cellular automata & dna computing

Cellular Automata & Cellular Automata & DNA ComputingDNA Computing

97300-199 97300-199 우정철우정철

Definition Of Cellular AutomataDefinition Of Cellular Automata

Von Neuman’s DefinitionVon Neuman’s Definition

Wolfram’s DefinitionWolfram’s Definition

Lyman Hurd’s DefinitionLyman Hurd’s Definition

Example of Cellular Example of Cellular AutomataAutomata

Ising ModelsIsing Models Conway’s Game of LifeConway’s Game of Life Lattice gasses and the Margolus Lattice gasses and the Margolus

Neighborhood Neighborhood Partitioning Cellular Automata. A Partitioning Cellular Automata. A

simulation an HPP Lattice gassimulation an HPP Lattice gas Biological and Chemical SystemsBiological and Chemical Systems

Features of Cellular Features of Cellular automataautomata

Nonlinear Cellular automataNonlinear Cellular automata Homoplectic and autoplectic systemsHomoplectic and autoplectic systems Particle like structuresParticle like structures

Computational UniversalityComputational Universality Turing machine Turing machine Cellular Automata Cellular Automata

ReversibilityReversibility

Nonlinear Cellular automataNonlinear Cellular automata

Homoplectic and AutoplecticHomoplectic and Autoplectic Homoplectic rule: Generally random input Homoplectic rule: Generally random input

states lead to random output states.states lead to random output states. Autoplectic rule: Non random input can lead Autoplectic rule: Non random input can lead

to random output states to random output states Non-linear CA Non-linear CA Wolfram’s rule 30.Wolfram’s rule 30.

Particle like structuresParticle like structures Class 3 automata.. The Rules of these CA Class 3 automata.. The Rules of these CA

may have following properties.may have following properties. Random walk.Random walk. Constant velocities.( Traffic simulation, Granular Constant velocities.( Traffic simulation, Granular

Model )Model )

Computational UniversalityComputational Universality

A lot earlier than I, Wolfram proved A lot earlier than I, Wolfram proved this. I have not studied his theory yet.this. I have not studied his theory yet. He postulates that infinite class four He postulates that infinite class four

cellular automata are capable of cellular automata are capable of Universal Computation.Universal Computation.

Even logic gates can be implemented Even logic gates can be implemented by Cellular Automataby Cellular Automata

Proof of TM Proof of TM CA(1)CA(1)

Def. of Turing MachineDef. of Turing Machine M = (Q,∑,Г,δ,qo,M = (Q,∑,Г,δ,qo, ㅁㅁ ,F),F)

Q:Q: a set of internal statesa set of internal states ∑∑: a set of: a set of input alphabetsinput alphabets Г: a set ofГ: a set of tape alphabetstape alphabets ㅁㅁ : blank symbol: blank symbol qo: initial stateqo: initial state F: final statesδF: final statesδ 는 는 transition functiontransition function 이다이다 . . δ: Q*Г δ: Q*Г Q*Г*{L,R} Q*Г*{L,R}

L,R direction of the headerL,R direction of the header of the TMof the TM


Let’s suppose following set of statesLet’s suppose following set of states {(0,x0),….(0,xn),(q0,x0),…,(q0,xn),{(0,x0),….(0,xn),(q0,x0),…,(q0,xn),

…………,(qn,xn)}…………,(qn,xn)} {(x,y)|x is the state of the header,0 {(x,y)|x is the state of the header,0

means that no header point the state, y means that no header point the state, y is the alphabet of the input tape.}is the alphabet of the input tape.}


The transition function is defined like this,The transition function is defined like this,

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

x(i),x’(i) ∈ ∑x(i),x’(i) ∈ ∑

0,q0,…,qn ∈ Q0,q0,…,qn ∈ Q

D ∈ {L,D}D ∈ {L,D}

And.. This can be translated like this,,And.. This can be translated like this,,


It could be It could be helpful to helpful to understand understand this to remind this to remind the Wolfram’s the Wolfram’s formal rules.formal rules.

And this And this means that means that the proof the proof ends.ends.


AssumptionsAssumptions There are infinite number of cells.There are infinite number of cells. TM’s input tape is the CA’s initial TM’s input tape is the CA’s initial

condition.condition. But at least, given TM, this proof But at least, given TM, this proof

shows CA can be constructed.shows CA can be constructed.

