Cellular%20Automata%20and%20Communication%20Complexity

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Cellular Automata and Communication Complexity

• Ivan Rapaport
• CMM, DIM, Chile

Christoph Dürr LRI, Paris-11, France
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Cellular Automata

• Local rules

Global dynamics
Wolfram numbered 0 to 255
fn(x,c,y)
f(x,c,y)
c?0,1
n
x,y?0,1n
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
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
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Matrices

• Fix center c0 (restrict to a single family of
  matrices)
• Possible measures
• number of different rows (rn)
• number of different columns (cn)
• rank
• discrepancy
• ...

Do these measures tell something about the
cellular automata?
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Communication Complexity

• Def necessary number of communication bits in
  order to compute a function when each party knows
  only part of the input

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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
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One-round communication
f(x,y)
Alice says x?1,2 or x?3,4,5
Alice says x1 or x2
Alice says x3 or x?4,5
Alice says x4 or x5
f 0
Bob knows
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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
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Example rule 105

• Dynamics
• Matrix

time
?
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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 bfn(x,0,0..0)
• computes b ? fn(0..0,0,y)fn(x,0,y)

?

?
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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
rn2,2,2,2,... cn2,2,2,2,...
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Different sequences (rn)
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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 linear rn a1na0
• 11, 14, 23, 33, 35, 43, 44, 50, 56, 58, 74, 77,
  142, 164, 168, 178, 184, 232
• Polynomial rn??(poly(n))
• 6, 9, 18, 22, 25, 26, 37, 40, 41, 54, 57, 62,
  73, 94, 104, 110, 122, 126, 134, 146
• Exponential rn??(2n)
• 30, 45, 106

Mostly experimental classification
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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

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Limited sensibility

• Example rule 172

f n(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
rn2,2,2,2,... cn2,2,2,2,...
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Cell. autom. with rn linear

• Example rule 23
• rn2,3,4,5,6,7,8,9,10,11,12,...

Matrix
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Other examples
Rule 184
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An interesting matrix
f(x,y)

• The function compares the lengths of the longest
  prefix in 1 of x and y

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Rule 132
center c1 rn2,3,4,...
center c0 rn1,1,1,...
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
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Realtime simulation
A is simulated by B in realtime if there are
constants l,k and an injection h0,1l?0,1k
such that h(fA(x,c,y))fB(h(x),h(c),h(y))
h
l
k
Joint work with Guillaume Thessier
0, 8, ...
rn constant
54, 110, ...
rn polynomial
30, 45, ...
rn exponential
lt
lt
If A is simulated by B in realtime then class(A)
? class(B)
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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