The Quest for
Small Universal
Cellular Automata
Nicolas Ollinger
LIP, ENS Lyon, France
8 july 2002 / ICALP’2002 / Málaga, Spain
◦I×
Cellular Automata
Definition. A d-CA A is a 4-uple Z , S, N, δ where:
d
• S is the finite state set of A;
• N ⊂ Zd , finite, is the neighborhood of A;
• δ : S|N| → S is the local rule of A.
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Cellular Automata
Definition. A d-CA A is a 4-uple Z , S, N, δ where:
d
• S is the finite state set of A;
• N ⊂ Zd , finite, is the neighborhood of A;
• δ : S|N| → S is the local rule of A.
A configuration C is a mapping from Zd to S.
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Cellular Automata
Definition. A d-CA A is a 4-uple Z , S, N, δ where:
d
• S is the finite state set of A;
• N ⊂ Zd , finite, is the neighborhood of A;
• δ : S|N| → S is the local rule of A.
A configuration C is a mapping from Zd to S.
The global rule applies δ uniformly according to N:
∀p ∈ Zd , G(C)p = δ (Cp+N1 , . . . , Cp+Nν )
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Space-Time Diagram
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Space-Time Diagram
3 CC C ◦ B BB ×
Space-Time Diagram
3 CC C ◦ B BB ×
Space-Time Diagram
3 CC C ◦ B BB ×
Space-Time Diagram
3 CC C ◦ B BB ×
Space-Time Diagram
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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.
• This notion is rather difficult to formalize...
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Computation Universality
Idea. A CA is computation universal if it can compute any
partial recursive function.
• This notion is rather difficult to formalize...
• In practice: step-by-step Turing machine simulation.
A. R. Smith III. Simple Computation-Universal Cellular Spaces. 1971
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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.
Definition. 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)
Definition. The hm, n, ki rescaling of A is defined 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)
Definition. The hm, n, ki rescaling of A is defined by:
hm,n,ki
GA
−m
= σk ◦ o m ◦ G n
◦
o
.
A
Ah4,4,1i
A
Definition. A 6 B if there exist hm, n, ki and hm 0 , n 0 , k 0 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.
Definition. 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.
Definition. A CA A is intrinsically universal if:
∀B, ∃ hm, n, ki ,
B ⊆ Ahm,n,ki
.
Proposition. Every intrinsically universal CA is computation 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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CA and Boolean Circuits (1)
• We decompose a CA local rule into k boolean functions
where k = dlog2 |S|e:
δi : {0, 1}
δ
|N|k
→ {0, 1} .
δ1
δ2
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CA and Boolean Circuits (2)
• To a boolean function we associate a leveled circuit:
δ1 (x, y, z, t)
δ2 (x, y, z, t)
L2
L2
L1
L1
variable
NAND
copy
L0
x
y
z
t
δ1 (x, y, z, t) = x | t
L0
x
y
z
t
δ2 (x, y, z, t) = y | y
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Boolean Circuit Simulator
A BC Simulator is a 1D dynamical system that simulate a
CA via its boolean circuits. Each cell contains:
• a boolean value;
• an operator (identity or NAND);
• the relative positions of the operands.
δ1
boolean value
operator
operand 1
operand 2
δ2
δ2
δ1
δ2
δ1
z }| { z }| { z }| { z }| { z }| { z }| {
··· ? x ? ? y ? z ? ? t ? u ? ? v ···
| C C C | | C C C | | C C C |
1 1 8 9 1 1 1 8 9 1 1 1 8 9 1
|9 {z 2} |9 {z 2} |9 {z 2}
s0
s1
s2
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Microscopic Description
We build a 3-state 1D-CA to move information.
Op
Val
Sig
• Sig cells transport boolean values between cells;
• Val cells encode current meta-cell value;
• Op vells encode the operation to execute.
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8 States
Direct encoding:
• Sig: 0 or 1 boolean value;
• Val: 0 or 1 boolean value;
• Op: Border, Copy, Follow or NAND operation.
Copy and NAND
Border
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7 States
New encoding:
• Sig: 0 or 1 boolean value;
• Val: 0 or 1 boolean value;
• Op: Border, Follow or NAND operation.
The Copy operation is emulated. We encode a signal x by
three consecutive signals 1, x, 0.
old Op
Border
Follow
NAND
Copy
new 3 Ops
Follow, Border, Follow
Follow, Follow, Follow
Follow, NAND, Follow
NAND, NAND, NAND
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6 States
More tricky!
New encoding with Op a boolean value which meaning
becomes position dependant...
right Op
left Op
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Going further
• We get 6 states as a product 3 × 2. What about 2 × 2?
• Cook and Wolfram have proven a 2 states rule computation universal. Is it also intrinsically universal ?
• Good formal definition of computation universality ?
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