The Intrinsic Universality
Problem of 1D CA
Nicolas Ollinger
LIP, ENS Lyon, France
STACS 2003 / Berlin
◦I×
Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
2 CC J ◦ B BB ×
Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
time
..
.
4
3
2
1
0
2 CC C ◦ B BB ×
Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
time
..
.
4
3
2
1
0
Z
. . . −4 −3 −2 −1 0
1
2
3
4 ...
2 CC C ◦ B BB ×
Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
time
..
.
S={
,
}
4
3
2
1
0
Z
. . . −4 −3 −2 −1 0
1
2
3
4 ...
• A con guration C is a mapping from Z to S.
2 CC C ◦ B BB ×
Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
time
..
.
4
3
2
1
0
Z
. . . −4 −3 −2 −1 0
1
2
3
S={
,
}
N ⊆ nite Z
4 ...
• A con guration C is a mapping from Z to S.
2 CC C ◦ B BB ×
Cellular Automata
• A 1D•CA A is a tuple Z, S, N , δ .
time
..
.
4
3
2
1
0
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.
2 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.
2 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.
2 CC C ◦ I BB ×
Computation Universality
Idea. Some CA can compute every recursive func•
tion, for example by simulating a universal TM.
B B b a b b b a B B
s2
B B b a b b b a B B
s1
B B b a a b b a B B
s0
time
3 CC J ◦ B BB ×
Computation Universality
Idea. Some CA can compute every recursive func•
tion, for example by simulating a universal TM.
B
·
B
·
B
·
B
·
B
·
B
·
b
·
b
·
b
·
a
·
a
·
a
·
b b
s2 ·
b b
· s1
a b
s0 ·
b
·
b
·
b
·
a
·
a
·
a
·
B
·
B
·
B
·
B
·
B
·
B
·
time
3 CC C ◦ B BB ×
Computation Universality
Idea. Some CA can compute every recursive func•
tion, for example by simulating a universal TM.
B
·
B
·
B
·
B
·
B
·
B
·
b
·
b
·
b
·
a
·
a
·
a
·
b b
s2 ·
b b
· s1
a b
s0 ·
b
·
b
·
b
·
a
·
a
·
a
·
B
·
B
·
B
·
B
·
B
·
B
·
time
3 CC C ◦ B BB ×
Computation Universality
Idea. Some CA can compute every recursive func•
tion, for example by simulating a universal TM.
B
·
B
·
B
·
B
·
B
·
B
·
b
·
b
·
b
·
a
·
a
·
a
·
b b
s2 ·
b b
· s1
a b
s0 ·
b
·
b
·
b
·
a
·
a
·
a
·
B
·
B
·
B
·
B
·
B
·
B
·
time
Problem. This is di cult to de ne formally and it
involves extrinsic notions of simulation. . .
3 CC C ◦ I BB ×
Intrinsic Universality
Idea. Some CA can simulate every possible CA.
#01@01@??@01|10|01|01#..#10@01@??@01|10|01|01#..
· · · · · · s0 · · · · · · · · · · · · · · · · · · · · · · · s0 · · · · · · · · · · · · · · · · ·
#01@??@??@01|10|01|01#..#10@??@??@01|10|01|01#..
s0 · · · · · · · · · · · · · · · · · · · · · · · s0 · · · · · · · · · · · · · · · · · · · · · · ·
(t)
(t)
(t+1)
Ci−1 Ci−2 Ci−1
transition table
(t)
ϕ(Ci−1 )
(t)
Ci
(t)
(t+1)
Ci−1 Ci
transition table
(t)
ϕ(Ci )
4 CC J ◦ I BB ×
Deciding Universality
• As computation universality for TM is certainly a prop•
erty of the computed function, undecidability follows
from Rice's theorem.
Theorem[Rice] No non•trivial property on the
computed function of TM is recursive.
• What about intrinsic universality for CA?
