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13th International Conference on Membrane Computing - MTA Sztaki

13th International Conference on Membrane Computing - MTA Sztaki

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D. Sburlan<br />

assuming that M is in a state q ∈ Q and given an observati<strong>on</strong> o ∈ O, then M will<br />

perform the acti<strong>on</strong> λ(q, o) (it increments register r if λ(q, o) = inc, it decrements<br />

register r if λ(q, o) = dec, and it does not modify the c<strong>on</strong>tent of r if λ(q, o) =<br />

skip). The output of M in resp<strong>on</strong>se to a sequence of observati<strong>on</strong>s o 1 , o 2 , . . . , o k<br />

c<strong>on</strong>sists in the applicati<strong>on</strong>s of the acti<strong>on</strong>s given by λ(q 0 , o 1 ), . . . , λ(q k−1 , o k ) <strong>on</strong><br />

register r, where q 0 , . . . , q n is the sequence of states such that δ(q i−1 , o i ) = q i ,<br />

1 ≤ i ≤ n; the sequence of observati<strong>on</strong>s o 1 , o 2 , . . . , o k is called accepted by M iff<br />

q n ∈ F .<br />

The system Φ = (Π, M) computes as follows. The systems Π and M run<br />

in parallel: at each passing from a c<strong>on</strong>figurati<strong>on</strong> C 1 to C 2 in a computati<strong>on</strong> of<br />

Π, based <strong>on</strong> the observati<strong>on</strong> {x 1 , . . . , x k } of the pair (C 1 , C 2 ), the system M<br />

changes its current state q to a new <strong>on</strong>e p = δ(q, {x 1 , . . . , x k }); in additi<strong>on</strong>,<br />

M performs the acti<strong>on</strong> defined by λ(q, {x 1 , . . . , x k }). A computati<strong>on</strong> of Φ is<br />

c<strong>on</strong>sidered successful if the above procedure is applied for each pair of c<strong>on</strong>secutive<br />

c<strong>on</strong>figurati<strong>on</strong>s in a halting computati<strong>on</strong> of Π and the system M accepts the<br />

sequence of observati<strong>on</strong>s determined by the computati<strong>on</strong> of Π; in this case, the<br />

result of the computati<strong>on</strong> is the number stored in register r at its end. Collecting<br />

all the values stored by r at the end of all possible successful computati<strong>on</strong>s of Φ<br />

<strong>on</strong>e obtains the set of integers N(Φ(Π, M)).<br />

In case of a n<strong>on</strong>-halting computati<strong>on</strong> of Π, the system Φ does not produce<br />

any output. The same outcome is obtained when M does not accept the sequence<br />

of observati<strong>on</strong>s determined by the underlying computati<strong>on</strong> of Π.<br />

The families of all sets of numbers generated by Observer/Interpreter P systems,<br />

having as core systems P systems with symbol objects, multiset rewriting<br />

rules, at most k catalysts and <strong>on</strong>e membrane is denoted by NOI(cat k ).<br />

Because the system M recalls the definiti<strong>on</strong> of a GSM, in what follows we<br />

will use a similar notati<strong>on</strong> for the transiti<strong>on</strong> graph.<br />

Example 1 Let Φ = (Π, M) such that Π = (O, C, µ, R 1 , w 1 , i 0 ) where O =<br />

{a, a, a, c}, C = {c}, µ = [ ] 1 , w 1 = a, i 0 = 1, and R 1 is defined as follows:<br />

R 1 = {a → aa,<br />

a → a,<br />

a → a,<br />

ca → c}.<br />

The system M is defined by the transiti<strong>on</strong> graph depicted in Figure 1.<br />

The computati<strong>on</strong> of Φ proceeds as follows. If the rule a → aa is the <strong>on</strong>ly<br />

rule applied in the first k c<strong>on</strong>secutive c<strong>on</strong>figurati<strong>on</strong>s of Π, then 2 k−1 objects a<br />

are produced. During this exp<strong>on</strong>ential generati<strong>on</strong> of objects a, the system M<br />

remains in state q 0 (this is because M detects that the number of objects a does<br />

not change). Assuming that in the k-th c<strong>on</strong>figurati<strong>on</strong> both the rules a → aa and<br />

a → a are applied, then M, being in state q 0 , can either remain in the same<br />

state q 0 and the computati<strong>on</strong> stops (an unsuccessful computati<strong>on</strong>; the case a−<br />

or a ↑, a ↑) or it can pass to state q 1 (the case a ↓, a ↑). However, there is no<br />

guarantee that all the objects a were rewritten by a → a; M will arrive in state<br />

412

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