Theoretical and Experimental DNA Computation (Natural ...

78 4 Complexity Issues
Ω = {∧, ∨, ¬}. In addition, S has some unique output node, s, withoutdegree
zero. An example Boolean circuit for the three-input majority function
is depicted in Fig. 4.1.
x 1
x x
2 3
Fig. 4.1. Boolean circuit for the three-input majority function
The two standard complexity measures for Boolean circuits are size and depth:
the size of a circuit is S, andm is the number of gates in S; its depth, d, is
the number of gates in the longest directed path connecting an input vertex
to an output gate. The circuit depicted in Fig. 4.1 has size 8 and depth 3.
In [113], Ogihara and Ray describe the simulation of Boolean circuits
within a model of DNA computation. The basic structure operated upon
is a tube, U, which contains strings representing the results of the output of
each gate at a particular depth. The initial tube contains strands encoding
the values of each of the inputs Xn.
In what follows, Ω = {∧, ∨}. Foreachi, 1≤ i ≤ m, astringσ[i] isfixed.
The presence of σ[i] inU signifies that gi evaluates to 1. The absence of σ[i] in
U signifies that gi evaluates to 0. The initial tube, U, (i.e., a tube representing
the inputs Xn is created as follows:
for each gate xi do
if xi =1then U ← U ∪ σ[i]
end for
S