Theoretical and Experimental DNA Computation (Natural ...

82 4 Complexity Issues
connecting a node at level k − 1 to a node at level k. We thus have a total, in
simulating a circuit C of size m and depth d, of
d�
|{g : depth(g) =k} |+ |{(g,h) :depth(h) =kand(g,h) ∈ Edges(C)} |
k=1
distinct pour operations being performed. Obviously (since every gate has a
unique depth)
d�
|{g : depth(g) =k} |= m
k=1
Furthermore,
d�
|{g : depth(g) =k} |+ |{(g,h) :depth(h) =kand(g, h) ∈ Edges(C)} |≥2m
k=1
since every gate has at least two inputs. It follows that the total number of
pour operations performed over the course of the simulation is at least 3m.
Despite these observations, Ogihara and Ray’s work is important because
it establishes the Turing-completeness of DNA computation. This follows from
the work of Fischer and Pippenger [60] and Schnorr [139], who described
simulations of Turing Machines by combinational networks. Although a Turing
Machine simulation using DNA has previously been described by Reif [128],
Ogihara and Ray’s method is simpler, if less direct.
4.5 An Alternative Boolean Circuit Simulation
Since it is well known [54, 75, 157] that the NAND function provides a complete
basis by itself, we restrict our model to the simulation of such gates. In
fact, the realization in DNA of this basis provides a far less complicated simulation
than using other complete bases. It is interesting to observe that the
fact that NAND offers the most suitable basis for Boolean network simulation
within DNA computation continues the traditional use of this basis as a fundamental
component within new technologies, from the work of Sheffer [143],
that established the completeness of NAND with respect to propositional
logic, through classical gate-level design techniques [75], and, continuing, in
the present day, with VLSI technologies both in nMOS [106], and CMOS [159,
pp. 9–10].
The simulation proceeds as follows: An n-input, m-output Boolean network
is modelled as a directed acyclic graph, S(V,E), in which the set of
vertices V is formed from two disjoint sets, Xn, theinputs of the network (of
which there are exactly n) andG, thegates (of which exactly m are distinguished
as output gates). Each input vertex has in-degree 0 and is associated
with a single Boolean variable, xi. Each gate has in-degree equal to 2 and is