular Automata M. Mitchell, J.P. Crutchfield, P.T. Hraber Fn,m = the relative number of correct classifications out of m initial states with a width of n cells.

18. Evolving Transition Rules for Multi Dimensional Cellular Automata M. Mitchell, J.P. Crutchfield, P.T. Hraber (copied experiment) (a) and (b) are block expanding rules(c) and (d) are particle based rules

19. Evolving Transition Rules for Multi Dimensional Cellular Automata M. Mitchell, J.P. Crutchfield, P.T. Hraber (copied experiment) Block expanding rules Very bad rules Particle communication based rules

20. Evolving Transition Rules for Multi Dimensional Cellular Automata M. Mitchell, J.P. Crutchfield, P.T. Hraber (copied experiment)

21. Evolving Transition Rules for Multi Dimensional Cellular Automata Two Dimensional Cellular Automata von Neumann neighborhood Moore neighborhood

22. Evolving Transition Rules for Multi Dimensional Cellular Automata Two Dimensional Cellular Automata r = 1 r = 2 r = 3

23. Evolving Transition Rules for Multi Dimensional Cellular Automata Multi Dimensional Cellular Automata In a CA with d dimensions {e1, e2, …, ed} and connected borders von Neumann neighborhood: Moore neighborhood:

24. Evolving Transition Rules for Multi Dimensional Cellular Automata Number of cells

25. Evolving Transition Rules forMulti Dimensional Cellular Automata Transition Rules • Rules are defined the similar way as in one dim. CA: • Cells are numbered • : {0,1}n {0,1} , n = S(d, r) • Bitstring length explodes in high dimensions22S(d, r)bits in a rule. Only small number of dimensions or small radius look seem doable. • Experiments will focus on two dimensional CA withr = 1 to simplify them.

26. Evolving Transition Rules for Multi Dimensional Cellular Automata Genetic Algorithm • Improved Genetic Algorithm: • Using tournament selection. • Using a gliding distribution instead of uniform. • Using crossover for only 60% of the population.

27. Evolving Transition Rules forMulti Dimensional Cellular Automata Experiments • Four multi-dimensional experiments: • Majority Problem with von Neumann neighborhood to compare two and three dim. with one dim. • AND and XOR problem to better show communication in 2D CA. • Checkerboard Problem to test robustness of algorithm and again compare dimensionality. • Bitmap generation to show the potential of multi dimensional CA and work towards a real-time application.

28. Evolving Transition Rules forMulti Dimensional Cellular Automata Multi Dimensional Majority Problem • The same pool size, evaluation method, selection method, crossover and mutation for d=1, 2 or 3. • CA with linked borders. • A von Neumann neighborhood with r = 1. For d=2: 25 = 32 bits in rule and 232 possible rules for d=2.(This is 296 times smaller then one dim.)

29. Evolving Transition Rules forMulti Dimensional Cellular Automata Multi Dimensional Majority Problem 1D 3D

30. Evolving Transition Rules forMulti Dimensional Cellular Automata Multi Dimensional Majority Problem

31. Evolving Transition Rules forMulti Dimensional Cellular Automata Checkerboard Problem Given a random initial state: Generate a checkerboard pattern. (alterning blank and white in every direction) Note that the CA must have even dimensions

32. Evolving Transition Rules forMulti Dimensional Cellular Automata Checkerboard Problem 1D

33. Evolving Transition Rules forMulti Dimensional Cellular Automata Checkerboard Problem 2D

34. Evolving Transition Rules forMulti Dimensional Cellular Automata Checkerboard Problem

35. Evolving Transition Rules forMulti Dimensional Cellular Automata AND and XOR Problem Given two special input cells v1 and v2 : AND problem: If (v1 ANDv2 == TRUE)  iterate to an ‘all ones’ stateelse  iterate to an ‘all zeros’ state. XOR problem: If (v1XOR v2 == TRUE)  iterate to an ‘all ones’ stateelse  iterate to an ‘all zeros’ state.

36. Evolving Transition Rules forMulti Dimensional Cellular Automata AND and XOR Problem v2 v1 • Borders of the CA are unlinked to increase the distance between v1 andv2. • I was set to 10 to increase challenge.

37. Evolving Transition Rules forMulti Dimensional Cellular Automata AND Problem (a) von Neumann, (b) Moore

38. Evolving Transition Rules forMulti Dimensional Cellular Automata XOR Problem (a) von Neumann, (b) Moore

39. Evolving Transition Rules forMulti Dimensional Cellular Automata Bitmap generation Given a target bitmap and an initial state: Find a transition rule that generates the target bitmap from the initial state. Initial state: Five target bitmaps:(5 x 5)

40. Evolving Transition Rules forMulti Dimensional Cellular Automata Bitmap generation

41. Evolving Transition Rules forMulti Dimensional Cellular Automata Bitmap generation

42. Evolving Transition Rules forMulti Dimensional Cellular Automata Bitmap generation A von neumann neighborhood trying to spell the name ‘RON’ in a 11x5 CA and ending up having only 3 errors.

43. Evolving Transition Rules forMulti Dimensional Cellular Automata Bitmap generation A CA using a Moore neighborhood generating a 9 x 9 image that looks like a gecco.

44. Conclusions • From the results can be concluded: • Multi dimensional CA are able to solve the Majority Problem with results similar to the one dimensional CA, but with shorter duration times and with d=2: smaller neighborhood. • Multi dimensional CA can be used to evolve transition rules that exhibit communicational behaviour. • Multi dimensional CA can be trained to exhibit very diverse behaviour and might well have real-world applications in Parallel Computing and modelling social / biological behaviour.

45. Further Work • Some ideas on continuation of this project: • Explore how far Bitmap Generation is possible. (is compression an option?) • Try this approach on real-world applications. • Investigate how other forms of crossover influence the results. • …

46. Questions? Any questions / ideas?