Universalité de la règle 110
vers une démonstration
Nicolas Ollinger
LIF, Univ. de Provence
Gaétan Richard
étudiant ENS Lyon
LITA, Metz
30 juin 2004
◦I×
Table of Content
1. Cellular Automata
2. Universalities
3. Rule 110 basics
4. Cook•Wolfram proof
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Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
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Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
time
..
.
4
3
2
1
0
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Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
time
..
.
Z
4
3
2
1
0
. . . −4 −3 −2 −1 0
1
2
3
4 ...
3 CC C ◦ B BB ×
Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
time
..
.
Z
S={
,
}
4
3
2
1
0
. . . −4 −3 −2 −1 0
1
2
3
4 ...
• A con guration C is a mapping from Z to S.
3 CC C ◦ B BB ×
Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
time
..
.
Z
4
3
2
1
0
. . . −4 −3 −2 −1 0
1
2
3
S={
}
,
N ⊆ nite
Z
4 ...
• A con guration C is a mapping from Z to S.
3 CC C ◦ B BB ×
Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
time
..
.
Z
4
3
2
1
0
. . . −4 −3 −2 −1 0
1
2
3
4 ...
S={
,
}
δ : S|N | → S
• A con guration C is a mapping from Z to S.
3 CC C ◦ B BB ×
Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
time
..
.
4
3
2
1
0
G
Z
. . . −4 −3 −2 −1 0
1
2
3
4 ...
S={
,
}
δ : S|N | → S
• A con guration C is a mapping from Z to S.
3 CC C ◦ B BB ×
Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
time
..
.
4
3
2
1
0
G
Z
. . . −4 −3 −2 −1 0
1
2
3
4 ...
S={
,
}
δ : S|N | → S
• A con guration C is a mapping from Z to S.
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Examples (1)
σ = (Z, {
, } , {−1} , q 7→ q)
Σ2 = (Z, {
, } , J−1, 0K, (q, q 0 ) 7→ q ⊕ q 0 ),
where ({
, } , ⊕) is isomorphic to (Z2 , +)
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Examples (2)
(Z, {
, } , J−1, 1K, maj),
where maj is majority between 3
(Z, {
, , , , , } , J−1, 1K, δ6 )
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Table of Content
1. Cellular Automata
2. Universalities
3. Rule 110 basics
4. Cook•Wolfram proof
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Computation Universality
Idea. A CA is computation universal if it can compute
any partial recursive function.
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Computation Universality
Idea. A CA is computation universal if it can compute
any partial recursive function.
• In practice : step•by•step Turing machine simulation.
A. R. Smith III. Simple Computation•Universal Cellular Spaces. 1971
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Universalities
B. Durand and Z. Róka, The game of life: universality
revisited, Cellular automata (Saissac, 1996) (Kluwer
Acad. Publ., Dordrecht, 1999), (pp. 51 74).
• Several di erent notions of universality :
Turing (computation universality) ;
Intrinsic (CA simulating all CA) ;
Circuits (CA simulating boolean circuits).
• Problems in the proof of universality of GOL.
• Discusses the di culty of formalization.
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Inducing an Order on CA (1)
Idea. A CA A is less complex than a CA B if, up
to some renaming of states and some rescaling, every
space•time diagram of A is a space•time diagram of B.
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Inducing an Order on CA (1)
Idea. A CA A is less complex than a CA B if, up
to some renaming of states and some rescaling, every
space•time diagram of A is a space•time diagram of B.
De nition. A ⊆ B if there exists an injective mapping
ϕ from SA into SB such that this diagram commutes :
C


GA y
ϕ
−−−−→
ϕ(C)

G
y B
GA (C) −−−−→ ϕ(GA (C))
ϕ
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Inducing an Order on CA (2)
De nition. The hm, n, ki rescaling of A is de ned
by :
hm,n,ki
GA
A
−m
.
= σk ◦ o m ◦ G n
◦
o
A
Ah4,4,1i
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Inducing an Order on CA (2)
De nition. The hm, n, ki rescaling of A is de ned
by :
hm,n,ki
GA
−m
.
= σk ◦ o m ◦ G n
◦
o
A
A
Ah4,4,1i
De nition. A 6 B if there exist hm, n, ki and
0
0
0
hm , n , k i such that A
hm,n,ki
⊆B
hm 0 ,n 0 ,k 0 i
.
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Inducing an Order on CA (3)
Proposition. The relation 6 is a quasi•order on CA.
• The induced order admits a maximal equivalence
class.
De nition. A CA A is intrinsically universal if :
∀B, ∃ hm, n, ki ,
B ⊆ Ahm,n,ki
.
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Inducing an Order on CA (3)
Proposition. The relation 6 is a quasi•order on CA.
• The induced order admits a maximal equivalence
class.
De nition. A CA A is intrinsically universal if :
∀B, ∃ hm, n, ki ,
B ⊆ Ahm,n,ki
.
Proposition. Every intrinsically universal CA is com•
putation universal. The converse is false.
