cellular automata [3ex] and universalityemc/15817-s12/lecture/ca-univerality.pdf · cellular...

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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.