Theoretical and Experimental DNA Computation (Natural ...

50 3 Models of Molecular Computation
Parallel filtering model
A detailed description of our parallel filtering model appears in [12]. This
model was the first to provide a formal framework for the easy description of
DNA algorithms for any problem in the complexity class NP. Lipton claims
some generalisation of Adleman’s style of computation in [98], but it is difficult
to see how algorithms for different problems may be elegantly and universally
expressed within his model. Lipton effectively uses the same operations as
Adleman, but does not explicitly describe the operation set. In addition, he
describes only one algorithm (3SAT), whereas in subsequent sections we show
how our model provides a natural description for any NP-complete problem
through many examples.
As stated earlier, within our model all computations start with the construction
of the initial set of strings. Here we define the basic legal operations
on sets within the model. Our choice is determined by what we know can
be effectively implemented by very precise and complete chemical reactions
within the DNA implementation. The operation set defined here provides the
power we claim for the model but, of course, it might be augmented by additional
operations in the future to allow greater conciseness of computation.
The main difference between the parallel filtering model and those previously
proposed lies in the implementation of the removal of strings. All other models
propose separation steps, where strings are conserved, andmaybeusedlater
in the computation. Within the parallel filtering model, however, strings that
are removed are discarded, and play no further part in the computation. This
model is the first exemplar of the so-called “mark and destroy” paradigm of
molecular computing.
• remove(U, {Si}). This operation removes from the set U, in parallel, any
string which contains at least one occurrence of any of the substrings Si.
• union({Ui},U). This operation, in parallel, creates the set U which is the
set union of the sets Ui.
• copy(U, {Ui}). In parallel, this operation produces a number of copies, Ui,
of the set U.
• select(U). This operation selects an element of U at random; if U is the
empty set then empty is returned.
From the point of view of establishing the parallel time complexities of algorithms
within the model, these basic set operations will be assumed to take
constant time. However, this assumption is reevaluated in Chap. 4.
A first algorithm
We now provide our first algorithmic description within the model. The problem
solved is that of generating the set of all permutations of the integers 1
to n. Apermutationisarearrangement of a set of elements, where none are
removed, added, or changed. The initial set and the filtering out of strings