13th International Conference on Membrane Computing - MTA Sztaki

Zs. Gazdag, G. Kolonits
membrane system by the string c 1 . . . c m ∈ O ∗ n, where, for every 1 ≤ i ≤ m,
ĉ i = C i .
A P system deciding SAT. Now we define a family Π := (Π(i)) i∈N of
recognizer P systems that solves SAT in weak linear time. For every n ∈ N, let
Π(n) := (O, H, µ, w 1 , . . . , w 3 , R), where:
– O := O n ∪ {d 1 , . . . , d n+3 , yes, no};
– H := {1, . . . , n + 3};
– µ := [[[ ] 3 ] 2 ] 1 , where the input membrane is [ ] 3 ;
– w 1 := ε, w 2 := d 1 and w 3 := ε;
– R is the set of the following rules (in some cases we also give explanations
of the presented rules):
(a) [c → c 1 c 2 ] i+2 , for every 1 ≤ i ≤ n and c, c 1 , c 2 ∈ O n with x i , ¯x i ∉ ĉ,
ĉ 1 = ĉ ∪ {x i } and ĉ 2 = ĉ ∪ {¯x i }
(for every 1 ≤ i ≤ n, these rules will replace those clauses in membrane
i + 2 that do not contain x i or ¯x i by two other clauses, a clause that
additionally contains x i , and another one that contains ¯x i );
(b) [ ] i+2 → [K i ] i+3 [O − K i ] i+3 , for every 1 ≤ i ≤ n and K i = {c ∈ O n |
x i ∈ ĉ}
(for every 1 ≤ i ≤ n, these rules will separate the objects in membranes
with label i + 2 according to that whether the clauses represented by the
objects contain x i or not; the new membranes will have label i + 3);
(c) [d i → d i+1 ] 2 , for every 1 ≤ i ≤ n + 2;
(d) [c] n+3 → ε, for every c ∈ O n such that ĉ is a complete clause in C n ;
(e) d n+2 [ ] n+3 → [yes] n+3 ,
[yes] n+3 → [ ] n+3 yes,
[yes] 2 → [ ] 2 yes,
[yes] 1 → [ ] 1 yes;
(f) [d n+2 ] 2 → [d n+3 ] 2 [no] 2 ,
[no] 2 → [ ] 2 no,
[no] 1 → [ ] 1 no.
Next we give an example to demonstrate how our P systems create new
clauses from the input and separate them into new membranes. We will refer to
this example also in Section 4, where we will discuss the possible improvements
of our P systems.
Example 1. We show the working of Π(3) on a formula with variables in X 3 .
For the better readability, we denote these variables by x, y and z, respectively.
Moreover, the objects in O 3 are denoted by sequences of literals occurring in
the corresponding clauses of the formula, i.e., the symbols in O 3 are now strings
over the set of literals.
Let the input formula be ϕ := {{x, y, z}, {¬x}, {¬y}, {¬z}}. Then Π(3) is
started with symbols xyz, ¯x, ȳ, ¯z in the input membrane, thus at the beginning
the membrane with label 3 looks as follows: [xyz, ¯x, ȳ, ¯z] 3 . In the first step, the
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