5 CC J ◦ I BB ×
Undecidability and CA
One time step properties
(2D+) Injectivity of the global rule [Kari 94];
(2D+) Surjectivity of the global rule [Kari 94];
6 CC J ◦ B BB ×
Undecidability and CA
One time step properties
(2D+) Injectivity of the global rule [Kari 94];
(2D+) Surjectivity of the global rule [Kari 94];
Long time behavior properties
(2D+) Nilpotency [ƒulik et al. 89];
(1D) Nilpotency [Kari 92];
(1D) Kind of Rice's theorem for limit sets [Kari 94];
(1D) Nilpotency on periodic [Mazoyer and Rapaport 99];
6 CC C ◦ B BB ×
Undecidability and CA
One time step properties
(2D+) Injectivity of the global rule [Kari 94];
(2D+) Surjectivity of the global rule [Kari 94];
Long time behavior properties
(2D+) Nilpotency [ƒulik et al. 89];
(1D) Nilpotency [Kari 92];
(1D) Kind of Rice's theorem for limit sets [Kari 94];
(1D) Nilpotency on periodic [Mazoyer and Rapaport 99];
CA•1D•NIL•PER
Input
A CA A and a state s of A
Question
Is A s•nilpotent on periodic ?
6 CC C ◦ I BB ×
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.
7 CC J ◦ B BB ×
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
7 CC C ◦ I BB ×
Inducing an Order on CA (2)
Idea. Rescaling corresponds to an information pre•
serving zoom out operation...
A sample h4, 4, 1i rescaling
8 CC J ◦ B BB ×
Inducing an Order on CA (2)
Idea. Rescaling corresponds to an information pre•
serving zoom out operation...
A sample h4, 4, 1i rescaling
8 CC C ◦ B BB ×
Inducing an Order on CA (2)
Idea. Rescaling corresponds to an information pre•
serving zoom out operation...
A sample h4, 4, 1i rescaling
8 CC C ◦ B BB ×
Inducing an Order on CA (2)
Idea. Rescaling corresponds to an information pre•
serving zoom out operation...
A sample h4, 4, 1i rescaling
8 CC C ◦ B BB ×
Inducing an Order on CA (2)
Idea. Rescaling corresponds to an information pre•
serving zoom out operation...
A sample h4, 4, 1i rescaling
8 CC C ◦ I BB ×
Inducing an Order on CA (3)
De nition. The hm, n, ki rescaling of A is Ahm,n,ki :
hm,n,ki
GA
A
−m
= σk ◦ o m ◦ G n
◦
o
.
A
Ah4,4,1i
9 CC J ◦ B BB ×
Inducing an Order on CA (3)
De nition. The hm, n, ki rescaling of A is Ahm,n,ki :
hm,n,ki
GA
−m
= σk ◦ o m ◦ G n
◦
o
.
A
Ah4,4,1i
A
De nition.
0
0
0
A 6 B if there exist hm, n, ki and
hm , n , k i such that A
hm,n,ki
⊆B
hm 0 ,n 0 ,k 0 i
.
9 CC C ◦ I BB ×
Inducing an Order on CA (4)
Proposition. The relation 6 is a quasi•order on CA.
• The induced order admits a maximal element.
De nition. A CA A is intrinsically universal if:
∀B, ∃ hm, n, ki ,
B ⊆ Ahm,n,ki
.
10 CC J ◦ B BB ×
Inducing an Order on CA (4)
Proposition. The relation 6 is a quasi•order on CA.
• The induced order admits a maximal element.
De nition. A CA A is intrinsically universal if:
∀B, ∃ hm, n, ki ,
B ⊆ Ahm,n,ki
.
Proposition. Every CA in the maximal equivalence
class of 6 is intrinsically universal.
10 CC C ◦ I BB ×
Sketch of the proof
• De ne a particular intrinsically universal CA U.
11 CC J ◦ B BB ×
Sketch of the proof
• De ne a particular intrinsically universal CA U.
• Map recursively each CA A to a CA A ~s U satisfying:
A ~s U is intrinsically universal
if and only if
A is not s•nilpotent on periodic
11 CC C ◦ B BB ×
Sketch of the proof
• De ne a particular intrinsically universal CA U.
