13th International Conference on Membrane Computing - MTA Sztaki

Sublinear-space P systems with active membranes
For the innermost membrane i 0 instead we use the following rules, which
add a binary counter of l(n) bits (starting from 0) as a superscript to the state
object:
q i,w [ ] 0 i 0
→ [q 0 i,w] − i 0
if the least signicant bit of i is 0 (3)
q i,w [ ] 0 i 0
→ [q 0 i,w] + i 0
if the least signicant bit of i is 1. (4)
These rules are replicated for all q ∈ Q, 0 ≤ i < n, 0 ≤ w < s(n).
When membrane i 0 becomes non-neutral, the input objects a i (for 0 ≤ i < n)
are sent out. Membranes i 0 , . . . , i l(n)−1 behave as lters in the following sense:
object a i may pass through i j only if the charge of the membrane corresponds to
the j-th bit of i (where positive denotes a 1, and negative a 0). Only one object
will traverse all of them and reach the outermost membrane, namely, the object
corresponding to the symbol under the tape head in the current conguration
of M. Formally, the required rules are:
[a i ] − i j
→ [ ] − i j
a i if the j-th bit of i is 0 (5)
[a i ] + i j
→ [ ] + i j
a i if the j-th bit of i is 1. (6)
These rules are replicated for all a ∈ Σ, 0 ≤ i < n, 0 ≤ j < l(n).
The single object that reaches the outermost membrane h is then used in
order to set to positive the charge of the corresponding membrane a i (thus
signalling that the symbol under the input tape head is a):
a i [ ] 0 a i
→ [a i ] 0 a i
(7)
[a i ] 0 a i
→ [ ] + a i
a i (8)
These rules are replicated for all a ∈ Σ, 0 ≤ i < n.
It can be shown that the number of steps required for these operations to
be carried out (starting from the moment membrane i 0 becomes non-neutral) is
bounded by n 2 + l(n) + 1. During this time, the head object waits inside i 0 by
using the following rules:
[qi,w t → q t+1
i,w ]α i 0
for 0 ≤ t < n + l(n) + 1 (9)
2
These rules are replicated for all q ∈ Q, 0 ≤ i < n, 0 ≤ w < s(n), α ∈ {+, −}.
(See Fig. 3.)
When the superscript t reaches n 2
+ l(n) + 1, the state object travels back to
membrane h while resetting the charges of i 0 , . . . , i l(n)−1 to neutral:
[q n i,w] α i 0
→ [ ] 0 i 0
q ′ i,w (10)
[q ′ i,w] α i j
→ [ ] 0 i j
q ′ i,w (11)
[q ′ i,w] α i l(n)−1
→ [ ] 0 i l(n)−1
q 0 i,w (12)
These rules are replicated for all q ∈ Q, 0 ≤ i < n, 0 ≤ w < s(n), 0 < j < l(n)−1,
α ∈ {+, −}.
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