13th International Conference on Membrane Computing - MTA Sztaki

One-membrane symport P systems with few extra symbols
We construct a P system simulating the counter automaton M ′ :
Π = (O, E, [ ] 1
, w, R, 1), where
O = E ∪ alph(w),
E = Q \ {q f } ∪ {x 2 , x 4 , #} ∪ {a i | i ∈ C} ∪ {p 2 | p ∈ P },
O ′ = {p i | i ∈ {1, 3}, p ∈ P } ∪ {A i | i ∈ C},
w = q f oq 0 x 1 x 3 x 5 ds, where s represents O ′ ,
R = {1 : (x 1 x 2 , in), 2 : (x 2 x 1 x 3 , out), 3 : (x 2 d, out),
4 : (x 3 x 4 , in), 5 : (x 4 x 3 x 5 , out), 6 : (x 4 d, out)}
∪ {7 : (A i a i x 5 , in), 8 : (a i a ī d, out) | i ∈ C}
∪ {9 : (qp 1 x 1 , out), 10 : (p 1 p 2 x 1 , in), 11 : (p 2 p 3 A i , out),
12 : (p 3 q ′ x 3 , in), 13 : (p 3 #q f , in) | p : (q → q ′ , i+) ∈ P }
∪ {14 : (qp 1 x 3 , out), 15 : (p 1 p 2 x 3 , in), 16 : (p 2 p 3 a i , out), 17 : (p 2 p 3 d, out),
18 : (p 3 q ′ x 5 , in), 19 : (p 3 #q f , in) | p : (q → q ′ , i−) ∈ P }
∪ {20 : (q f bx, out) | x ∈ O ′ } ∪ {21 : (q f b, in),
22 : (#d, out), 23 : (#d, in), 24 : (oq f , out), 25 : (od, out)}.
We now explain the “correct” work of Π. If non-determinism allows to apply a
different multiset of rules, then one of rules from the group
T = {3, 6, 8, 13, 17, 19, 25}
is also applied, and then the computation applies rules 22, 23 forever, without
producing the result. Notice that even if multiple applications of rules from T
happen, this would only add further objects # to the skin, and the computation
would still be unable to halt. In the “correct” computations of Π, rules from
T are not applied, their role is only to ensure that symbols x 2 , x 4 , p 2 , o or pairs
(a i , a ī ) are never idle in the skin, as well as that symbols p 3 are never idle in the
environment.
Rule 24 is applied at the beginning of the computation, in parallel with
simulation of the first instruction of M ′ . If rule 21 is applied instead, then rule
25 forces an infinite computation. Throughout the simulation of M ′ , object q f
stays in the environment (available in a single copy unlike other objects from
Q).
Increment is performed by a sequence of multisets of rules 9,1,2,(10,4),(5,11),
(7,12). We remark that the system seems to have a choice between rules 1 and
10. However, if rule 10 is applied immediately after rule 9, then rule 11 is applied,
followed by rule 13, forcing a non-ending computation. Yet, if rule 1 is applied
again immediately after rule 2, then rule 2 is no longer applicable, forcing rule
3 and a non-ending computation.
Decrement is performed by a sequence of multisets of rules 14,4,5,15,16,18.
We remark that the system seems to have a choice between rules 4 and 15.
However, if rule 15 is applied immediately after rule 14, then rule 16 or 17 is
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