Cellular Automata & DNA Computing

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 ofinput alphabets • Г: a set oftape alphabets • ㅁ: blank symbol • qo: initial state • F: final statesδ는 transition function이다. • δ: Q*Г  Q*Г*{L,R} • L,R direction of the headerof 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 aBCA. • 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.

Cellular Automata & DNA Computing