13th International Conference on Membrane Computing - MTA Sztaki

Sublinear-space P systems with active membranes
those symbols as superscripts:
q i,w ′′ [ ] + a i
→ [q i,w] ′′ a i
(20)
[q i,w] ′′ a i
→ [ ] 0 a i
qi,w a (21)
q a i,w [ ] + b w
→ [q a i,w] + b w
(22)
[q a i,w] + b w
→ [ ] 0 b w
q a,b
i,w (23)
These rules are replicated for all q ∈ Q, 0 ≤ i < n, 0 ≤ w < s(n), a, b ∈ Σ.
Now the state object possesses all the information needed in order to simulate
the transition of M, namely, the state itself and the two symbols currently
scanned by the Turing machine. Let
δ : Q × Σ 2 → Q × Σ × {+1, −1} 2
be the transition function of M; here we assume δ is only dened for non-nal
states, and that the head movements are represented by ±1. Assume that
δ(q, a, b) = (r, c, d 1 , d 2 ).
Then, the following rules produce the object representing the new work tape
symbol that replaces a:
[q a,b
i,w → ˆqa,b i,w c′ ] 0 h (24)
These rules are replicated for all q ∈ Q, 0 ≤ i < n, 0 ≤ w < s(n), a, b ∈ Σ.
The object c ′ is sent to the membrane simulating the tape cell it is written
on, i.e., the only negatively charged membrane w w , and it resets its charge to
neutral (while losing the prime):
c ′ [ ] − w w
→ [c] 0 w w
(25)
This rule is replicated for all 0 ≤ w < s(n), c ∈ Σ.
Simultaneously, the state object has to update three pieces of information
(state and positions on the tapes) in order to complete the simulation of the
current step of M:
[ˆq a,b
i,w → r i ′ ,w ′]0 h where i ′ = i + d 1 , w ′ = w + d 2 (26)
These rules are replicated for all q ∈ Q, 0 ≤ i < n, 0 ≤ w < s(n), a, b ∈ Σ.
The conguration of Π x now encodes the conguration of M after having
simulated the step performed by the Turing machine. The simulation may now
proceed with the next step of M.
If M reaches an accepting state q, then the following rule is applied:
while the following one is applied for a rejecting state:
[q i,w ] 0 h → [ ] 0 h yes (27)
[q i,w ] 0 h → [ ] 0 h no (28)
379