13th International Conference on Membrane Computing - MTA Sztaki

Time-varying sequential P systems
where G = (O, O T , w, P, =⇒ G ) is a grammar of type X; g = (H, E, K) is a
labeled graph where H is the set of node labels identifying the nodes of the
graph in a one-to-one manner, E ⊆ H × {Y, N} × H is the set of edges labeled
by Y or N, K : H → 2 P is a function assigning a subset of P to each node
of g; H i ⊆ H is the set of initial labels, and H f ⊆ H is the set of final labels.
The derivation relation =⇒ GC is defined based on =⇒ G and the control graph
g as follows: For any i, j ∈ H and any u, v ∈ O, (u, i) =⇒ GC (v, j) if and only if
either
– u =⇒ p v by some rule p ∈ K (i) and (i, Y, j) ∈ E (success case), or
– u = v, no p ∈ K (i) is applicable to u, and (i, N, j) ∈ E (failure case).
The language generated by G GC is defined by
L(G GC ) = { v ∈ O T | (w, i) =⇒ ∗ G GC
(v, j) , i ∈ H i , j ∈ H f
}
.
If H i = H f = H, then G GC is called a programmed grammar. The families of
languages generated by graph-controlled and programmed grammars of type X
are denoted by L (X-GC ac ) and L (X-P ac ), respectively. If the set E contains
no edges of the form (i, N, j), then the graph-controlled grammar G GC is said
to be without appearance checking; the corresponding families of languages are
denoted by L (X-GC) and L (X-P ), respectively. If (i, Y, j) ∈ E if and only if
(i, N, j) ∈ E for all i, j ∈ H, then G GC is said to be a graph-controlled grammar
or programmed grammar with unconditional transfer, the corresponding families
of languages are denoted by L (X-GC ut ) and L (X-P ut ), respectively. In the case
of string grammars, it is well-known (e.g., see [6]) that
RE = L (CF -GC ac ) = L (CF -P ac ) = L (CF -GC ut ) = L (CF -P ut )
L (CF -GC) = L (CF -P ) .
2.4 Matrix Grammars
A matrix grammar (with appearance checking) of type X is a construct
G M = (G, M, F, =⇒ GM )
where G = (O, O T , w, P, =⇒ G ) is a grammar of type X, M is a finite set of
sequences of the form (p 1 , . . . , p n ), n ≥ 1, of rules in P , and F ⊆ P . For w, z ∈ O
we write w =⇒ GM z if there are a matrix (p 1 , . . . , p n ) in M and objects w i ∈ O,
1 ≤ i ≤ n + 1, such that w = w 1 , z = w n+1 , and, for all 1 ≤ i ≤ n, either
– w i =⇒ G w i+1 or
– w i = w i+1 , p i is not applicable to w i , and p i ∈ F .
L(G M ) = { v ∈ O T | w =⇒ ∗ G M
v } is the language generated by G M . The
family of languages generated by matrix grammars of type X is denoted by
L (X-MAT ac ). If the set F is empty (or if F = P ), then the grammar is said to
be without appearance checking (with unconditional control); the corresponding
family of languages is denoted by L (X-MAT ) (L (X-MAT ut )).
103