Statistical mechanics of cellular automata by 3HegVOHP

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									Cellular Automata &
 DNA Computing




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Definition Of Cellular Automata


   Von Neuman’s Definition

   Wolfram’s Definition

   Lyman Hurd’s Definition
Example of Cellular Automata

   Ising Models
   Conway’s Game of Life
   Lattice gasses and the Margolus
    Neighborhood
       Partitioning Cellular Automata. A simulation
        an HPP Lattice gas
   Biological and Chemical Systems
Features of Cellular automata
   Nonlinear Cellular automata
       Homoplectic and autoplectic systems
       Particle like structures

   Computational Universality
       Turing machine  Cellular Automata

   Reversibility
     Nonlinear Cellular automata
   Homoplectic and Autoplectic
       Homoplectic rule: Generally random input
        states lead to random output states.
       Autoplectic rule: Non random input can lead to
        random output states  Non-linear CA
            Wolfram’s rule 30.
   Particle like structures
       Class 3 automata.. The Rules of these CA may
        have following properties.
            Random walk.
            Constant velocities.( Traffic simulation, Granular
             Model )
     Computational Universality

   A lot earlier than I, Wolfram proved this.
    I have not studied his theory yet.
       He postulates that infinite class four
        cellular automata are capable of Universal
        Computation.


   Even logic gates can be implemented
    by Cellular Automata
             Proof of TM CA(1)
   Def. of Turing Machine
   M = (Q,∑,Г,δ,qo,ㅁ,F)
       Q: a set of internal states
       ∑: a set of input alphabets
       Г: a set of tape alphabets
       ㅁ: blank symbol
       qo: initial state
       F: final statesδ는 transition function이다.
       δ: Q*Г  Q*Г*{L,R}
            L,R direction of the header of the TM
           Proof of TM CA(2)

   Let’s suppose following set of states
       {(0,x0),….(0,xn),(q0,x0),…,(q0,xn),………
        …,(qn,xn)}
       {(x,y)|x is the state of the header,0 means
        that no header point the state, y is the
        alphabet of the input tape.}
          Proof of TM CA(3)
   The transition function is defined like this,

       δ(q(i),x(i))  δ(q(i+1),x’(i),D)

           x(i),x’(i) ∈ ∑
           0,q0,…,qn ∈ Q
           D ∈ {L,D}
And.. This can be translated like this,,
          Proof of TM CA(4)
   It could be
    helpful to
    understand this
    to remind the
    Wolfram’s
    formal rules.
   And this means
    that the proof
    ends.
           Proof of TM CA(5)

   Assumptions
       There are infinite number of cells.
       TM’s input tape is the CA’s initial condition.
   But at least, given TM, this proof shows
    CA can be constructed.
        Partitioning CA(BCA)


   DNA Computing with BCA
   pca.html
               CABCA(1)




   The rule table must be changed.
   And the time step can be doubled.
                     CABCA(2)

   Let’s suppose a 1-dim multi-state CA.
   And it has this set of states and rules.
       {….Sa,Sb,…..Si,Sj…..}
       {….o(Sa,Sb,Si)……o(Sb,Si,Sj)……}
           You can think of the Wolfram’s 1 dim cellular
            automata.
                    CABCA(3)
   The set of states of the BCA of the CA should
    have the joined states.
       (Si,Sj),(Sa,Sb) for all pair of the states of the
        original CA.
       That is, the result set will be
        {..Sa,Sb,…(Si,Sj),(Sa,Sb)….} like this.
   And then add following rules to the rule table
    of the BCA
       Si,Sj((Si,Sj),(Si,Sj)) Sa,Sb((Sa,Sb),(Sa,Sb))
       (Si,Sj) ,(Sa,Sb)  (o(Si,Sj,Sa),o(Sj,Sa,Sb))
              CABCA(4)


   It is proved that any given Turing
    Machine can be transformed into a BCA.
   And BCA can be directly used as the
    model of the DNA Computing.(Winfree
    96’).
Winfree’s DNA Computing(1)
    Winfree’s DNA Computing(2)

   This is so explicitly described in the first
    part of his thesis.
   He uses only “Ligation” to implement a
    BCA.
Winfree’s DNA Computing(3)
    Winfree’s DNA Computing(4)
    First express your problem via computer program. Convert
     that program into a blocked cellular automaton.
    Create small molecules (H-shaped and linear) which self-
     assemble to create the initial molecule( or initial molecules, if
     search over a FSA=generated set of strings is desired.)
    Create small H-shaped molecules encoding the rule table for
     your program.
    Mix the molecules created in steps 2 and 2 together in a test
     tube, and keep under precise conditions (temperature, salt
     concentrations) as the DNA lattice crystallizes.
    When the solution turns blue, ligate, cut the crossovers, and
     extract the strand with the halting symbol.
    Sequence the answer.
    Winfree’s DNA Computing(5)

   Limits of this method.
   Shortly speaking, this is another
    approach to the crystal computation.
    This is thought to be another hardware
    for the cellular automata. Winfree just
    implements this technique with DNA…..
   But not that good.
                 Future Work
   Study crystal computation, study ligation and
    try winfree’s work again.
   In my opinion, to successfully compute with
    DNA using the winfree’s method, we should
    have more knowledge about Nano technology
    to control more. So.. ,until then, we may find
    another approach to using DNA molecules.
    And if possible I’ll study about its possibilities.

								
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