13th International Conference on Membrane Computing - MTA Sztaki

A new approach for solving SAT by P systems with active membranes
(c) [a] h → [ ] h b, for h ∈ H, a, b ∈ O
(communication rules; an object is sent out of the membrane, possibly
modified during this process; the label cannot be modified);
(d) [a] h → b, for h ∈ H, a, b ∈ O
(dissolving rules; in reaction with an object, a membrane can be dissolved,
while the object specified in the rule can be modified);
(e) [a] h → [b] h [c] h , for h ∈ H, a, b, c ∈ O
(division rules for elementary membranes; in reaction with an object,
the membrane is divided into two membranes with the same label; the
object a specified in the rule is replaced in the two new membranes by
(possibly new) objects b, c, and the remaining objects are duplicated);
(f) [ ] h1 → [K] h2 [O − K] h3 , for h 1 , h 2 , h 3 ∈ H, K ⊂ O
(2-separation rules for elementary membranes, with respect to a given
set of objects; the membrane is separated into two new membranes with
possibly different labels; the objects from each set of the partition of the
set O are placed in the corresponding membrane).
As usual, Π works in a maximal parallel manner: rules of type (a) are executed
in parallel, while at most one rule out of all rules of types (b)-(f) can
be applied to the same membrane in a step of the system. We say that Π is a
recognizing P system if (1) Π has a designated input membrane i 0 , (2) a string
w, called the input of Π, can be added to the system by placing it into the region
i 0 in the initial configuration, (3) O has two designated objects yes and no, and
(4) every computation of Π halts and sends out to the environment either yes
or no. Moreover, Π is called deterministic if, for every input w placed into i 0 ,
there is only a single computation of Π.
Let Π := (Π(i)) i∈N be a family of P systems. Π is uniform (by Turing
machines) if it can be constructed by a deterministic Turing machine. Moreover,
Π is polynomially uniform (by Turing machines) if, for every n ∈ N, Π(n) can
be constructed in polynomial time in n by a deterministic Turing machine (see
e.g. Section 12.2.1 in [9] for further details).
We say that Π solves SAT if, for every formula ϕ in CNF with size n, starting
Π(n) with a polynomial time encoding of ϕ, Π(n) sends out to the environment
yes if and only if ϕ is satisfiable. Finally, Π solves SAT in linear time if it is (1)
polynomially uniform and (2) the computation of Π(n) always halts in linear
number of steps in the size of the input formula. If only condition (2) holds, then
we say that Π solves SAT in weak linear time.
3 The Main Result
Encoding SAT instances. Here we describe how the input formulas are encoded
in the input membrane of our SAT solver P system. In fact, we employ a
trivial encoding, using symbols that are in one-to-one correspondence with the
clauses in C n (n ∈ N). For every n ∈ N, let O n be an alphabet with a bijection
between C n and O n . For a symbol c ∈ O n , we denote the corresponding clause
in C n by ĉ. Thus, a formula ϕ = {C 1 , . . . , C m } (m ∈ N) will be encoded in our
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