13th International Conference on Membrane Computing - MTA Sztaki

A. Alhazov, R. Freund, H. Heikenwälder, M. Oswald, Yu. Rogozhin, S. Verlan
Simulation of ADD-instruction j : (ADD (a) , k, l)
P 1 : p j → p ′ j
P 2 : ˜p j → λ
P 3 : ˜p j → #, p j → #
P 4 : p ′ j → ¯p j
P 5 : ¯p j → ¯p ′ j
P 6 : p ′ j → #, ¯p j → #
P 7 : ¯p ′ j → ˆp j
P 8 : ˆp j → ˆp ′ j
P 9 : ¯p ′ j → #, ˆp j → #
P 10 : ˆp ′ j → ˆp′′ j
P 11 : ˆp ′′
j → o ap k ˜p k , ˆp ′′
j → o ap l ˜p l
P 12 : ˆp ′ j → #, ˆp′′ j → #
The trap rules introduced in the rule sets P 3 , P 6 , P 9 , and P 12 guarantee that
the rules in the rule sets P 1 , P 2 , P 4 , P 5 , P 7 , P 8 , P 10 , and P 11 have to be applied
in a correct way to avoid the introduction of the trap symbol #.
Without loss of generality, we may assume that the last instruction applied in
the register machine M is a SUB-instruction (labeled by j) being applied to the
empty register 1; instead of taking the rule p ′ j → p f ˜p f we take the rule p ′ j → λ
into P 10 . If until then the actions of the register machine have been simulated
correctly in Π C , only the terminal results consisting of specific numbers of copies
of the symbols o i , 3 ≤ i ≤ m, remain in the membrane region. The P system
therefore finally stops before entering a new cycle P 1 · · · P 12 ; hence, in sum we
have shown that the language generated by the register machine is also generated
by the time-varying P system Π C with delay 2, i.e.,
P sRE ⊆ L (ncoo-T V ut OP 1 (12) , 2) .
As weak control is the less restrictive control variant, we immediately infer
P sRE ⊆ L (ncoo-wT V ut OP 1 (12) , 2) .
too.
□
As a challenge for future research it remains to search for a proof which eventually
allows to obtain computational completeness with delay one only. Another
parameter to be improved is the period of the control language. Eventually there
might also be a trade-off between these two parameters: It is easy to see that
using p j only instead of the pair p j ˜p j we could save the second rule set at the
beginning of a simulation; then, in addition, we might even omit P 3 and P 12 ,
but with these rather obvious changes we would increase the delay to 3.
The construction given in the preceding proof shows that any action in the
time-varying P system can be seen as simple multiset rewriting, hence, we obviously
get the following result:
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