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Reversible Cellular Automata CS240 Kwnaghyun Paek

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Agenda Introduction Gas Theory The Second Law Entropy Reversible Cellular Automata Rule 37R Further Study

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Introduction How was the Universe created? We dont 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? Lets see!

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Gas Theory The Universe consists of gases Many people have tried to reveal the behavior of molecules of gases CelLab, Rudy Rucker

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Gas Theory Contd Why 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)

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The Second Law

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The Second Law Contd Heat is motion of molecules The motion of molecules becomes more and more random Randomness of the system grows The Entropy of system increases

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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

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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… 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?

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Reversible CA Contd 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 Contd Lets see some Reversible CA Rule 204Rule 170 Rule 51Rule 15 Rule 240 Rule 85

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Reversible CA Contd Thats it? YES… Come on… How about Rule 30 and 225? They are just the opposite to each other. Arent they inverse each other?

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Reversible CA Contd Rule 30 Rule 225

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Reversible CA Contd Why doesnt it work… Because the reverse step is not deterministic When the current cell is 0, there are four possible ancestors Then whats the Reversible CA in Wolframs book? They are newly blended, not originally by Wolfram

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Reversible CA Contd Second-order by Toffoli and Margolus Present PastFuture

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Reversible CA Contd The new state of each cell is determined as a linear combination of the current and the previous states q t+1 = q t – q t-1 q t-1 = q t – q t+1 Rule 30R

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Reversible CA Contd

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Rule 30R

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Rule 37R What is so special with Rule 37R? Rule 13RRule 30RRule 214R

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Rule 37R Contd

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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???

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Rule 37R Contd If the universe followed the Second Law, it should have evolved to…

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Rule 37R Contd 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

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Rule 37R Contd However, 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…

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Rule 37R Contd Total number of black cells is conserved The complexity starts when it hits the boundaries Without the boundaries, it would show a trivial behavior

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Further Study Wolframs 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

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Further Study Contd Conventional CA Margolus neighborhood

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Thank you very much Any Question?

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