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Cellular Automata and Communication Complexity

Cellular Automata and Communication Complexity. Ivan Rapaport CMM, DIM, Chile. Christoph Dürr LRI, Paris-11, France. c. 00. 00. 0. 0. 0. 0. 00. x. y. 01. 01. 0. 0. 0. 0. 0. 0. 10. 0. 0. 1. 1. 10. 10. 1. 1. 1. 1. 1. 1. 01. 1. 1. 1. 1. 11. 11. 11. 1. 1.

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Cellular Automata and Communication Complexity

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  1. Cellular Automata and Communication Complexity Ivan Rapaport CMM, DIM, Chile Christoph Dürr LRI, Paris-11, France

  2. c 00 00 0 0 0 0 00 x y 01 01 0 0 0 0 0 0 10 0 0 1 1 10 10 1 1 1 1 1 1 01 1 1 1 1 11 11 11 1 1 1 00 1 0 10 01 11 Cellular Automata Global dynamics Local rules Wolfram numbered 0 to 255 fn(x,c,y) f(x,c,y) c{0,1} n x,y{0,1}n x c y x,c,y{0,1} Example: rule 54 0 1 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0

  3. Fix center c=0 (restrict to a single family of matrices) Possible measures number of different rows (rn) number of different columns (cn) rank discrepancy ... Matrices Do these measures tell something about the cellular automata?

  4. y x f(x,y) f(x,y) Communication Complexity Def: necessary number of communication bits in order to compute a function when each party knows only part of the input

  5. 1 2 3 4 5 y f 1 2 3 4 5 1 0 0 0 1 0 1 1 1 0 x 1 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 if x=20 if x=1 1 if x=10 if x=2 1 if x=30 else 1 if x=40 else f= f= f= f= Example Alice says x{1,2} or x{3,4,5} Bob says y{1,5} or y{2,3,4} Bob says y{1,2} or y{3,4,5} Alice knows

  6. 1 2 3 4 5 y f 1 2 3 4 5 1 0 0 0 1 0 1 1 1 0 x 1 1 0 0 0 1 1 0 0 1 0 0 0 0 0 x y 1 if x{3,4,5}0 else 1 if x{1,2}0 else 1 if x{2,3,4}0 else f= f= f= 1 if x{1,5}0 else f= One-round communication f(x,y) Alice says x{1,2} or x{3,4,5} Alice says x=1 or x=2 Alice says x=3 or x{4,5} Alice says x=4 or x=5 f= 0 Bob knows

  7. Communication Complexity Complexity measures are captured by measures on the matrix defined by f 1-round comm. comp.  min(rn,cn)comm. comp.  rankdistributional comm. comp.  discrepancy Communication Complexity Eyal Kushilevitz and Noam NisanCambridge Univ. Press, 1997

  8. Example: rule 105 Dynamics Matrix time ?

  9. x x c c y y x’ x’ c’ c’ y’ y’ Communication protocol for additive rules Def: Automaton f is additive if n , fn(x,c,y)  fn(x’,c’,y’) = fn(xx’,cc’,yy’) Protocol: computes and communicates b=fn(x,0,0..0) computes b  fn(0..0,0,y)=fn(x,0,y)  = 

  10. Rule 105 is additive f105(x,c,y) = x  c  y  1 A single bit has to be communicated so there are only 2 different rows rn=2,2,2,2,... cn=2,2,2,2,...

  11. Different sequences (rn)

  12. By-product: a classification Constant rn(1) 0, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 19, 24, 27, 28, 29, 32, 34, 36, 38, 42, 46, 51, 60, 71, 72, 76, 78, 90, 105, 108, 128, 130, 132, 136, 138, 140, 150, 152, 154, 156, 160, 162, 170, 172, 200, 204 Exact linearrn= a1·n+a0 11, 14, 23, 33, 35, 43, 44, 50, 56, 58, 74, 77, 142, 164, 168, 178, 184, 232 Polynomialrn(poly(n)) 6, 9, 18, 22, 25, 26, 37, 40, 41, 54, 57, 62, 73, 94, 104, 110, 122, 126, 134, 146 Exponentialrn(2n) 30, 45, 106 Mostly experimental classification

  13. Cell. autom. with rn constant Constant by additivity 15, 51, 60, 90, 105, 108, 128, 136, 150, 160, 170, 204 Constant by limited sensibility 0, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 19, 28, 29, 34, 36, 38, 42, 46, 72, 76, 78, 108, 138, 140, 172, 200 Constant for any other reason 27, 32, 130, 132, 152, 154, 156, 162

  14. Limited sensibility Example rule 172 fn(x,c,y) depends only on a fixed number of cells (bits) in x x y A constant number of bits has to be communicated so there are only a constant number of different rows rn=2,2,2,2,... cn=2,2,2,2,...

  15. l k Protocol: communicates k knows that after min(k,l) the cell alternates Cell. autom. with rnlinear Example: rule 23 rn=2,3,4,5,6,7,8,9,10,11,12,... Matrix

  16. Rule 164 Rule 33 Rule 44 Rule 50 Rule 14 Rule 35 Rule 168 Other examples Rule 184

  17. 110... 10... 0... ...0 ...01 ...011 An interesting matrix The function compares the lengths of the longest prefix in 1* of x and y f(x,y)

  18. Rule 132 center c=0 rn=1,1,1,... center c=1 rn=2,3,4,... A white cell remains white forever A black cell is part of a block. Blocks shrink by two cells at each step, exept the isolated black cell. Only even width blocks will vanish. Protocol: Compare k and l k l

  19. Realtime simulation A is simulated by B in realtime if there are constants l,k and an injection h:{0,1}l{0,1}k such that h(fA(x,c,y))=fB(h(x),h(c),h(y)) h l k Joint work with Guillaume Thessier rn constant 0, 8, ... rn polynomial 54, 110, ... rn exponential 30, 45, ... < < If A is simulated by B in realtime then class(A)  class(B)

  20. To do • Prove the behavior of rn for remaining rules • Compare with Wolfram classification • Consider many round communication complexity • Study the rank of the matrices • Study the discrepancy • Analyse quasi-randomness of matrices

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