L ECTURE 17: C ELLULAR A UTOMATA 2 / D ISCRETE -T IME D YNAMICAL S - - PowerPoint PPT Presentation

LECTURE 17: CELLULAR AUTOMATA 2 / DISCRETE-TIME DYNAMICAL SYSTEMS 5

TEACHER:

GIANNI A. DI CARO

15-382 COLLECTIVE INTELLIGENCE – S19

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DESIGN CHOICES TO STUDY CAS

§ Being multidimensional iterated maps, CAs are very complex entities, therefore, to study them, let’s make a few reasonably simplifying assumptions: § Homogeneous CAs: § Lattice is a regular grid, in 1D or 2D § All cell functions ! have the same (relatively simple) neighborhood mapping "(!) à they all have the same number of neighbors defined according to the lattice § All cell functions have the same state transition function, %(!, " ! ) § States are encoded in a few bits, typically, 2 or 3 § Synchronous updating

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1D CA

§ Simplest case: State variables / Cells are Boolean units, ! = {0,1} § The neighborhood of a cell (), * () corresponds to the one or two closest neighbors in both left and right directions § à Transition function + is a Boolean function of , = 3 or , = 5 arguments +((), ()01, ()21, ()03, ()23) = 51 if () + ()01 + ()21 + ()03 + ()23 > 2

• therwise

§ à A 1D Boolean CA with , cells is an ,-dimensional binary vector B(t), the state vector of the CA, that evolves over time by the iterated application of the map + : B t + 1 = +(B t ) § State space of the CA: All possible configurations of the vector B

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1D BOOLEAN CA, SOME NUMBERS

§ ! = # = number of (cell) states § # = {0,1} à ! = 2 § * = number of cells à 2+ possible configurations of CA’s state vector, § * = 100, ! = 2 à 2,-- ≈ 10/- !!!!

One specific function 0

212, = 8

§ 4 = range = 5 6 /2 (assuming a symmetric neighborhood) § !182, = !|5|2, possible configurations of neighbor set § If 4 = 1, ! = 2 à 8 possible neighbor configurations § If 4 = 2, ! = 2 à 32 possible neighbor configurations § !:;<=> = !:|5|=> = possible evolution functions for the CA § If 4 = 1, ! = 2 à 256 possible Boolean evolution functions § If 4 = 2, ! = 2 à 4 @ 10A possible Boolean functions!

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ELEMENTARY CA: WOLFRAM CODE

§ ! = 0,1 , & = 1 à ' = 2, ) + 1 = 8, 256 possible Boolean functions

Transition function . (rule of the CA) Example: Rule 30 This is a bit string à Decimal number Rule 30: (00011110) à 30 Wolfram code

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SOME RULES …

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STUDYING CAS: NON-LINEAR BUSINESS AS USUAL

Direct problem (Prediction): Given the function, what’s the behavior?

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RULE 30

Class 3 cellular automata: overall the evolution presents regularities, however, the state sequence generated by the central cell is used as random generator in Mathematica! (randomness deriving from a purely deterministic process with no external ’noisy’ inputs)

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A ZOO OF BEHAVIORS: ANY REGULARITY?

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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 à Identification of parameter of structure Strong dependence à Chaotic behaviors Complex elaboration, à Hard to predict

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DEPENDENCE ON INITIAL STATE

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LYAPUNOV EXPONENTS

Two Lyapunov 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 !: 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