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Cellular Automata and Communication

ComplexityIvan Rapaport

CMM, DIM, Chile

Christoph DürrLRI, Paris-11, France

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Cellular AutomataLocal rules Global dynamics

x c y

f(x,c,y)

x,c,y{0,1}

fn(x,c,y)

x,y{0,1}n

c{0,1}

x yc

n

0110

101

1 1

10111

0

10

10111

0

1000 01 1

1

0 0

0000

00

01 1

11000

0111

01

00

10

1100

1000

0111

11

11

10 00 01

11

10 00 0

01

00

10

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Example: rule 54

Wolfram numbered 0 to 255

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Matrices

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– ...

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

yx

f(x,y)f(x,y)

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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}

f=1 if x=10 if x=2

f=1 if x=20 if x=1

f=1 if x=30 else

f=1 if x=40 else

Alice knows

f1

2

3

4

5

1 2 3 4 5x

y

0

1 0 0 0 1

01 1 1

1 1 0 0 0

0 0

0 0 0 0 0

1 1 1

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One-round communicationf1

2

3

4

5

1 2 3 4 5x

y

0

1 0 0 0 1

01 1 1

1 1 0 0 0

0 0

0 0 0 0 0

1 1 1Alice says x{1,2} or x{3,4,5}

Alice says x=1 or x=2

Alice says x=3 or x{4,5}

Bob knows

Alice says x=4 or x=5

f=1 if x{1,5}0 else

f=1 if x{2,3,4}0 else

f=1 if x{1,2}0 else

f= 0f=1 if x{3,4,5}0 else

yx

f(x,y)

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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 ?

tim

e

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Def: Automaton f is additive if n ,fn(x,c,y) fn(x’,c’,y’) = fn(xx’,cc’,yy’)

Protocol: computes and communicates

b=fn(x,0,0..0) computes b fn(0..0,0,y)=fn(x,0,y)

Communication protocol for additive rules

x’ y’c’x yc

=

x’ y’c’x yc

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Rule 105 is additive

A single bit has to be communicated so there are only 2 different rows

rn=2,2,2,2,...

cn=2,2,2,2,...

f105(x,c,y) = x c y 1

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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= a1·n+a0 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 sensibility0, 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 reason27, 32, 130, 132, 152, 154, 156, 162

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Limited sensibility

Example rule 172

x y

f n(x,c,y) depends only on a fixed number of cells (bits) in

x

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,...

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Cell. autom. with rn linear

Matrix

Protocol:communicates kknows that after min(k,l)

thecell alternates

k l

Example: rule 23 rn=2,3,4,5,6,7,8,9,10,11,12,...

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Other examples

Rule 33 Rule 44 Rule 50 Rule 164

Rule 184 Rule 14 Rule 35 Rule 168

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An interesting matrix

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

...0

...01...011

0..

.

10

... 1

10

...

f(x,y)

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Rule 132

center c=1rn=2,3,4,...

center c=0rn=1,1,1,...

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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If A is simulated by B in realtime then class(A) class(B)

0, 8, ...

Realtime simulationA 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

rn constant54, 110, ...

rn polynomial30, 45, ...

rn exponential< <

Joint work with Guillaume Thessier

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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