13th International Conference on Membrane Computing - MTA Sztaki

D. Sburlan
assuming that M is in a state q ∈ Q and given an observation o ∈ O, then M will
perform the action λ(q, o) (it increments register r if λ(q, o) = inc, it decrements
register r if λ(q, o) = dec, and it does not modify the content of r if λ(q, o) =
skip). The output of M in response to a sequence of observations o 1 , o 2 , . . . , o k
consists in the applications of the actions given by λ(q 0 , o 1 ), . . . , λ(q k−1 , o k ) on
register r, where q 0 , . . . , q n is the sequence of states such that δ(q i−1 , o i ) = q i ,
1 ≤ i ≤ n; the sequence of observations o 1 , o 2 , . . . , o k is called accepted by M iff
q n ∈ F .
The system Φ = (Π, M) computes as follows. The systems Π and M run
in parallel: at each passing from a configuration C 1 to C 2 in a computation of
Π, based on the observation {x 1 , . . . , x k } of the pair (C 1 , C 2 ), the system M
changes its current state q to a new one p = δ(q, {x 1 , . . . , x k }); in addition,
M performs the action defined by λ(q, {x 1 , . . . , x k }). A computation of Φ is
considered successful if the above procedure is applied for each pair of consecutive
configurations in a halting computation of Π and the system M accepts the
sequence of observations determined by the computation of Π; in this case, the
result of the computation is the number stored in register r at its end. Collecting
all the values stored by r at the end of all possible successful computations of Φ
one obtains the set of integers N(Φ(Π, M)).
In case of a non-halting computation of Π, the system Φ does not produce
any output. The same outcome is obtained when M does not accept the sequence
of observations determined by the underlying computation of Π.
The families of all sets of numbers generated by Observer/Interpreter P systems,
having as core systems P systems with symbol objects, multiset rewriting
rules, at most k catalysts and one membrane is denoted by NOI(cat k ).
Because the system M recalls the definition of a GSM, in what follows we
will use a similar notation for the transition graph.
Example 1 Let Φ = (Π, M) such that Π = (O, C, µ, R 1 , w 1 , i 0 ) where O =
{a, a, a, c}, C = {c}, µ = [ ] 1 , w 1 = a, i 0 = 1, and R 1 is defined as follows:
R 1 = {a → aa,
a → a,
a → a,
ca → c}.
The system M is defined by the transition graph depicted in Figure 1.
The computation of Φ proceeds as follows. If the rule a → aa is the only
rule applied in the first k consecutive configurations of Π, then 2 k−1 objects a
are produced. During this exponential generation of objects a, the system M
remains in state q 0 (this is because M detects that the number of objects a does
not change). Assuming that in the k-th configuration both the rules a → aa and
a → a are applied, then M, being in state q 0 , can either remain in the same
state q 0 and the computation stops (an unsuccessful computation; the case a−
or a ↑, a ↑) or it can pass to state q 1 (the case a ↓, a ↑). However, there is no
guarantee that all the objects a were rewritten by a → a; M will arrive in state
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