13th International Conference on Membrane Computing - MTA Sztaki

Observer/interpreter P systems
Proof. The inclusion NOI(cat 1 ) ⊆ NRE is supposed to be true by invoking the
Turing-Church thesis. The opposite inclusion NOI(cat 1 ) ⊇ NRE can be shown
by simulating an arbitrary register machine M = (n, P, l 0 , l h ) with 3 registers
(n = 3) with an Observer/Interpreter P system Φ(Π, M); the P system Π
uses non-cooperative and/or catalytic rules with one catalyst. The core system
Π = (O, C, µ = [ ] 1 , R 1 , w 1 , i 0 = 1) is defined as follows:
O = lab(P) ∪ {p | p ∈ lab(P)} ∪ {a 1 , a 2 , a 3 , a 1 , a 2 , a 3 , a 1 , a 2 , a 3 }
∪ {X 1 , X 2 , X 3 } ∪ {c},
C = {c},
w 1 = l 0 ,
and the set of rules R 1 is defined below:
• the following rules are added to R 1
a i → a i , for 1 ≤ i ≤ 3
a i → a i , for 1 ≤ i ≤ 3
• for any instruction l 1 : (sub(r), l 2 , l 3 ) ∈ P the following rules are added to R 1
ca r → cX r
X r → λ
l 1 → l 2
l 2 → l 2
l 1 → l 3
l 3 → l 3
In case of M, the states and the transitions between them are defined as
follows.
{a r↓,a r↑,l 1↓,l 2↑,a 1−,a 2−,a 3−},skip
{l 2↓,l 2↑,X r↑},op
where op = inc if r = 1
and op = skip if r ≠ 1
l 1
l 2 l 2
{a r−,l 1↓,l 3↑,a 1−,a 2−,a 3−},skip
l 3 l 3
{X 1, X 2, X 3} \ {X r}, skip
• for any instruction l 1 : (add(r), l 2 , l 3 ) ∈ P
l 1 → l 1
l 1 → l 2 a r
l 1 → l 3 a r
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