Theoretical and Experimental DNA Computation (Natural ...

60 3 Models of Molecular Computation
We now sort the remaining strands according to how many bags they encode.
(1) for i =0toq − 1 do begin
(2) for j − 1 down to 0 do begin
(3) separate(+(Nj,i+1), −(Nj,i+1))
(4) Nj+1 ← merge(+(Nj,i+1),Nj+1)
(5) Nj ←−(Nj,i+1)
(6) end for
(7) end for
Line 3 separates each tube according to the value of i, and line 4 performs the
right shift of selected strands. We then search for a final output:
(1) Read N1
(2) else if empty then read N2
(3) else if empty then read N3
(4) ...
3.3 Splicing Models
Since any instance of any problem in the complexity class NP may be expressed
in terms of an instance of any NP-complete problem, it follows that
the multi-set operations described earlier at least implicitly provide sufficient
computational power to solve any problem in NP. We do not believe they provide
the full algorithmic computational power of a Turing Machine. Without
the availability of string editing operations, it is difficult to see how the transition
from one state of the Turing Machine to another may be achieved using
DNA. However, as several authors have recently described, one further operation,
the so-called splicing operation, will provide full Turing computability.
Here we provide an overview of the various splicing models proposed.
Let S and T be two strings over the alphabet α. Then the splice operation
consists of cutting S and T at specific positions and concatenating the resulting
prefix of S with the suffix of T and concatenating the prefix of T with
the suffix of S (Fig. 3.5). This operation is similar to the crossover operation
employed by genetic algorithms [68, 96].
Splicing systems date back to 1987, with the publication of Tom Head’s
seminal paper [77] (see also [78]). In [50], the authors show that the generative
power of finite extended splicing systems is equal to that of Turing Machines.
For an excellent review of splicing systems, the reader is directed to [120].
Reif’s PAM model
In [128], Reif within his so-called Parallel Associative Memory Model describes
a Parallel Associative Matching (PA-Match) operation. The essential con-