LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

12 Artiom Alhazov, Carlos Martín-Vide, and Linqiang Pan
– From every fi (with 1 ≤ i ≤ m) the object fi+1 is obtained in some membranes
after the execution of 2 steps.
– The checking stage ends as soon as the object d3n+2m appears in the skin.
Therefore, the total number of steps of this stage is 2m.
The output stage starts immediately after the appearance of the object
d3n+2m in the skin and it is controlled by the objects fm+1 and fm+2.
– To produce the output yes the object fm+1 must have been produced in
some membrane with label 2 of the configuration C 5n+2m−1 . Then, after 4
steps the system returns yes to the environment, through the evolution of
objects fm+2 present in the skin, and when it has positive charge.
– To produce the output no, no object fm+1 appears in any membrane with
label 2 of the configuration C 5n+2m−1 . Then after 4 steps the system returns
no to the environment, through the evolution of objects d3n+2m+3 present
in the skin, and when it has neutral charge.
Therefore, the total number of steps in the output stage is 4.
Let us see that the family Π is linearly bounded, with regard to (g, h). For
that, it is enough to note that the time of the stages of the execution of Π(h(G))
with input g(G) is: (a) generation stage, 3n − 1 steps; (b) synchronization stage,
2n steps; (c) checking stage, 2m steps; and (d) output stage, 4 steps. Hence, the
total execution time of Π(h(G)) with input g(G) is5n +2m +3∈ O(n + m).
Now, let us see that the family Π is VCP-sound and VCP-complete, with
respect to the polynomial encoding (g, h). For that it is sufficient to verify that
the following results are true:
1. If s is a subset with cardinality k covering all edges, then at the end of
the checking stage (that is, in the configuration C5n+2m−1 ), the object fm+1
appears in the membrane associated with s.
2. If s is not a subset with cardinality k covering all edges, then at the end of
the checking stage (that is, in the configuration C5n+2m−1 ), the object fm+1
does not appear in the membrane associated with s.
Next we justify that the designed P systems Π(t), with t ∈ N, are recognizing
devices.
By inductive processes it can be proved that the configuration C5n+2m−1 verifies the following properties:
(a) It has 2n membranes with label 2. The membranes with label 2 encoding
subsets of vertices with cardinality exactly k have negative charge. The other
membranes with label 2 have neutral charge.
(b) The skin has neutral charge and its content is d 2n
3n+2m .
(c) If the object fm+1 is present in a membrane with label 2, then the subset of
vertices encoded by it is a vertex cover.
Proposition 51 Suppose that there exist exactly q membranes with label 2 (with
1 ≤ q ≤ � � n 5n+2m−1
k )ofC containing some object fm+1. Then, in the last step
of the computation, the P system sends out the object yes to environment.