Theoretical and Experimental DNA Computation (Natural ...

4.2 An Existing Model of DNA Computation 73
match those needed for a conventional computation (the product of the running
time and the number of processors used).
It is clearly crucial, especially when judging the candidacy of a proposed
DNA computation for the role of “killer application,” to have a firm grasp of
the computational resources that it requires. It is often the case that there is
not an agreed-upon model of computation in the literature within which we
may agree what the required resources are for any particular computation.
This chapter attempts to address these issues in a realistic way.
Traditional computational complexity theory [5, 63] is concerned with
quantifying the resources (generally time and space) needed to solve computational
problems. Meaningful analysis of the complexity of algorithms may
only take place in the context of an agreed-upon model of computation, or
machine model. Many different machine models have been proposed in the
past, including the Deterministic Turing Machine, Boolean circuit [54, 75],
and P-RAM [61, 64]. The field of DNA computing also suffers from the problem
of proliferation of machine models. Several models have been proposed,
within which we may construct algorithms for the solution of computational
problems. However, complexity analyses of algorithms within different models
of DNA computation are meaningless, since there are no uniform definitions
of the concepts of time and space. Furthermore, if we are to compare a DNAbased
model with a more traditional machine model, we require a way of
demonstrating equivalence between the two.
4.2 An Existing Model of DNA Computation
We may describe abstract models of computation without necessarily considering
their implementation. In [64], for example, for the sake of emphasizing
what is inherently parallelizable within problems, the authors disregard constraints
of implementation. However, in what follows we are naturally constrained
by what is feasible in the intended mode of implementation; in this
case, what is possible in the molecular biology laboratory. We consider models
that operate upon sets of strings. It is generally the case that a DNA computation
starts and ends with a single set of strings. An algorithm is composed
of a sequence of operations upon one or more sets of strings. At the end of
the algorithm’s execution, a solution to the given problem is encoded as a
string in the final set. We use the term computational substrate to describe
the substance that is acted upon by the implementation of a model. Since
DNA is the underlying computational substrate of all models described (as
we shall see), we may naturally assume that all abstract models operate on
strings over a four-letter alphabet {A,G,C,T}. For a detailed description of
the structure of the DNA molecule, the reader is referred to [156, 155], as well
as to Chap. 1.