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Simple Universal CA
year
1966
1968
1970
author
von Neumann
Codd
Banks
1971
Smith III
1987
1990
2002
2002
Albert & Culik II
Lindgren & Nordhal
NO
Cook & Wolfram
d
2
2
2
1
2
1
1
1
1
1
|N|
5
5
5
3
7
3
3
3
3
3
states
29
8
2
18
7
18
14
7
6
2
universality
intrinsic
intrinsic
intrinsic
intrinsic
computation
computation
intrinsic
computation
intrinsic
computation
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Banks' Universal 2D•CA
, ,
Z , 2
!
,δ
E. R. Banks. Universality in Cellular Automata. 1970
Idea. Emulate logical circuits by building :
wires transporting binary signals
logical gates AND, OR and NOT
wires crossing
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Table of Content
1. Cellular Automata
2. Universalities
3. Rule 110 basics
4. Cook•Wolfram proof
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Point of View
• We want to construct huge space•time diagrams.
• We need to prove their existence.
• We cannot simply draw some basis of them because
of the size of diagrams involved (squares of millions of
cells on a side).
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Tiling the plane
Space•time diagrams as tiling
of the plane by triangles
Changing the point of view
from 1D to 2D
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Particles
Particles repeats themselves
in a uniform background
+
*
0
7
A=
,
+
*
B=
C=
0
7
D 2
3
,
,
E
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Collisions
Particles collide when meeting
0
0
0
−4
0
5
Γ:
C+
A`
B
+ some pertubation pattern F
We are given a set of valid
elementary particles and
elementary collisions
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Bindings
• To combine collisions we use one operation : binding.
0
Γ =
α1
α2
αn
Γ1 +
Γ2 + · · · +
Γn
β1
β2
βn
bind
Principle Merge incoming and outgoing particles
when possible. Some bindings are not valid !
• Binding is easy to construct and validate.
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Table of Content
1. Cellular Automata
2. Universalities
3. Rule 110 basics
4. Cook•Wolfram proof
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Sketch of the proof
• We prove that rule 110 is Turing•universal.
1. Reduce Turing Machines to Post Tag Systems.
2. Reduce Tag Systems to Cyclic Tag Systems.
3. Encode Cyclic Tag Systems with collisions.
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Post Tag Systems
M. Minsk y, Computation : Finite and In nite Machines
(Prentice Hall, Englewoods Cli s, 1967).
• A classical model used to prove universality of small
Turing Machines.
• Con gurations are words on Σ, a system is given by
(k, v1 , . . . , v|Σ| ). A transition from u is done as follows :
u1 . . . uk uk+1 . . . um ` uk+1 . . . um · vu1
• When the rule cannot be applied, the system accepts.
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Cyclic Tag systems
• A cyclic tag system acts only on the binary alphabet.
• A con guration is given by a word u and a set of nite
words (v1 , . . . , vn ).
• A transition is done as follows :
1. if the rst letter of u is 1 then catenate v1 to u ;
2. erase the rst letter of u ;
3. rotate the list of words as (v2 , . . . , vn , v1 ).
• Such systems can simulate any Post Tag System.
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A Local Dynamical System
Idea Replace the nite set of words by a periodic one.
Idea Make the rst letter cross the word letter by letter.
• A transition is done as follows :
1. the rst letter of u crosses the word to the right ;
2. when it meets a boundary, it destroys it ;
3. it begins either to erase of unfreeze letters ;
4. when it meets the second boundary, it stops.
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A Sample CA
• 16 states, a large neighborhood (−1, 0, 1, 2).
• Locally it can simulate the cyclic Tag system.
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A Sample CA
• 16 states, a large neighborhood (−1, 0, 1, 2).
• Locally it can simulate the cyclic Tag system.
Claim This CA may not work ! Why ?
27 CC C ◦ B BB ×
A Sample CA
• 16 states, a large neighborhood (−1, 0, 1, 2).
• Locally it can simulate the cyclic Tag system.
Claim This CA may not work ! Why ?
• Synchronization problems may appear. Be careful.
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Roadmap
• Now we need to exhibit the gadgets for rule 110.
• This is very technical and requires an Oracle.
• M. Cook and S. Wolfram tour de force .
S. Wolfram, A New Kind of Science, 2002
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information active
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information passive
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info passive (x2) (x3)
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synchronisation
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(E) synchronisation
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(E) pré•bits N B
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(E) bits passifs N B
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(E) bits actifs N B
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croisement1
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croisement2
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(E) croisement NN NB
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(E) croisement BN BB
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(E) poubelle N B
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(E) poubelle sync
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redressement
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(E) redressement N B
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passage1
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passage2a
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passage2b
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(E) passage N B
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blocage1
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blocage2a
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blocage2b
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(E) blocage N B
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délimiteur 1
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délimiteur 2
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délimiteur 3
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délimiteur 4
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(E) pré•bits N B n
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(E) analyse N B
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(E) passage nal N B
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(E) blocage nal N B
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Combining the gadgets
Claim Only synchronization invariants are missing.
Idea 1 Combine groups of particles.
Idea 2 Express synchronization as a big system of
linear equalities and solve it.
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Test Page (+ pdfTEX & Acrobat issue)
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