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13th International Conference on Membrane Computing - MTA Sztaki

13th International Conference on Membrane Computing - MTA Sztaki

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Sublinear-space P systems with active membranes<br />

those symbols as superscripts:<br />

q i,w ′′ [ ] + a i<br />

→ [q i,w] ′′ a i<br />

(20)<br />

[q i,w] ′′ a i<br />

→ [ ] 0 a i<br />

qi,w a (21)<br />

q a i,w [ ] + b w<br />

→ [q a i,w] + b w<br />

(22)<br />

[q a i,w] + b w<br />

→ [ ] 0 b w<br />

q a,b<br />

i,w (23)<br />

These rules are replicated for all q ∈ Q, 0 ≤ i < n, 0 ≤ w < s(n), a, b ∈ Σ.<br />

Now the state object possesses all the informati<strong>on</strong> needed in order to simulate<br />

the transiti<strong>on</strong> of M, namely, the state itself and the two symbols currently<br />

scanned by the Turing machine. Let<br />

δ : Q × Σ 2 → Q × Σ × {+1, −1} 2<br />

be the transiti<strong>on</strong> functi<strong>on</strong> of M; here we assume δ is <strong>on</strong>ly dened for n<strong>on</strong>-nal<br />

states, and that the head movements are represented by ±1. Assume that<br />

δ(q, a, b) = (r, c, d 1 , d 2 ).<br />

Then, the following rules produce the object representing the new work tape<br />

symbol that replaces a:<br />

[q a,b<br />

i,w → ˆqa,b i,w c′ ] 0 h (24)<br />

These rules are replicated for all q ∈ Q, 0 ≤ i < n, 0 ≤ w < s(n), a, b ∈ Σ.<br />

The object c ′ is sent to the membrane simulating the tape cell it is written<br />

<strong>on</strong>, i.e., the <strong>on</strong>ly negatively charged membrane w w , and it resets its charge to<br />

neutral (while losing the prime):<br />

c ′ [ ] − w w<br />

→ [c] 0 w w<br />

(25)<br />

This rule is replicated for all 0 ≤ w < s(n), c ∈ Σ.<br />

Simultaneously, the state object has to update three pieces of informati<strong>on</strong><br />

(state and positi<strong>on</strong>s <strong>on</strong> the tapes) in order to complete the simulati<strong>on</strong> of the<br />

current step of M:<br />

[ˆq a,b<br />

i,w → r i ′ ,w ′]0 h where i ′ = i + d 1 , w ′ = w + d 2 (26)<br />

These rules are replicated for all q ∈ Q, 0 ≤ i < n, 0 ≤ w < s(n), a, b ∈ Σ.<br />

The c<strong>on</strong>gurati<strong>on</strong> of Π x now encodes the c<strong>on</strong>gurati<strong>on</strong> of M after having<br />

simulated the step performed by the Turing machine. The simulati<strong>on</strong> may now<br />

proceed with the next step of M.<br />

If M reaches an accepting state q, then the following rule is applied:<br />

while the following <strong>on</strong>e is applied for a rejecting state:<br />

[q i,w ] 0 h → [ ] 0 h yes (27)<br />

[q i,w ] 0 h → [ ] 0 h no (28)<br />

379

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