LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

10 Artiom Alhazov, Carlos Martín-Vide, and Linqiang Pan
16. [ 2 fm+1] −
2 → [ 2
] −
2 fm+1.
The rule of type 16 sends to the skin membrane the objects fm+1 appearing
in the membranes with label 2.
17. [ 1 fm+1 → fm+2t] 0
1 .
By using the rule of type 17 the objects fm+1 in the skin evolve to objects
fm+2t. The objects t in the skin are produced simultaneously with the appearance
of the objects d3n+2m+2 in the skin, and they will show that there
exists some subset of vertices which is a vertex cover with cardinality k.
18. [ 1t] 0
1 → [ +
1 ] 1 t.
The rule of type 18 sends out of the system an object t changing the polarization
of the skin to positive, then objects t remaining in the skin are
not able to evolve. Hence, an object fm+2 can exit the skin producing an
object yes. This object is then sent out to the environment through the rule
of type 19, telling us that there exists a vertex cover with cardinality k, and
the computation halts.
19. [ 1fm+2] +
1 → [ −
1 ] 1 yes.
The applicability of the rule of type 19 changes the polarization in the skin
membrane to negative in order that the objects fm+2 remaining in it are not
able to continue evolving.
20. [ 1d3n+2m+3] 0
1 → [ +
1 ] 1 no.
Bytheruleoftype20theobjectd3n+2m+3 only evolves when the skin has
neutral charge (this is the case when there does not exist any vertex cover
with cardinality k). Then the system will evolve sending out to the environment
an object no and changing the polarization of the skin to positive, in
order that objects d3n+2m+3 remaining in the skin do not evolve.
From the previous explanation of the use of rules, one can easily see how these
P systems work. It is easy to prove that the designed P systems are deterministic.
Now, we prove that the family Π =(Π(t))t∈N solves the vertex cover problem
in linear time.
The above description of the evolution rules is computable in an uniform
way. So, the family Π =(Π(t))t∈N is polynomially uniform because:
– The total number of objects is 2nm +4nk +9n +4m − k +8∈ O(n 3 ).
– The number of membranes is 2.
– The cardinality of the initial multisets is 2.
– The total number of evolution rules is O(n 3 ).
– The maximal length of a rule (the number of symbols necessary to write a
rule, both its left and right sides, the membranes, and the polarizations of
membranes involved in the rule) is 13.
We consider the (input) function g : IV CP →∪t∈NI Π(t), defined as follows:
g(G) ={ex,(i,j)|1 ≤ x ≤ m, 1 ≤ i, j ≤ n,
the two vertices of edge ex are vi and vj}.