Cellular Automata & DNA Computing 우정철

Definition Of Cellular Automata Von Von Neuman’s Neuman’s Definition Wolfram’s Wolfram’s Definition Lyman Lyman Hurd’s Hurd’s Definition

Example of Cellular Automata Ising Models Ising Models Conway’s Game of Life Conway’s Game of Life Lattice gasses and the Margolus Neighborhood Lattice gasses and the Margolus Neighborhood Partitioning Cellular Automata. A simulation an HPP Lattice gas Partitioning Cellular Automata. A simulation an HPP Lattice gas Biological and Chemical Systems Biological and Chemical Systems

Features of Cellular automata Nonlinear Cellular automata Nonlinear Cellular automata Homoplectic and autoplectic systems Homoplectic and autoplectic systems Particle like structures Particle like structures Computational Universality Computational Universality Turing machine  Cellular Automata Turing machine  Cellular Automata Reversibility Reversibility

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

Computational Universality A lot earlier than I, Wolfram proved this. I have not studied his theory yet. 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. He postulates that infinite class four cellular automata are capable of Universal Computation. Even logic gates can be implemented by Cellular Automata Even logic gates can be implemented by Cellular Automata

Proof of TM  CA(1) Def. of Turing Machine Def. of Turing Machine M = (Q,∑,Г,δ,qo, ㅁ,F) M = (Q,∑,Г,δ,qo, ㅁ,F) Q: a set of internal states Q: a set of internal states ∑: a set of input alphabets ∑: a set of input alphabets Г: a set of tape alphabets Г: a set of tape alphabets ㅁ : blank symbol ㅁ : blank symbol qo: initial state qo: initial state F: final statesδ 는 transition function 이다. F: final statesδ 는 transition function 이다. δ: Q*Г  Q*Г*{L,R} δ: Q*Г  Q*Г*{L,R} L,R direction of the header of the TM L,R direction of the header of the TM

Proof of TM  CA(2) Let’s suppose following set of states Let’s suppose following set of states {(0,x0),….(0,xn),(q0,x0),…,(q0,xn),…………,(q n,xn)} {(0,x0),….(0,xn),(q0,x0),…,(q0,xn),…………,(q n,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.} {(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, 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. It could be helpful to understand this to remind the Wolfram’s formal rules. And this means that the proof ends. And this means that the proof ends.

Proof of TM  CA(5) Assumptions Assumptions There are infinite number of cells. There are infinite number of cells. TM’s input tape is the CA’s initial condition. TM’s input tape is the CA’s initial condition. But at least, given TM, this proof shows CA can be constructed. But at least, given TM, this proof shows CA can be constructed.

Partitioning CA(BCA) DNA Computing with BCA DNA Computing with BCA pca.html pca.html

CA  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.

CA  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 rules. And it has this set of states and 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 automata. 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. 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. (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. 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 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  ((Si,Sj),(Si,Sj)) Sa,Sb  ((Sa,Sb),(Sa,Sb)) (Si,Sj),(Sa,Sb)  (o(Si,Sj,Sa),o(Sj,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. 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’). 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. This is so explicitly described in the first part of his thesis. He uses only “Ligation” to implement a BCA. 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. 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 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. 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. 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. When the solution turns blue, ligate, cut the crossovers, and extract the strand with the halting symbol. Sequence the answer. Sequence the answer.

Winfree’s DNA Computing(5) Limits of this method. 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….. 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. But not that good.

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