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Cellular Automata
and Universality
Klaus Sutner
Cellular Automata: 2
ECA 1
Cellular Automata: 3
ECA 1
Cellular Automata: 4
ECA 28
Cellular Automata: 5
ECA 28
Cellular Automata: 6
ECA 54
Cellular Automata: 7
Multiple Seeds
Cellular Automata: 8
Disaster
n = 23, t = 24, p = 690
Cellular Automata: 9
ECA 73
Cellular Automata: 10
ECA 45
Cellular Automata: 11
ECA 90
Cellular Automata: 12
ECA 30
Cellular Automata: 13
ECA 30
Cellular Automata: 14
782359
Cellular Automata: 15
782340
Cellular Automata: 16
782353
Cellular Automata: 17
722797
Cellular Automata: 18
Langton’s CA
CA and Correctness Proofs: 19
� Cellular Automata
2 CA and Correctness Proofs
CA and Correctness Proofs: 20
A Hard Problem
How does one prove properties of cellular automata?
The short term behavior (first-order logic) is decidable via automatatheory.
Almost any question about the long term behavior is undecidable; thereis no standardized argument.
CA and Correctness Proofs: 21
Moews
13 states
CA and Correctness Proofs: 22
Waksman
9 states
CA and Correctness Proofs: 23
Mazoyer
6 states
CA and Correctness Proofs: 24
Mazoyer’s Proof
The paper in TCS is 54 pages long and uses lots of diagrams.
To be sure, the proof is almost certainly correct, but it contains lots oflittle, unrelated combinatorial facts that are difficult for a human tocheck – eyes glaze over very quickly.
The proof should be machine checked.
CA and Correctness Proofs: 25
Universality of ECA 110
ECA 110 is given by the following local rule:
000 → 0 100 → 0001 → 1 101 → 1010 → 1 110 → 1011 → 1 111 → 0
As a Boolean function this comes down to
ρ(x, y, z) = (x ∧ y) ∨ (y ⊕ z)
CA and Correctness Proofs: 26
Another Look
Here is the table again, but ordered differently:
000 → 0 010 → 1001 → 1 011 → 1100 → 0 110 → 1101 → 1 111 → 0
The left column has “control bit” y = 0 and amounts to a left shift of z.
The right column has “control bit” y = 1 and amounts to NAND(x, z).
Since NAND is a functionally complete set of Boolean operations . . .
CA and Correctness Proofs: 27
One-Point Seed
We get a half-light-cone: the configuration grows to the left only, atspeed 1. Does not look particularly complicated.
CA and Correctness Proofs: 28
Finite Seed
CA and Correctness Proofs: 29
Random Seed
CA and Correctness Proofs: 30
After 1000 Steps
CA and Correctness Proofs: 31
Background
A background pattern naturally evolves; think of this as vacuum.
CA and Correctness Proofs: 32
Particles
Particles can move in this vacuum.
CA and Correctness Proofs: 33
More Particles
CA and Correctness Proofs: 34
Interactions
CA and Correctness Proofs: 35
Interactions
CA and Correctness Proofs: 36
Interactions
CA and Correctness Proofs: 37
Interactions
and interact . . .
CA and Correctness Proofs: 38
Interactions
and interact . . .
CA and Correctness Proofs: 39
Interactions
and interact . . .
CA and Correctness Proofs: 40
Putting it Together
In the mid-90’s, Matthew Cook, working at WRI, developed a fancysimulation system that allowed him to design and experiment with largeconfigurations on ECA 110.
Based on his observations, he was able to show universality assumingalmost periodic configurations by simulating cyclic tag systems, a variantof Post tag systems.
This is arguable the most interesting result in the study of computationaluniversality in the last two or three decades. It’s also the only result thatlead to a law suit.
CA and Correctness Proofs: 41
Verification
A number of people have checked the proof in great detail and there areeven attempts at producing a more formalized versions of it.
Again, the proof is probably correct but it is really impossible to check allthe details by hand – we need a computer-checkable version.