LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

58 Francesco Bernardini, Marian Gheorghe, and Mike Holcombe
Proof. Let γ be a formula in conjunctive normal form with m clauses, C1,...,Cm,
each one being a disjunction, and the variables used are x1,...,xn. The following
EOPP system, Π =(µ, X), may be then constructed:
µ =[1[2...[m+1]m+1 ...]2]1,
X =(V,Q,M1,...Mm+1,Φ,F,I),
where:
– V = {(i1,...,ik) | ij ∈{0, 1}, 1 ≤ j ≤ k, 1 ≤ k ≤ n};
– Q = {q};
– M1 = ...= Mm = ∅, Mm+1 = {(0), (1)};
– Φ = {φ0,φ1,φ2};
• φ0 =(∅,...,∅, {(i1,...,ik) → (i1,...,ik, 0) |
ij ∈{0, 1}, 1 ≤ j ≤ k, 1 ≤ k ≤ n − 1}),
• φ1 =(∅,...,∅, {(i1,...,ik) → (i1,...,ik, 1) |
ij ∈{0, 1}, 1 ≤ j ≤ k, 1 ≤ k ≤ n − 1}),
• φ2 =({(i1,...,ij =1,...,in) → ((i1,...,ij =1,...,in),out) |
xj is present in C1, 1 ≤ j ≤ n}∪
{(i1,...,ij =0,...,in) → ((i1,...,ij =0,...,in),out) |
¬xj is present in C1, 1 ≤ j ≤ n},
...,
{(i1,...,ij =1,...,in) → ((i1,...,ij =1,...,in),out) |
xj is present in Cm, 1 ≤ j ≤ n}∪
{(i1,...,ij =0,...,in) → ((i1,...,ij =0,...,in),out) |
¬xj is present in Cm, 1 ≤ j ≤ n},
∅);
where (i1,...,ij =1,...,in) and(i1,...,ij =0,...,in) denoteelements
of V having the j−th component equal to 1 and 0, respectively;
– F (q, φk) ={q}, 0 ≤ k ≤ 2;
– I = {q}.
The system Π starts from state q with ∅,...,∅, {(0), (1)}. By applying n
times φ0 and φ1 in parallel one generates all truth values for the n variables in
the form of 2n symbols (i1,...,in) withij =1orij = 0 indicating that variable
xj is either true or false. All these combinations are obtained in n steps in state
q. In the next m steps φ2 checks whether or not at least one truth-assignment
satisfies all clause; this, if exists, will exit the system. The SAT problem is solved
in this way in n + m steps. ✷
There are some important similarities between the above theorem and Theorem
5 in [1]:
– the same membrane structure;
– the first m initial regions empty;
– the truth and false values introduced in parallel;
but also relevant distinct features:
– less states and a simpler definition of F in the above theorem;
– linear (in n) number of symbols in V and rules, in the case of Theorem 5 [1],
but exponential number of corresponding components, in the above theorem.