13th International Conference on Membrane Computing - MTA Sztaki

D. Sburlan
q 1
{t−, T 1 ↓, T 2−}, skip
{t−, T 1 ↑, T 2−}, skip
{t ↓, T 1−, T 2−}, skip
q 0
q 3
{t−, T 2 ↑, T 1−}, skip
{t−, T 2 ↓, T 1−}, skip
q 2
Fig. 2. The system M that is used to regulate the computation of Π.
The finite state machine M is defined in Figure 2.
Assuming that M is in state q 0 and Π is in a configuration tw where w ∈ V ∗
(w corresponds to a string derived by H), then M passes from state q 0 to state
q 1 if Π executes the rules t → t, the rules corresponding to the Table 1 of system
H (i.e., rules from the set {A → h(α)T 1 | A → α ∈ T 1 , A ∈ V \ ∆}, and no rules
corresponding to the Table 2 (recall that the observation set is {t−, T 1 ↑, T 2 −}).
Next, if M is in state q 1 , then the only way for M to comeback to the state
q 0 is that Π executes the rules t → t, T 1 → λ, and the rules from the set
{A → A | A ∈ V \ ∆}.
The applications of rules of Π in these two steps (”regulated” in a certain
sense by the actions of M) correspond to an application of Table 1 of H. Moreover,
if M is in the state q 0 and Π executes the rule t → λ and at least one
rule from the set {A → h(α)T 1 | A → α ∈ T 1 , A ∈ V \ ∆} ∪ {A → h(α)T 2 |
A → α ∈ T 2 , A ∈ V \ ∆} then M will halt in state q 0 by rejecting; if instead Π
executes t → λ and no rule that produces object(s) T 1 or T 2 (that is, the number
of objects T 1 and T 2 does not grow between consecutive configurations) then M
passes from the state q 0 to q 3 and accepts (actually, Π halts by having in its
region a multiset composed only by terminals and which correspond to a string
generated by H). However, N(Φ(Π, M)) = {0} and Φ(Π, M) halts by having 0
in its register iff w ∈ L(H).
The following result shows the computational power of the Observer/ Interpreter
systems when P systems with symbol objects and multiset rewriting rules
(with one catalyst) are used as core systems.
Theorem 4. NOI(cat 1 ) = NRE.
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