reversible cellular automata cs240 kwnaghyun paek

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Reversible Cellular Automata CS240 Kwnaghyun Paek

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Page 1: Reversible Cellular Automata CS240 Kwnaghyun Paek

Reversible Cellular Automata

CS240

Kwnaghyun Paek

Page 2: Reversible Cellular Automata CS240 Kwnaghyun Paek

Agenda

Introduction

Gas Theory

The Second Law

Entropy

Reversible Cellular Automata

Rule 37R

Further Study

Page 3: Reversible Cellular Automata CS240 Kwnaghyun Paek

Introduction

How was the Universe created? We don’t know… Maybe the God did? Big Bang?

How has the Universe evolved? Observing molecules of gases might tell us something…

Have we found the secret of Gases? Yes, we believe in The First and Second Laws of

Thermodynamics!Can we show and prove it with Cellular Automata? Let’s see!

Page 4: Reversible Cellular Automata CS240 Kwnaghyun Paek

Gas Theory

The Universe consists of gasesMany people have tried to reveal the behavior of molecules of gases

CelLab, Rudy Rucker

Page 5: Reversible Cellular Automata CS240 Kwnaghyun Paek

Gas Theory Cont’dWhy is the gas model loved so much?Fundamental Physics Rules work well with Gas Model Uniformity – Every molecule follows one rule Locality – Each molecule affects its neighbors only.

Not others far away from it. Reversibility – All information, the motion of

molecules, is backward traceable

Thermodynamics Laws work well too The First Law – Energy is conserved

The number of molecules of gas The Second Law – Heat does not spontaneously flow

from a colder body to a hotter one (Clausius and Thomson)

Page 6: Reversible Cellular Automata CS240 Kwnaghyun Paek

The Second Law

Page 7: Reversible Cellular Automata CS240 Kwnaghyun Paek

The Second Law Cont’d

Heat is motion of molecules

The motion of molecules becomes more and more randomRandomness of the system growsThe Entropy of system increases

Page 8: Reversible Cellular Automata CS240 Kwnaghyun Paek

Entropy

What is Entropy? The amount of information about a system that is

still unknown For example, all the positional information of

molecules of gas If certain parts of a system are hotter than the rest,

it is easier to represent the positional information of molecules

For example, we can compress the information when there are some repetition or order

As heat disperses, it becomes more difficult to represent those information.

Entropy increases until the system reaches complete equilibrium – total disorder

Page 9: Reversible Cellular Automata CS240 Kwnaghyun Paek

Reversible Cellular Automata

Why do we want Reversible Cellular Automata?Cellular Automata can be completely closedTo simulate the motion of gas, a system should have… Uniformity Locality Reversibility

By definition, Uniformity and Locality are already built in Cellular AutomataReversibility does not come along automatically, but can be programmed inWhy is Reversibility so important?

Page 10: Reversible Cellular Automata CS240 Kwnaghyun Paek

Reversible CA Cont’d

Why is Reversibility so important?“Reversible” means all information has been preserved, and the system follows a deterministic rule If part of heat has flowed out of a system, we

cannot restore the past There is no boundary yet (as far as we know)

in the Universe; thus, all information should be somewhere in the Universe.

If the universe has evolved non-deterministically,

There is no hope

Page 11: Reversible Cellular Automata CS240 Kwnaghyun Paek

Reversible CA Cont’d

Let’s see some Reversible CA

Rule 204Rule 170

Rule 51Rule 15

Rule 240

Rule 85

Page 12: Reversible Cellular Automata CS240 Kwnaghyun Paek

Reversible CA Cont’d

That’s it? YES…

Come on… How about Rule 30 and 225? They are just the opposite to each other. Aren’t they inverse each other?

Rule 30

Rule 225

Page 13: Reversible Cellular Automata CS240 Kwnaghyun Paek

Reversible CA Cont’d

Rule 30 Rule 225

Page 14: Reversible Cellular Automata CS240 Kwnaghyun Paek

Reversible CA Cont’d

Why doesn’t it work… Because the reverse step is not deterministic

Rule 30Forward

When the current cell is 0, there are four possible ancestors

Then what’s the Reversible CA in Wolfram’s book?They are newly blended, not originally by Wolfram

Page 15: Reversible Cellular Automata CS240 Kwnaghyun Paek

Reversible CA Cont’d

Second-order by Toffoli and Margolus

PresentPast Future

Page 16: Reversible Cellular Automata CS240 Kwnaghyun Paek

Reversible CA Cont’d

The new state of each cell is determined as a linear combination of the current and the previous states

qt+1 = qt – qt-1 qt-1 = qt – qt+1

Rule 30R

Page 17: Reversible Cellular Automata CS240 Kwnaghyun Paek

Reversible CA Cont’d

Rule 30R

Page 18: Reversible Cellular Automata CS240 Kwnaghyun Paek

Reversible CA Cont’d

Rule 30R

Page 19: Reversible Cellular Automata CS240 Kwnaghyun Paek

Rule 37R

What is so special with Rule 37R?

Rule 13R Rule 30R Rule 214R

Page 20: Reversible Cellular Automata CS240 Kwnaghyun Paek

Rule 37R Cont’d

Page 21: Reversible Cellular Automata CS240 Kwnaghyun Paek

Rule 37R Cont’d

Rule 37R is reversible

Preserves all past information

Randomness does not simply increase, it fluctuates, self organized

Does not evolve to complete equilibrium

Does not obey the Second Law!

OK! Now what???

Page 22: Reversible Cellular Automata CS240 Kwnaghyun Paek

Rule 37R Cont’d

If the universe followed the Second Law, it should have evolved to…

Page 23: Reversible Cellular Automata CS240 Kwnaghyun Paek

Rule 37R Cont’d

Great! Wolfram has revealed the secret of the Universe!!!Not yet…The rule 37R has to remember the

state of two steps back. Not very general

37R does not obey the Second Law, nor does it obey the First Law!!!

Wolfram noticed this issue, and tried to find rules following the First Law

Page 24: Reversible Cellular Automata CS240 Kwnaghyun Paek

Rule 37R Cont’dHowever, Wolfram could not find any non-trivial Cellular Automaton with two colors and nearest-neighbor rules, which follows the First Law but not the Second Law

With next-nearest-neighbor rules, what he found is…

Page 25: Reversible Cellular Automata CS240 Kwnaghyun Paek

Rule 37R Cont’d

Total number of black cells is conservedThe complexity starts when it hits the boundariesWithout the boundaries, it would show a trivial behavior

Page 26: Reversible Cellular Automata CS240 Kwnaghyun Paek

Further Study

Wolfram’s idea was nice, but still many things to solve

Three cells determine the next state of one cell Reversibility fails, one cell cannot contain all the information of the previous three cells

Margolus neighborhood

Page 27: Reversible Cellular Automata CS240 Kwnaghyun Paek

Further Study Cont’d

Conventional CA Margolus neighborhood

Page 28: Reversible Cellular Automata CS240 Kwnaghyun Paek

Thank you very much

Any Question?