# Statistical mechanics of cellular automata by 3HegVOHP

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```									Cellular Automata &
DNA Computing

97300-199 우정철
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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