Theoretical and Experimental DNA Computation (Natural ...

46 3 Models of Molecular Computation
the time taken to construct the initial set of strings and read out the final
solution may be very time consuming, and must also be taken into
account.
3. Each laboratory technique has associated with it a nonzero error rate.
Some techniques are far more error-prone than others, so the choice of
laboratory techniques directly affects the probability of success of a DNAbased
algorithm.
We therefore focus on abstract models in this chapter, and consider their physical
implementation in Chap. 5. The models fall into four natural categories:
• Filtering
• Splicing
• Constructive
• Membrane
3.2 Filtering Models
In all filtering models (motivated by Adleman [3] and contemporaneously
generalized by Lipton [98] and Amos et al. [94, 12]), a computation consists of
a sequence of operations on finite multi-sets of strings. Multi-sets are sets that
may contain more than one copy of the same element. It is normally the case
that a computation begins and terminates with a single multi-set. Within the
computation, by applying legal operations of a model, several multi-sets may
exist at the same time. We define operations on multi-sets shortly, but first
consider the nature of an initial set.
An initial multi-set consists of strings which are typically of length O(n)
where n is the problem size. As a subset, the initial multi-set should include
all possible solutions (each encoded by a string) to the problem to be solved.
The point here is that the superset, in any implementation of the model, is
supposed to be relatively easy to generate as a starting point for a computation.
The computation then proceeds by filtering out strings which cannot
be a solution. For example, the computation may begin with a multi-set containing
strings representing all possible three-colorings of a graph, and then
proceed by removing those that encode illegal colorings.
To give another example, if the problem is to generate a permutation of
the integers 1,...,n, then the initial multi-set might include all strings of the
form p1i1p2i2 ...pnin where each ik maybeanyoftheintegersintherange
[1,...,n]andpk encodes the information “position k.” Here, as will be typical
for many computations, the multi-set has cardinality which is exponential in
the problem size. For our example of finding a permutation, we should filter
out all strings in which the same integer appears in at least two locations pk.
Any of the remaining strings is then a legal solution to the problem.
We now describe the important features of the various filtering models.