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Cellular Automata, Part 2

Cellular Automata, Part 2. What is the relationship between the dynamics of cellular automata and their ability to compute? . control parameter. temperature. Phase transitions. steam water ice. gas fluid solid. chaotic complex ordered.

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Cellular Automata, Part 2

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  1. Cellular Automata, Part 2

  2. What is the relationship between the dynamics of cellular automata and their ability to compute?

  3. control parameter temperature Phase transitions steam water ice gas fluid solid chaotic complex ordered

  4. Hypothesis (Langton, Packard, Kauffman, others): Need maximally “fluid” state to maximize potential for • information processing • complexity of dynamics • ability to adapt

  5. Recall the “logistic map” model of population dynamicsOrder parameter is population growth rate, r

  6. Recall the “logistic map” model of population dynamicsOrder parameter is population growth rate, r Fixed point

  7. Recall the “logistic map” model of population dynamicsOrder parameter is population growth rate, r Fixed point Period 2

  8. Recall the “logistic map” model of population dynamicsOrder parameter is population growth rate, r Period 4

  9. Recall the “logistic map” model of population dynamicsOrder parameter is population growth rate, r Period 4 Period 8

  10. Recall the “logistic map” model of population dynamicsOrder parameter is population growth rate, r Period 16

  11. Recall the “logistic map” model of population dynamicsOrder parameter is population growth rate, r Period 16 Chaos The onset (or “edge”) of chaos is r = 3.569946

  12. Langton devised an “order parameter” for cellular automata called λ. For binary-state CAs, λ is defined as follows: The ‘edge of chaos’ in cellular automata(C. Langton, Physica D, 42:12-37, 1990)

  13. Rule 110 Rule: λ = ?

  14. Building on Wolfram’s classes of behavior for CA, Langton found evidence that cellular automata can be “ordered” or “chaotic” roughly according to lambda He showed evidence that the “complexity” of patterns formed by cellular automata is maximized at the transition between order and chaos He argued that the potential for computation must be maximized at this “phase transition” The ‘edge of chaos’ in cellular automata(C. Langton, Physica D, 42:12-37, 1990)

  15. http://math.hws.edu/xJava/CA/EdgeOfChaosCA.html

  16. How to define “potential for computation”?

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