LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Solving Graph Problems by P Systems 19
and the set R contains the following rules (we also give explanations about the
use of these rules during the computations):
1. [ 2ai] 0
2 → [ 0
2ti] 2 [ 2
2. [ 2 di → di+1] 0
2
0
fi] 2 , 1 ≤ i ≤ n.
, 1 ≤ i ≤ n − 1.
3. [ 2 dn → qqh0] 0
2 .
We omit the explanations about the use of the rules of types 1, 2, and 3,
because they are the same as the explanations in the proof of Theorem 2.
Now we pass to the next phase of computation – counting the number of
objects ti (1 ≤ i ≤ n) in each membrane with label 2, which corresponds to
the cardinality of each subset.
4. [ 2 q] 0
2 → [ 2
5. [ 2 q] −
2 → [ 2
] −
2 u.
] 0
2 u.
6. [ 2ti → ab] −
2 , 1 ≤ i ≤ n.
7. [ 2hi → h ′ 0
i ] 2 , 0 ≤ i ≤ k.
8. [ 2h ′ +
i → hi+1]
2 , 0 ≤ i ≤ k − 1.
9. [ 2a] 0
2 → [ +
2 ] 2 u.
10. [ 2b] +
2 → [ 0
2 ] 2u. The rules of types 4-10 are used in the same way as the corresponding rules
from the proof of Theorem 2, the only difference is that at step n +2, ti
evolves to ab.
The rules of types 7, 8, 9, and 10 are applied as many times as possible (in
one step rules of types 7 and 9, in the next one rules of types 8 and 10, and
then we repeat the cycle). Clearly, at step n +2+2k, a membrane contains
object hk if and only if the cardinality of the corresponding subset is at least
k. Atstepn +3+2k, in the membrane whose corresponding subset has
cardinality more than k, hk evolves to h ′ k , and one copy of a changes its
polarization to positive. This membrane will no longer evolve, as no further
rule can be applied to it. In the membrane whose corresponding subset has
cardinality exactly k, hk evolves to h ′ k , and its polarization remains neutral,
because there is no copy of a which can be used. We pass to the next phase
of computation – checking whether a subset with cardinality k is a clique.
(A subset A of vertices is a clique if and only if for each edge (vi,vj) ∈ E ′ ,
xi ∈ V − A or xj ∈ V − A, i.e., E ′ ⊆ V × (V − A) ∪ (V − A) × V . The process
of checking whether a subset with cardinality k is a clique is based on this
fact.)
11. [ 2h ′ 0
k → qqg] 2 .
12. [ 2fi → yizi] −
2 ,1≤ i ≤ n.
13. [ 2g → g0] −
2 ,
[ 2ei,j,l → ei,j,l−1] α
2 ,1≤ i, j ≤ n, 1 ≤ l ≤ n +2k +5,α∈{+, −, 0}.
At step n +4+2k, in the membranes with label 2 and polarization 0, h ′ k
evolves to qqg. Atstepn+5+2k,onecopyofqexits the membrane, changing
its polarization to negative. At step n +6+2k, in parallel, fi (1 ≤ i ≤ n)
evolves to yizi, g evolves to g0 and an object ei,j,0 appears for each (vi,vj) ∈