13th International Conference on Membrane Computing - MTA Sztaki

One-membrane symport P systems with few extra symbols
– An increment instruction is performed by replacing q by q ′ in two steps with
the help of object p 1 , also bringing an object a i in.
– A decrement instruction is performed by replacing q by q ′ in four steps with
the help of objects p 1 , p 2 , p 3 , also bringing an object a i out. Moreover, rules
6,7 are optionally applied an arbitrary number of steps. If a i is not present,
then the computation never exits the loop of applying rules 6,7.
– The conflicting counter semantics is implemented by rule 9.
– Once the simulation of M is finished, object q f enters the system. Each
application of the group of rules 11,10,12 leads to removal from the skin of
one copy of c and one object from O ′ . Notice that objects from O ′ cannot
reenter the skin without object T . The first application of rule 11 actually
happens in the first step of the computation, but the rest of this process
takes place after the simulation of M ′ is finished. The multiplicity of objects
c has been chosen to be |O ′ | − 1 so that objects q f , b stop reentering the skin
exactly when no more objects from O ′ remain there. The computation halts
with the skin containing the result and objects T , N.
□
Hence, the known bounds can be described as follows.
NOP 1 (sym 2 ) ⊇ SEG 1 ∪ SEG 2 . (1)
NOP 1 (sym 2 ) ⊆ NF IN. (2)
5⋃
NOP 1 (sym 3 ) ⊇ NF IN 1 ∪ (N k F IN k ∪ N k REG k ) ∪ N 6 RE. (3)
k=0
NOP 1 (sym 4 ) ⊇ NF IN 0 ∪ NF IN 1 ∪ N 1 REG 1 ∪ N 2 RE. (4)
NOP 1 (sym ∗ ) ⊆ NF IN ∪ N 1 RE. (5)
3.5 Sequential Mode
Recently, in [4] we obtained the counterpart of these results for sequential systems.
We recall them here.
NOP sequ
1 (sym 2 ) ⊇ SEG 1 ∪ SEG 2 . (6)
NOP sequ
4 Discussion
1 (sym 2 ) ⊆ NF IN. (7)
∞⋃
NOP sequ
1 (sym 3 ) ⊇ NF IN 1 ∪ (N k F IN k ∪ N k REG k ). (8)
k=0
NOP sequ
1 (sym ∗ ) = NF IN ∪ N 1 REG. (9)
We have improved the best known lower bounds of the computational power of
one-membrane P systems with symport only. Using symport of weight at most 3,
computational completeness holds with 6 extra objects. With symport of weight
at most 4, 2 extra objects suffice. Since 1 extra object is known to be necessary
to generate infinite sets by any symport-only P system, a particularly interesting
open question is whether
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