• Map recursively each CA A to a CA A ~s U satisfying:
A ~s U is intrinsically universal
if and only if
A is not s•nilpotent on periodic
Remark. We suppose U is universal in the strong meaning
and prove that A ~s U is universal in the weak meaning.
11 CC C ◦ I BB ×
Introducing Boiler CA
energy consumption
SU
energy di usion
{·, ∗}
energy production
SA
A ~s U = (Z, SA × {·, ∗} × SU , NA ∪ {−1, 0} ∪ NU , δ)
12 CC J ◦ I BB ×
If A is not s•nilpotent on periodic
• There exists a space•time pattern P of A, periodic
both in time and space, which is not s•monochromatic.
s 6=
?
?
?
•
?
?
?
?
?
?
?
?
13 CC J ◦ B BB ×
If A is not s•nilpotent on periodic
• There exists a space•time pattern P of A, periodic
both in time and space, which is not s•monochromatic.
s 6=
?
?
?
•
?
?
?
?
?
?
?
?
• This pattern can be used to produce energy in a
uniform way to the U layer in such a way that this layer
simulates U behavior up to some slowdown factor.
13 CC C ◦ I BB ×
If A is s•nilpotent on periodic
• Every periodic con guration of A evolves to the s•
monochromatic con guration in nite time.
14 CC J ◦ B BB ×
If A is s•nilpotent on periodic
• Every periodic con guration of A evolves to the s•
monochromatic con guration in nite time.
Idea. The U layer do not get enough energy to com•
pute everything needed...
14 CC C ◦ B BB ×
If A is s•nilpotent on periodic
• Every periodic con guration of A evolves to the s•
monochromatic con guration in nite time.
Idea. The U layer do not get enough energy to com•
pute everything needed...
• In fact, A ~s U cannot simulate (Z, {◦, •} , {−1, 0} , ⊕)
Two key patterns:
◦••
•◦•
••◦
and
◦••••••••••••••••◦
◦•◦•◦•◦•◦•◦•◦•◦•◦◦
◦••◦◦••◦◦••◦◦••◦◦◦
◦•◦◦◦•◦◦◦•◦◦◦•◦◦◦◦
◦••••◦◦◦◦••••◦◦◦◦◦
◦•◦•◦◦◦◦◦•◦•◦◦◦◦◦◦
◦••◦◦◦◦◦◦••◦◦◦◦◦◦◦
◦•◦◦◦◦◦◦◦•◦◦◦◦◦◦◦◦
◦••••••••◦◦◦◦◦◦◦◦◦
◦•◦•◦•◦•◦◦◦◦◦◦◦◦◦◦
◦••◦◦••◦◦◦◦◦◦◦◦◦◦◦
◦•◦◦◦•◦◦◦◦◦◦◦◦◦◦◦◦
◦••••◦◦◦◦◦◦◦◦◦◦◦◦◦
◦•◦•◦◦◦◦◦◦◦◦◦◦◦◦◦◦
◦••◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦
◦•◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦
14 CC C ◦ I BB ×
Future Work
• Adapt this energy driven proof technic to other properties.
• Can we extend the analogy between computation univer•
sality for TM and intrinsic universality for CA to derive some
kind of Rice's theorem for non trivial properties on space•
time diagrams?
15 CC J ◦ I BB ×
Test Page (+ pdfTEX & Acrobat issue)
abcdefghijklmnopqrstuvwxyz
ABCDEFGHIJKLMNOPQRSTUVWXYZ
012345789
abcdefghijklmnopqrstuvwxyz
ABCDEFGHIJKLMNOPQRSTUVWXYZ
012345789
abcdefghijklmnopqrstuvw x y z
ABCDEFGHIJKLMNOPQRSTUVWXYZ
012345789
abcdefghijklmnopqrstuvwxyz
ABCDEFGHIJKLMNOPQRSTUVWXYZ
012345789
16 CC J ◦ I BB ×