Theoretical and Experimental DNA Computation (Natural ...

48 3 Models of Molecular Computation
(5) for all vertices adjacent to this vertex do
(6) From red, remove strings encoding
adjacent vertex red
(7) From blue, remove strings encoding
adjacent vertex blue
(8) From green, remove strings encoding
adjacent vertex green
(9) end for
(10) Merge red, green and blue tubes to form new tube T
(11) end for
(12) Read what is left in T
Or more formally:
(1) Input(T)
(2) for i =1ton do begin
(3) Tr ← +(T,ri) and Tbg ←−(T,ri)
(4) Tb ← +(Tbg,bi) and Tg ←−(Tbg,bi)
(5) for all j such that ∈ E do begin
(6) Tr ←−(Tr,rj)
(7) Tg ←−(Tg,gj)
(8) Tb ←−(Tb,bj)
(9) end for
(10) T ← merge(Tr,Tg,Tb)
(11) end for
(12) Output(detect(T ))
At Step 1 we input all possible colorings of the graph. Then, for each vertex
vi ∈ V we perform the following steps: split T into three sets, Tr,Tg,Tb, where
Tr contains only strings containing ri, Tg contains only strings containing gi,
and Tb contains only strings containing bi (Steps 3–4). Then, for each edge
∈ E, we remove from these sets any strings containing ci = cj (i.e.,
those strings encoding colorings where adjacent vertices i and j are colored
the same) (Steps 5–9). Then, these sets are merged, forming the new set T
(Step 10), and the algorithm proceeds to the next vertex (Step 11). After
the coloring constraints for each vertex have been satisfied, we perform a
detection (Step 12). If T is nonempty then any string in T encodes a proper
3-vertex-coloring of G.
Satisfiability model
Lipton [98] described a solution to another NP-complete problem, namely the
so-called satisfiability problem (SAT). SAT may be phrased as follows: given
a finite set V = {v1,v2,...,vn} of logical variables, we define a literal to be a