reversible cellular automata cs240 kwnaghyun paek
TRANSCRIPT
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 gasesMany people have tried to reveal the behavior of molecules of gases
CelLab, Rudy Rucker
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)
The Second Law
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
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
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?
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
Reversible CA Cont’d
Let’s see some Reversible CA
Rule 204Rule 170
Rule 51Rule 15
Rule 240
Rule 85
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 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
Reversible CA Cont’d
Second-order by Toffoli and Margolus
PresentPast 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+1 = qt – qt-1 qt-1 = qt – 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’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…
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
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?