1. Cellular Automata (Reading: Chapter 10, Complexity: A Guided Tour)

2. What is a cellular automaton? light bulbs pictures relation to Turing machines – “non-von-Neumann-style architecture” invented by von Neumann CAs and universal computation

3. What is a cellular automaton? Circular (“toroidal”) boundary conditions

4. time = 1time = 2

5. Example: Game of Life (John Conway, 1970s) Neighborhood: 2 dimensional 3x3 neighborhood: Rules: – A dead cell with exactly three live neighbors becomes a live cell (birth). – A live cell with two or three live neighbors stays alive (survival). – In all other cases, a cell dies or remains dead (overcrowding or loneliness).

6. Demo: http://golly.sourceforge.net A “glider”

7. Netlogo models library: Computer science –> Cellular Automata –> Life Go through code See http://www.bitstorm.org/gameoflife/http://www.bitstorm.org/gameoflife/ See http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

8. Is there a general way (a “definite procedure”) to predict the behavior of Life from a given initial configuration?

9. Relation to the Halting Problem.

10. Is there a general way (a “definite procedure”) to predict the behavior of Life from a given initial configuration? Relation to the Halting Problem. Answer: No.

11. Is there a general way (a “definite procedure”) to predict the behavior of Life from a given initial configuration? Relation to the Halting Problem. Answer: No. Reason “Life is Universal.” http://rendell- attic.org/gol/tm.htmhttp://rendell- attic.org/gol/tm.htm

12. Elementary cellular automata One-dimensional, two states (black and white)

13. Rule:

14. Elementary cellular automata One-dimensional, two states (black and white)

15. Rule: Elementary cellular automata One-dimensional, two states (black and white)

16. Rule:

17. Elementary cellular automata One-dimensional, two states (black and white) Rule:

18. Elementary cellular automata One-dimensional, two states (black and white) Rule:

20. http://mathworld.wolfram.com/ElementaryCellularAutom aton.html See Netlogo models library –> Computer Science –> Cellular Automata –> CA 1D Elementary

21. Wolfram’s Four Classes of CA Behavior Class 1: Almost all initial configurations relax after a transient period to the same fixed configuration (e.g., all black). Class 2: Almost all initial configurations relax after a transient period to some fixed point or some temporally periodic cycle of configurations, but which one depends on the initial configuration Class 3: Almost all initial configurations relax after a transient period to chaotic behavior. (The term ``chaotic'‘ here refers to apparently unpredictable space-time behavior.) Class 4: Some initial configurations result in complex localized structures, sometimes long-lived.

22. Rule: ECA 110 is a universal computer (Matthew Cook, 2002) Wolfram’s numbering of ECA: 0 1 1 0 1 1 1 0 = 110 in binary

23. – Transfer of information: moving particles From http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html

24. – Transfer of information: moving particles From http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html

25. – Transfer of information: moving particles – Integration of information from different spatial locations: particle collisions From http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html

26. – Transfer of information: moving particles – Integration of information from different spatial locations: particle collisions From http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html

27. Outline of proof 1.Define “cyclic tag systems” and prove they are universal (they can emulate Turing machines). 2.Show ECA 110 can emulate a cyclic tag system.

28. Wolfram’s hypothesis: All class 4 CAs can support universal computation

29. Outline of Wolfram’s A New Kind of Science (from MM review, Science, 2002) Simple programs can produce complex, and random-looking behavior – Complex and random-looking behavior in nature comes from simple programs. Natural systems can be modeled using cellular-automata-like architectures Cellular automata are a framework for understanding nature Principle of computational equivalence

30. Principle of Computational Equivalence 1.The ability to support universal computation is very common in nature. 2.Universal computation is an upper limit on the sophistication of computations in nature. 3.Computing processes in nature are almost always equivalent in sophistication.