13th International Conference on Membrane Computing - MTA Sztaki

(Tissue) P systems with decaying objects
of the cell number, i.e., instead of writing (w, 1) we simply write w and instead
of writing (E : (x 1 , 1) → (y 1 , 1)) we may write E : x 1 → y 1 ; moreover, if E is a
condition which is always fulfilled, we may only write x 1 → y 1 .
Example 2. Consider the P systems
Π (d) = (1, {s, a} , {a} , s, {s → as, s → λ} , 1)
for d > 1. Then the only computations consist of applying n times the rule
s → as and finally ending up with applying the rule s → λ. For n = 0, we get
s [d] =⇒ (Π(d) [d] ,ϑ)
λ, for 1 ≤ n ≤ d, we obtain the sequence of configurations
s [d] =⇒ n (Π(d) [d] ,ϑ) a[d−n+1] . . . a [d] s [d] =⇒ (Π(d) [d] ,ϑ) a[d−n] . . . a [d−1] ,
whereas for n > d we get
s [d] =⇒ d+1
(Π(d) [d] ,ϑ) a[1] a [2] . . . a [d] s [d]
=⇒ ∗ (Π(d) [d] ,ϑ) a[1] a [2] . . . a [d] s [d] =⇒ (Π(d) [d] ,ϑ) a[1] . . . a [d−1] .
Hence, in sum we obtain
N [d] (Π (d) , ϑ, γ, ρ) = {n | 0 ≤ n < d} ,
for ρ ∈ {N, T } ∪ {−l | l ∈ N} and any of the transition modes ϑ as defined in the
preceding section as well as with γ denoting total halting, partial halting (the
whole rule set forms the only partition), or halting with final states (defined by
the regular set of multisets {a} ◦ , which in fact means the same as taking ρ = T ).
Therefore, the family of P systems Π (d) with d ∈ N forms a very simple infinite
hierarchy with respect to the decay d in any of these cases.
Example 3. Let M be a finite subset of T ◦ . Consider the P system
Π (M) = (1, {s} ∪ T, T, s, {s → w | w ∈ M} , 1) .
Obviously, P s [d] (Π (M) , ϑ, γ, ρ) = M for ρ ∈ {N, T } ∪ {−l | l ∈ N} and any
of the transition modes ϑ as defined in the preceding section as well as with
γ ∈ {H, h, F }; hence, for all n, d ≥ 1,
P sO ∗
[d] C n (ϑ, γ, ρ) [ncoo] ⊇ P sF IN,
where ncoo indicates (the use of) noncooperative rules.
Example 4. Let G = (N, T, P, S) be a regular grammar (without loss of generality,
we assume G to be reduced, i.e., from every nonterminal symbol a terminal
string can be derived). Consider the P system
Π (G) = (1, N ∪ T, T, S, R, 1)
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