Transcript Reversible Cellular Automata

Reversible Cellular
Automata
CS240
Kwnaghyun Paek
Agenda
Introduction
Gas Theory
The Second Law
Entropy
Reversible Cellular Automata
Rule 37R
Further Study
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!
Gas Theory
The Universe consists of gases
Many people have tried to reveal the
behavior of molecules of gases
CelLab, Rudy Rucker
Gas Theory Cont’d
Why is the gas model loved so much?
Fundamental Physics Rules work well with Gas
Model
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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

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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)
The Second Law
The Second Law Cont’d
Heat is motion of molecules
The motion of molecules becomes more and more
random
Randomness of the system grows
The Entropy of system increases
Entropy
What is Entropy?

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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
Reversible Cellular Automata
Why do we want Reversible Cellular Automata?
Cellular Automata can be completely closed
To simulate the motion of gas, a system should
have…
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Uniformity
Locality
Reversibility
By definition, Uniformity and Locality are already
built in Cellular Automata
Reversibility does not come along automatically, but
can be programmed in
Why is Reversibility so important?
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
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Reversible CA Cont’d
Let’s see some Reversible CA
Rule 15
Rule 51
Rule 85
Rule 170
Rule 204
Rule 240
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
Reversible CA Cont’d
Rule 30  Rule 225
Reversible CA Cont’d
Why doesn’t it work…

Because the reverse step is not deterministic
Rule 30
Forward
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
Reversible CA Cont’d
Second-order by Toffoli and Margolus
Past
Present
Future
Reversible CA Cont’d
The new state of each cell is determined as a linear
combination of the current and the previous states
= qt – qt-1
 qt-1 = qt – qt+1
 qt+1
Rule 30R
Reversible CA Cont’d
Rule 30R
Reversible CA Cont’d
Rule 30R
Rule 37R
What is so special with Rule 37R?
Rule 13R
Rule 30R
Rule 214R
Rule 37R Cont’d
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???
Rule 37R Cont’d
If the universe followed the Second Law,
it should have evolved to…
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

Rule 37R Cont’d
However, Wolfram could not find any non-trivial
Cellular Automaton with two colors and nearestneighbor rules, which follows the First Law but not
the Second Law
With next-nearest-neighbor rules, what he found is…
Rule 37R Cont’d
Total number of black
cells is conserved
The complexity starts
when it hits the
boundaries
Without the
boundaries, it would
show a trivial
behavior
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
Further Study Cont’d
Conventional CA
Margolus neighborhood
Thank you very much
Any Question?