13th International Conference on Membrane Computing - MTA Sztaki

Observer/interpreter P systems
• δ(q, λ) = {(q, λ)}
• if x ∈ Σ ∗ and a ∈ Σ then
δ(q, xa) = {(p, w) | w = w 1 w 2 and (∃) p ′ ∈ Q, such that
(p ′ , w 1 ) ∈ δ(q, x) and (p, w 2 ) ∈ δ(p ′ , a)}.
If M is a GSM defined as above and x ∈ Σ ∗ then M(x) denotes the set
{y | (∃)p ∈ F such that (p, y) ∈ δ(q 0 , x)}.
If L ⊆ Σ ∗ is a language, then M(L) = ⋃
w∈L
M(w).
A GSM always maps a regular language to a regular language.
Register Machine
A register machine is a tuple M = (n, P, l 0 , l h ), where n ≥ 1 is the number of
registers (each register stores a natural number), P is a finite set of uniquely
labeled instructions (P is called the program and the labels of the instructions
are from a set lab(P)), l 0 is the initial label, and l h is the halting label.
The instructions can be of the following forms:
• l 1 : (add(r), l 2 , l 3 ) – where l 1 , l 2 , l 3 ∈ lab(P), adds 1 to register r and nondeterministically
proceeds to one of the instructions l 2 or l 3 .
• l 1 : (sub(r), l 2 , l 3 ) – where l 1 , l 2 , l 3 ∈ lab(P), subtracts 1 from register r if the
number stored by register r is greater than zero and goes to the instruction with
the label l 2 , otherwise goes to the instruction with the label l 3 .
• l h : halt – where l h ∈ lab(P), halts the machine.
M starts with all registers being empty and runs the program P, starting
from the instruction with the label l 0 . Considering the content of register 1 for all
possible computations of M which are ended by the execution of the instruction
labeled l h , one gets the set N(M) ⊆ IN – the set generated by M.
The following result concerns the computational power of register machines.
Theorem 1. For any recursively enumerable set Q ⊆ IN there exists a nondeterministic
register machine with 3-registers generating Q such that when
starting with all registers being empty, M non-deterministically computes and
halts with n in register 1, and registers 2 and 3 being empty iff n ∈ Q.
Lindenmayer Systems
Lindenmayer systems are parallel computing devices representing a development
model inspired by multicellular organisms growth. A 0L system is a tuple G =
(V, ω, P ) where V is an alphabet, ω ∈ V ∗ is the axiom, and P ⊆ V × V ∗ is a
complete finite set of rules. For w 1 , w 2 ∈ V ∗ we write w 1 ⇒ w 2 if w 1 = a 1 . . . a n ,
w 1 = x 1 . . . x n , for a i → x i ∈ P , 1 ≤ i ≤ n. The language generated by G
is L(G) = {x ∈ V ∗ | ω ⇒ ∗ x} where ⇒ ∗ denotes the reflexive and transitive
closure of ⇒.
An ET0L system is a tuple H = (V, T, ω, ∆) where T = {P 1 , . . . , P k } is a
finite nonempty set of tables and such that each triple G i = (V, ω, P i ), 1 ≤ i ≤ k,
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