Reversible Cellular Automata

1
Reversible Cellular Automata

• CS240
• Kwnaghyun Paek

2
Agenda

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

3
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!

4
Gas Theory

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

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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

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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

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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
• qt1 ?qt qt-1
• qt-1 ?qt qt1
• Rule 30R

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Reversible CA Contd
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Reversible CA Contd
Rule 30R
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Rule 37R

• What is so special with Rule 37R?

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Rule 37R Contd
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Rule 37R Contd

• 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
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Thank you very much

• Any Question?