15-382 C OLLECTIVE I NTELLIGENCE – S19 L ECTURE 17: C ELLULAR A UTOMATA 2 / D ISCRETE -T IME D YNAMICAL S YSTEMS 5 TEACHER : G IANNI A. D I C ARO
D ESIGN CHOICES TO STUDY CA S 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 § 2
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 0 otherwise 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 § 3
1D B OOLEAN 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 § 4 = range = 5 6 /2 (assuming a symmetric neighborhood) function 0 § ! 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 2 12, = 8 § If 4 = 1, ! = 2 à 256 possible Boolean evolution functions § If 4 = 2, ! = 2 à 4 @ 10 A possible Boolean functions! 4
E LEMENTARY CA: W OLFRAM 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 Wolfram code Rule 30 : (00011110) à 30 5
S OME RULES … 6
S TUDYING CA S : NON - LINEAR BUSINESS AS USUAL Direct problem (Prediction): Given the function, what’s the behavior? 7
R ULE 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) 8
A ZOO OF BEHAVIORS : A NY REGULARITY ? 9
C LASS 1 10
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 11
C LASS 3 Rule 184 12
C LASS 4 Universal computation! 13
R ULE 110 14
R ULE 110: S PACE -T IME SCALES 15
D EPENDENCE ON THE INITIAL STATE Dependence ~ Elaboration of initial conditions Structure does not depend but lines do No dependence à Identification of parameter of structure Trivial elaboration Strong dependence Complex elaboration, à Chaotic behaviors à Hard to predict 16
D EPENDENCE ON INITIAL STATE 17
L YAPUNOV 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 18
U NIVERSAL COMPUTATION 19
U NIVERSAL COMPUTATION 20
L OCAL COMMUNICATIONS VS . GLOBAL BEHAVIORS 21
I NVERSE PROBLEM 22
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 ! : K. M. Lee, Hao Xu, and H. F. Chau , Parity problem with a cellular automaton solution , Phys. Rev. E 64 , 026702, 2001 23
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