1. 1 Cellular Automata What could be the simplest systems capable of wide-ranging or even universal computation? Could it be simpler than a simple cell?

2. GEK1530 2 Perhaps one can expect that strange and complex behavior results from very complicated rules. But what are the simplest systems that display complex behavior? This is an important question when we want to figure out whether relatively simple rules could underlie the complexity of life. As it turns out, probably the simplest systems that display complex behaviors are the so-called cellular automata. Cellular Automata

3. GEK1530 3 Born in 1959 in London First paper at age 15 Ph.D. at 20 Youngest recipient of MacArthur ‘young genius’ award Worked at Caltech and Princeton Owner of Mathematica (Wolfram Research) Fantastic publication record … until … 1988 when he stopped publishing in scientific journals From his web site … Stephen Wolfram Cellular Automata

4. GEK1530 4 A (one-dimensional) cellular automaton consists of a line of ‘cells’ (boxes) each with a certain color like e.g. black or grey and a rule on how the colors of the cells change from one time step to the next. Rule Line The first line is always given. This is what is called the ‘initial condition’. This rule is trivial. It means black remains black and grey remains grey. Time 0 Time 1 Time 2 This is how the Cellular Automaton evolves Cellular Automata

5. GEK1530 5 A (one-dimensional) cellular automaton consists of a line of ‘cells’ (boxes) each with a certain color like e.g. black or grey and a rule on how the colors of the cells change from one time step to the next. Rule Line The first line is always given. This is what is called the ‘initial condition’. Another simple rule. It means black turns into grey and grey turns into black. Time 0 Time 1 Time 2 This is how the Cellular Automaton evolves Cellular Automata

6. GEK1530 6 Like this, the rules are a bit boring of course because there is no spatial dependence. That is to say, neighboring cells have no influence. Therefore, let us look at rules that take nearest neighbors into account. or With 3 cells and 2 colors, there are 8 possible combinations. Cellular Automata

7. GEK1530 7 The 8 possible combinations: Of course, for each possible combination we’ll need to state to which color it will lead in the next time step. Let us look at a famous rule called rule 254 (we’ll get back to why it has this name later). Cellular Automata

8. GEK1530 8 Rule 254: We can of course apply this rule to the initial condition we had before but what to do at the boundary? Rule 254: Cellular Automata

9. GEK1530 9 Often one starts with a single black dot and takes all the neighbors on the right and left to be grey (ad infinitum). Now, let us apply rule 254. This is quite simple, everything, except for three neighboring grey cells will lead to a black cell. 254: Cellular Automata

10. GEK1530 10 Continuing the procedure: 254: Time 0 Time 1 Time 2 Time 3 Cellular Automata

11. GEK1530 11 Of course we don’t really need those arrows and the time so we might just as well forget about them to obtain: 254: Nice, but well … not very exciting. Cellular Automata

12. GEK1530 12 So let us look at another rule. This one is called rule 90. That doesn’t look like it’s very exciting either. What’s the big deal? Rule 90: Cellular Automata

13. GEK1530 13 Applying rule 90. 90: At least it seems to be a bit less boring than before…. After one time step: After two time steps: Cellular Automata

14. GEK1530 14 Applying rule 90. 90: Hey! This is becoming more fun…. After three time steps: Cellular Automata

15. GEK1530 15 Applying rule 90. 90: Hmmmm After four time steps: Cellular Automata

16. GEK1530 16 Applying rule 90. 90: It’s a Pac Man! After five time steps: Cellular Automata

17. GEK1530 17 Applying rule 90. 90: Which is a fractal! Well not really. It’s a Sierpinsky gasket: Cellular Automata

18. GEK1530 18 Applying rule 90. 90: Well not really. It’s a Sierpinsky gasket: From S. Wolfram: A new kind of Science. Cellular Automata

19. GEK1530 19 So we have seen that simple cellular automata can display very simple and fractal behavior. Both these patterns are in a sense highly regular. One may wonder now whether ‘irregular’ patterns can also exist. Rule 30: Surprisingly they do! Rule 30 Note that I’ve only changed the color of two boxes compared to rule 90. Cellular Automata

20. GEK1530 20 Applying rule 30. 30: Cellular Automata

21. GEK1530 21 Applying rule 30. 30: While one side has repetitive patterns, the other side appears random. From S. Wolfram: A new kind of Science. Cellular Automata

22. GEK1530 22 Now let us look at the numbering scheme The first thing to notice is that the top is always the same. This is the part that changes. Now if we examine the top more closely, we find that it just is the same pattern sequence that we obtain in binary counting. Cellular Automata

23. GEK1530 23 If we say that black is one and grey is zero, then we can see that the top is just counting from 7 to 0. Good. Now we know how to get the sequence on the top. Value 1 Value 4 Value 2 Value 1 Value 4 Value 2 = 4 = 3 Cellular Automata

24. GEK1530 24 How about the bottom? We can do exactly the same thing but since we have 8 boxes on the bottom it’s counting from 0 to 255. = 2+8+16+64 = 90 Value 128 Value 64 Value 16 Value 32 Value 8 Value 4 Value 2 Value 1 Cellular Automata

25. GEK1530 25 Like this we can number all the possible 256 rules for this type of cellular automaton. Cellular Automata

26. GEK1530 26 Like this we can number all the possible 256 rules for this type of cellular automaton. Cellular Automata

27. GEK1530 27 Like this we can number all the possible 256 rules for this type of cellular automaton. And of course, one does not need to restrict oneself to two colors and two neighbors … Cellular Automata

28. GEK1530 28 Wrapping up Key Points of the Day Give it some thought Can you think of any ‘real-life’ cellular automata? Simple dynamical rules can lead to complex behavior Cellular automaton rule 90: