Theoretical and Experimental DNA Computation (Natural ...

x 1
e 1,5
(a)
g 5
4.4 Ogihara and Ray’s Boolean Circuit Model 81
g5 g7 g6
e e
5,7
(b)
Fig. 4.3. (a) Splinting for ∨-gate. (b) Splinting for ∧-gate
described in the previous section. The results obtained were ambiguous, although
Ogihara and Ray claim to have identified the ambiguity as being
caused by pipetting error.
Complexity analysis
Within the strong model, it appears that the methods of [113] require a number
of pour operations where linearity in the size of the circuit simulated is
easily demonstrated. We recall that the basic circuit model considered in their
paper is that of “semi-unbounded fan-in circuits”:
Definition 1. A semi-unbounded fan-in Boolean circuit of n inputs is a labelled,
directed acyclic graph whose nodes are either inputs or gates. Inputs,
of which there are exactly 2n, have fan-in 0 and each is labelled with a unique
Boolean literal xi or xi (1 ≤ i ≤ n). Gates are either ∧ (conjunction) or ∨
(disjunction) gates. The former have fan-in of exactly 2; the latter may have
an arbitrary fan-in. There is a unique gate with fan-out of 0, termed the output.
The depth of a gate is the length of the longest directed path to it from an
input. The size of such a circuit is the number of gates; its depth is the depth
of the output gate.
We note, in passing, that there is an implicit assumption in the simulation of
[113] that such circuits are levelled, i.e. every gate at depth k (k >0) receives
its inputs from nodes at depth k-1. While this is not explicitly stated as being
a feature of their circuit model, it is well known that circuits not organized in
this way can be replaced by levelled circuits with at most a constant factor
increase in size.
The simulation associates a unique DNA pattern, σ[i], with each node of
the circuit, the presence of this pattern in the pool indicating that the i th
node evaluates to 1. For a circuit of depth d the simulation proceeds over
d rounds: during the k th round only strands associated with nodes at depth
k-1 (which evaluate to 1) and gates at depth k are present. The following
pour operations are performed at each round: in performing round k (k ≥ 1)
there is an operation to pour the strand σ[gi] for each gate gi at depth k.
Furthermore, there are operations to pour a “linker” for each different edge
7,6