LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

112 Erzsébet Csuhaj-Varjú and Arto Salomaa
We now define the D0L productions for each node. We begin with the letters
of V2. In all nodes Mi,M ′ i , 1 ≤ i ≤ v, as well as in the node P ,wehavethe
productions
tj → tj, fj→ fj, 1 ≤ j ≤ v.
In all nodes Ni, 1 ≤ i ≤ c, we have the productions
tj → tjtj, fj → fjfj, tj → λ, fj → λ, 1 ≤ j ≤ v,
except in the case that tj or fj is characteristic for the clause Ci the barred
letters are removed from the former productions, giving rise to the production
tj → tj or fj → fj.
We then define the productions for the remaining letters. Each node Mi (resp.
M ′ i ), 1 ≤ i ≤ v − 1, has the production Si−1 → tiSi (resp. Si−1 → fiSi). The
node Mv (resp. M ′ v) has the production Sv−1 → tvR1 (resp. Sv−1 → fvR1).
Each node Ni, 1 ≤ i ≤ c − 1, has the production Ri → Ri+1. The node Nc has
the production Rc → E. The node P has the production E → λ. Each letter x,
barred or nonbarred, whose production has not yet been defined in some node,
has in this node the production x → G v . This completes the definition of the
standard network Γ .
The formula α is satisfiable exactly in case, after v+c+1 computation steps, a
word w over the alphabet V2 appears in the node P . Each such word w indicates
a truth-value assignment satisfying α. Moreover, because of the communication,
each such word w appears in all nodes in the next steps.
The verification of this fact is rather straightforward. There is only one
“proper” path of computation. In the first v steps a truth-value assignment
is created. (Actually all possible assignments are created!) In the next c steps it
is checked that each of the clauses satisfies the assignment. In the final step the
auxiliary letter is then eliminated. Any deviation from the proper path causes
the letter G to be introduced. This letter can never be eliminated. We use the
production G → G v instead of the simple G → G to avoid the unnecessary
communication of words leading to nothing. Thus, we have completed the proof
of our theorem. ✷
Coming back to our example, the required network possesses 65 nodes. The
alphabet V1 has 46 and the alphabet V2 40 letters. The productions for the node
M1, for instance, are
S0 → t1S1, tj → tj, fj → fj, 1 ≤ j ≤ 20,
and x → G 20 forallotherlettersx. ForthenodeN1 the productions are
R1 → R2, f3 → f3, f16 → f16, t18 → t18, x → xx,
for other letters x in V2. Furthermore,N1 has the production x → λ, for all
letters x in V2, and the production x → G 20 for all of the remaining letters x.