Partitioning CA(BCA)Partitioning CA(BCA)

DNA Computing with BCADNA Computing with BCA pca.htmlpca.html

CACABCA(1)BCA(1)

The rule table must be changed.The rule table must be changed. And the time step can be doubled. And the time step can be doubled.

CACABCA(2)BCA(2)

Let’s suppose a 1-dim multi-state CA.Let’s suppose a 1-dim multi-state CA. And it has this set of states and And it has this set of states and

rules.rules. {….Sa,Sb,…..Si,Sj…..}{….Sa,Sb,…..Si,Sj…..} {….o(Sa,Sb,Si)……o(Sb,Si,Sj)……}{….o(Sa,Sb,Si)……o(Sb,Si,Sj)……}

You can think of the Wolfram’s 1 dim cellular You can think of the Wolfram’s 1 dim cellular automata.automata.

CACABCA(3)BCA(3)

The set of states of the BCA of the CA The set of states of the BCA of the CA should have the joined states. should have the joined states. (Si,Sj),(Sa,Sb) for all pair of the states of the (Si,Sj),(Sa,Sb) for all pair of the states of the

original CA. original CA. That is, the result set will be {..Sa,Sb,…(Si,Sj),That is, the result set will be {..Sa,Sb,…(Si,Sj),

(Sa,Sb)….} like this. (Sa,Sb)….} like this. And then add following rules to the rule And then add following rules to the rule

table of the BCAtable of the BCA Si,SjSi,Sj((Si,Sj),(Si,Sj)) Sa,Sb((Si,Sj),(Si,Sj)) Sa,Sb((Sa,Sb),(Sa,Sb)) ((Sa,Sb),(Sa,Sb)) (Si,Sj) ,(Sa,Sb) (Si,Sj) ,(Sa,Sb) (o(Si,Sj,Sa),o(Sj,Sa,Sb)) (o(Si,Sj,Sa),o(Sj,Sa,Sb))

CACABCA(4)BCA(4)

It is proved that any given It is proved that any given Turing Machine can be Turing Machine can be transformed into atransformed into a BCA.BCA.

And BCA can be directly used as the And BCA can be directly used as the model of the DNA Computing.model of the DNA Computing.(Winfree 96’).(Winfree 96’).

Winfree’s DNA Winfree’s DNA Computing(1)Computing(1)


This is so explicitly described in the This is so explicitly described in the first part of his thesis.first part of his thesis.

He uses only “Ligation” to implement He uses only “Ligation” to implement a BCA.a BCA.


First express your problem via computer program. First express your problem via computer program. Convert that program into a blocked cellular automaton.Convert that program into a blocked cellular automaton.

Create small molecules (H-shaped and linear) which self-Create small molecules (H-shaped and linear) which self-assemble to create the initial molecule( or initial assemble to create the initial molecule( or initial molecules, if search over a FSA=generated set of strings molecules, if search over a FSA=generated set of strings is desired.)is desired.)

Create small H-shaped molecules encoding the rule table Create small H-shaped molecules encoding the rule table for your program.for your program.

Mix the molecules created in steps 2 and 2 together in a Mix the molecules created in steps 2 and 2 together in a test tube, and keep under precise conditions test tube, and keep under precise conditions (temperature, salt concentrations) as the DNA lattice (temperature, salt concentrations) as the DNA lattice crystallizes.crystallizes.

When the solution turns blue, ligate, cut the crossovers, When the solution turns blue, ligate, cut the crossovers, and extract the strand with the halting symbol.and extract the strand with the halting symbol.

Sequence the answer.Sequence the answer.


Limits of this method.Limits of this method. Shortly speaking, this is another Shortly speaking, this is another

approach to the crystal computation. approach to the crystal computation. This is thought to be another This is thought to be another hardware for the cellular automata. hardware for the cellular automata. Winfree just implements this Winfree just implements this technique with DNA….. technique with DNA…..

But not that good.But not that good.

Future WorkFuture Work

Study crystal computation, study ligation Study crystal computation, study ligation and try winfree’s work again.and try winfree’s work again.

In my opinion, to successfully compute In my opinion, to successfully compute with DNA using the winfree’s method, we with DNA using the winfree’s method, we should have more knowledge about Nano should have more knowledge about Nano technology to control more. So.. ,until technology to control more. So.. ,until then, we may find another approach to then, we may find another approach to using DNA molecules. And if possible I’ll using DNA molecules. And if possible I’ll study about its possibilities.study about its possibilities.

cellular automata & dna computing

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