LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

8 Artiom Alhazov, Carlos Martín-Vide, and Linqiang Pan
1. [ 2di] 0
2 → [ +
2di] 2 [ −
2di] 2 , 1 ≤ i ≤ n.
By using a rule of type 1, a membrane with label 2 is divided into two
membranes with the same label, but with different polarizations. These rules
allow us to duplicate, in one step, the total number of membranes with
label 2.
2. [ 2ex,(1,j) → rx,1] +
2 ,1≤ x ≤ m, 2≤ j ≤ n.
[ e 2 x,(1,j) → λ] −
,1≤ x ≤ m, 2≤ j ≤ n.
2
The rules of type 2 try to implement a process allowing membranes with
label 2 to encode whether vertex v1 appears in a subset of vertices, in such a
way that if vertex v1 appears in a subset of vertices, then the objects ex,(1,j) encoding edges ex which are adjacent to vertex v1 will evolve to objects rx,1
in the corresponding membranes with label 2; otherwise, the objects ex,(1,j) will disappear.
3. [ 2ex,(i,j) → ex,(i−1,j)] +
2 ,1≤ x ≤ m, 2≤ i ≤ n, i ≤ j ≤ n.
[ 2ex,(i,j) → ex,(i−1,j)] −
2 ,1≤ x ≤ m, 2≤ i ≤ n, i ≤ j ≤ n.
The evolving process described previously is always made with respect to
the vertex v1. Hence, the rules of type 3 take charge of making a cyclic path
through all the vertices to get that, initially, the first vertex is v1, thenv2,
and so on.
4. [ 2ci,j → ci+1,j] +
2 ,0≤ i ≤ k − 1, 1 ≤ j ≤ 2n − 1.
[ 2ck,j → λ] +
2 ,1≤ j ≤ 2n − 1.
The rules of type 4 supply counters in the membranes with label 2, in such a
way that we increase the first subscript of ci,j, when the membrane has not
more than k vertices; when the membrane has more than k vertices, then
the counter-object ck,j will disappear. So when the process of generating
all subsets of vertices is finished, the object ck,j (note the first subscript is
k) will appear only in the membrane with label 2 encoding a subset with
cardinality exactly k.
5. [ 2di] +
2 → [ 0
2 ] 2di, 1≤ i ≤ n.
[ 2di] −
2 → [ 0
2 ] 2di, 1≤ i ≤ n.
di[ 2 ] 0
2 → [ 0
2di+1] 2 ,1≤ i ≤ n − 1.
The rules of type 5 are used as controllers of the generating process of all
subsets of vertices and the listing of adjacent edges: the objects d are sent to
the membrane with label 1 at the same time the listing of adjacent edges and
the counting of vertices are made, and they come back to the membranes
with label 2 to start the division of these membranes.
6. [ ri,j → ri,j+1]
2 0
,1≤ i ≤ m, 1≤ j ≤ 2n − 1.
2
The use of objects r in the rules of types 12, 13, and 14 makes necessary to
perform a “rotation” of these objects. This is the mission of the rules of type
6.
7. [ ci,j → ci,j+1]
2 0
,0≤ i ≤ k, 1≤ j ≤ 2n − 1.
2
The second subscript of ci,j (also in the rule of type 9) is used to control
when the process of checking whether a subset of vertices with cardinality k
is a vertex cover will start.