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L ECTURE 12: C ELLULAR A UTOMATA 2 / D ISCRETE -T IME D YNAMICAL S - - PowerPoint PPT Presentation
L ECTURE 12: C ELLULAR A UTOMATA 2 / D ISCRETE -T IME D YNAMICAL S - - PowerPoint PPT Presentation
15-382 C OLLECTIVE I NTELLIGENCE S18 L ECTURE 12: C ELLULAR A UTOMATA 2 / D ISCRETE -T IME D YNAMICAL S YSTEMS 4 I NSTRUCTOR : G IANNI A. D I C ARO A ZOO OF BEHAVIORS : A NY REGULARITY ? 2 C LASS 1 3 C LASS 2 Rule 2 The direction and
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CLASS 1
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CLASS 2
≪
Rule 2
The direction and location of the lines depend on the initial conditions, but the structural fact that we will have lines in a certain direction is independent from initial conditions Sierpinski gasket
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CLASS 3
Rule 184
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CLASS 4
Universal computation!
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RULE 110
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RULE 110: SPACE-TIME SCALES
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DEPENDENCE ON THE INITIAL STATE
Dependence ~ Elaboration of initial conditions
No dependence Trivial elaboration Structure does not depend but lines do Identification of parameter of structure Strong dependence Chaotic behaviors Complex elaboration, Hard to predict
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DEPENDENCE ON INITIAL STATE
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LYAPOUNOV EXPONENTS
Two Lyapounov exponents: measuring information propagation on initial conditions along the two directions
Both 0 exponents, information doesn’t travel Positive exponents, initial information travels far away Positive exponents, going to zero
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UNIVERSAL COMPUTATION
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UNIVERSAL COMPUTATION
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LOCAL COMMUNICATIONS VS. GLOBAL BEHAVIORS
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INVERSE PROBLEM
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RULE 184: PARITY PROBLEMS
No single CA can solve the parity problem, but applying a sequence of elementary CAs can do it, for instance the following operator applied to a lattice of length 𝑀: Why could 184 be a good candidate for parity detection problems?
- K. M. Lee, Hao Xu, and H. F. Chau, Parity problem with a cellular automaton solution, Phys. Rev. E 64,
026702, 2001