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Cellular Automata Cellular Automata Mus270C Cellular Automata • Discrete Dynamical System – Space and 9me • Dimension – 1-D, 2-D, … • States – Neighborhood and neighbors • Rules 2 1-D Cellular Automata • Neighborhood 3 1-D Cellular Automata • Neighborhood • Black & White neighbors (States) 4 1-D Cellular Automata • Neighborhood • Black & White neighbors (States) • Rules 5 1-D Cellular Automata • Neighborhood • Black & White neighbors (States) • Rules • Example t = 1 6 1-D Cellular Automata • Neighborhood • Black & White neighbors (States) • Rules • Example t = 1 t = 2 7 1-D Cellular Automata • Neighborhood • Black & White neighbors (States) • Rules • Example t = 1 t = 2 8 1-D Cellular Automata • Neighborhood • Black & White neighbors (States) • Rules • Example t = 1 t = 2 9 1-D Cellular Automata • Neighborhood • Black & White neighbors (States) • Rules • Example t = 1 t = 2 10 1-D Cellular Automata • Neighborhood • Black & White neighbors (States) • Rules • Example t = 1 t = 2 11 Wolfram Code Every cell and its two neighbors will be one of the following types Represent a black cell with 1 and a white cell with 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 Every yellow cell above can be filled out with a 0 or a 1 giving a total of 2 8=256 possible update rules. This allows any string of eight 0s and 1’s to represent a dis9nct update rule Example: Consider the string 0 1 1 0 1 0 1 0 it can be taken to represent the update rule 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 1 0 1 0 1 0 Or equivalently Now think of 01101010 as the binary expansion of the number 0 27+ 1 26+ 1 25+ 0 24+1 23+ 0 22+ 1 21+ 0 20 = 64+32+8+2=106. So the update rule is rule # 106 Rule #45=32+8+4+1 = 0 27+ 0 26+ 1 25+ 0 24+1 23+ 1 22+ 0 21+ 1 20 =0 0 1 0 1 1 0 1 Rule #30=16+8+4+2 = 0 27+ 0 26+ 0 25+ 1 24+1 23+ 1 22+ 1 21+ 0 20 =0 0 0 1 1 1 1 0 This is a naming conven9on of the 256 dis9nct update rules by Stephen Wolfram, New Kind of Science. Notable rules in this class include rule 30, rule 110, and rule 184. Rule 90 is also interes9ng because it creates Pascal's Triangle modulo 2. Rule 110 is known to be Turing complete. This implies that, in principle, any calculaon or computer program can be simulated using this automaton. hp://en.wikipedia.org/wiki/Elementary_cellular_automaton Rule 30 Rule 90 Classes of cellular automata (Wolfram) Class 1: aer a finite number of 9me steps, the CA tends to achieve a unique state from nearly all possible star9ng condi9ons (limit points) Class 2: the CA creates paerns that repeat periodically or are stable (limit cycles) – probably equivalent to a regular grammar/finite state automaton Class 3: from nearly all star9ng condi9ons, the CA leads to aperiodic- chao9c paerns, where the stas9cal proper9es of these paerns are almost iden9cal (aer a sufficient period of 9me) to the star9ng paerns (self-similar fractal curves) – computes ‘irregular problems’ Class 4: aer a finite number of steps, the CA usually dies, but there are a few stable (periodic) paerns possible (e.g. Game of Life) - Class 4 CA are believed to be capable of universal computaon Conway’s Game of Life live dead 18 Conway’s Game of Life live dead 1. Any dead cell with exactly three live neighbors comes to life. 19 Conway’s Game of Life live dead 1. Any dead cell with exactly three live neighbors comes to life. 20 Conway’s Game of Life live dead 1. Any dead cell with exactly three live neighbors comes to life. 21 22 23 CA in AthenaCL • Standard: f{s} Discrete cell values, rules match cell formaons (neighborhoods) • Totalis9c: f{t} Discrete cell values, rules match the sum of the neighborhood • Connuous: f{c} Real-number cell values within unit interval, rules specify values added to the average of previous cell formaon • Float: f{f} Like con9nuous, but implemented with floats (it makes a difference) CA in AthenaCL • For f{s,t}: the k value provides the number of possible values • For f{c,f}: the k value is zero • r: number of neighbor cells taken into account (neighborhood = 2r+1) • x: size of CA space • y: number of generaons • sub-table is defined with width (w), center (c), and skip (s) • i: ini9alizaon as center or random Examples • auca f{s} 380 0 • auca f{s} 379 0 • auca f{t} 37 0 • auca f{t} 39 0 • auca f{c} .8523 0 • auca f{f} .254 0 • tpmap 100 cl,f{s},12,0,fria • auca f{s} 12 0 cl: use ac9ve cell index posi9ons to indicate ac9ve posi9ons of a scale C.1.17. caList (cl) fria: flatRowIndexAc9ve sra: sumRowAcve Example (cont.) emo m pin a 5@0|7@2,c2,c7 tmo ha 9n a 27 e d0 c,0 9e d1 cl,f{s},30,0,fria,oc e d2 c,1 e d3 c,1 e r pt,(c,1),(c,1),(c,1),(c,1) Creang a direct mapping • A common technique extracts two dimensional data from ac9ve (non-zero) cells: table columns are treated as discrete pitch values, or a scale, and table rows are treated as discrete 9me steps. An ac9ve cell is interpreted as an event with a pitch specified by the cell’s column; this pitch is sustained for as many rows as the specific column is ac9ve. 9e d1 cl,f{s},30,0,fria,oc 9e d2 cl,f{s},30,0,sra,oc 9e d3 cl,f{s},30,0,sra,oc tpmap 100 cl,f{s},30,0,fria,oc tpmap 100 cl,f{s},30,0,sra,oc See also • F.1.20. script07a.py: The CA as a Generator of Melodies • F.1.21. script07B.py: The CA as a Generator of Rhythms Addendum: Sieve • Residual class consists of two integer values, a modulus (M) and a shid (I): M@I • Logic operators are used to combine residual classes “|” OR, “&” AND, “^” XOR, “-” NOT 3@0 = [ . , –6, –3, 0, 3, 6, . ] 3@0 | 4@0 = [ . , 0, 3, 4, 6, 8, 9, 12, . ] 3@0 & 4@0 = [ . , 0, 12, 24, . ] 3@0 ^ 4@0 = [ . , –3, 3, 4, 6, 8, 9, 15, . ] Logic vs. Sets • XOR: (¬P∧Q)∨(P∧¬Q) • Symmetric Difference: (A∩BC)∪(B∩AC) • Let A△B denote the symmetric difference of the sets A and B. Given an object x, x∈A△B⟺(x∈A) XOR (x∈B). • In general, one has a correspondence between statements in set theory and statements in logic, e.g. • x∈A∪B⟺(x∈A) OR (x∈B) • x∈A∩B⟺(x∈A) AND (x∈B) • x∈Ac⟺NOT (x∈A) .
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