13th International Conference on Membrane Computing - MTA Sztaki

Time-varying sequential P systems
symbol is introduced the computation can never stop, as at least in every third
step this rule is applicable, because due to the halting condition with delay 2
the system enters an infinite loop and never halts.
Simulation of SUB-instruction j : (SUB (a) , k, l) in case the contents of
register a is non-empty
register a is empty
P a : p j → ˆp j ˆp ′ j p j → ¯p j ¯p ′ j ¯p′′ j
P 3−a : ˜p j → λ ˜p j → λ
P 3 : p j → #, ˜p j → # p j → #, ˜p j → #
P 3+a : o a → o ′ a, ˆp ′ j → # ¯p j → λ
P 6−a : ˆp j → λ ¯p ′′
j → p′′ j
P 6 : ˆp j → # ¯p j → #, ¯p ′′
j → #
P 6+a : o ′ a → o ′′
a
o a → o ′ a
P 9−a : ˆp ′ j → ˆp′′ j p ′′
j → p′ j
P 9 : ˆp ′ j → #, o′ a → # p ′′
j → #, o′ a → #
P 9+a : ˆp ′′
j → p k ˜p k
p ′ j → p l ˜p l
P 12−a : o ′′
a → λ
P 12 : o ′′
a → #, ˆp ′′
j → # ¯p ′ j → λ
j → #, ¯p′ j → #
In case register a is assumed to be non-empty and the guess was wrong,
ˆp ′ j → # has to be applied instead of o a → o ′ a from P 3+a , hence, the symbol
ˆp ′ j cannot wait to be applied with the rule ˆp′ j → ˆp′′ j in P 9−a. In the other case,
when assuming register a to be empty, the rule o a → o ′ a should not be applicable
from rule set P 6+a , as then o ′ a → # would become applicable from rule set P 9 .
Observe that these arguments only work because we interchange the rule sets
for a = 1 and a = 2, e.g., o 1 → o ′ 1 is in P 4 and o 2 → o ′ 2 is in P 5 .
For an ADD-instruction j : (ADD (a) , k, l), with j, k, l ∈ B\{l h }, 1 ≤ a ≤ m,
it would be sufficient to just use the rules ˜p j → λ and p j → p k ˜p k or p j → p l ˜p l
in a sequence of two steps, but we have to extend this to a sequence of total
length 12 in order to have the same period as in the case of the simulation of a
SUB-instruction. Hence, for each ADD-instruction j : (ADD (a) , k, l), we take
the following rules into the rule sets P i , 1 ≤ i ≤ 12; in this case, we need not
interchange the rule sets for different registers a, a ∈ {1, 2}:
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