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

Cellular Automata (Reading: Chapter 10, Complexity: A Guided Tour ). 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. What is a cellular automaton?.

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

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  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 = 1 time = 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. A “glider” Demo: http://golly.sourceforge.net

  7. Netlogo models library: Computer science –> Cellular Automata –> Life Go through code See http://www.bitstorm.org/gameoflife/ See 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. Is there a general way (a “definite procedure”) to predict the behavior of Life from a given initial configuration? • 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.htm

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

  13. Elementary cellular automataOne-dimensional, two states (black and white) Rule:

  14. Elementary cellular automataOne-dimensional, two states (black and white) Rule:

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

  16. Elementary cellular automataOne-dimensional, two states (black and white) Rule:

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

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

  19. http://mathworld.wolfram.com/ElementaryCellularAutomaton.htmlhttp://mathworld.wolfram.com/ElementaryCellularAutomaton.html See Netlogo models library –> Computer Science –> Cellular Automata –> CA 1D Elementary

  20. 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.

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

  22. Transfer of information: movingparticles From http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html

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

  24. Transfer of information: movingparticles • Integration of information from different spatial locations: particle collisions From http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html

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

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

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

  28. 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

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

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