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 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 4

C LASS 3 Rule 184 5

C LASS 4 Universal computation! 6

R ULE 110 7

R ULE 110: S PACE -T IME SCALES 8

D EPENDENCE ON THE INITIAL STATE Dependence ~ Elaboration of initial conditions Structure does not depend but lines do No dependence  Identification of parameter of structure Trivial elaboration Strong dependence Complex elaboration,  Chaotic behaviors  Hard to predict 9

D EPENDENCE ON INITIAL STATE 10

L YAPOUNOV 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 11

U NIVERSAL COMPUTATION 12

U NIVERSAL COMPUTATION 13

L OCAL COMMUNICATIONS VS . GLOBAL BEHAVIORS 14

I NVERSE PROBLEM 15

R ULE 184: P ARITY PROBLEMS Why could 184 be a good candidate for parity detection 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 : K. M. Lee, Hao Xu, and H. F. Chau , Parity problem with a cellular automaton solution , Phys. Rev. E 64 , 026702, 2001 16