13th International Conference on Membrane Computing - MTA Sztaki

A. Alhazov, R. Freund, H. Heikenwälder, M. Oswald, Yu. Rogozhin, S. Verlan
labeled by q represented by q ′ in a string in L (G M ). In that way we get
a language L (G ′ M ) for a matrix grammar G′ M , as L (CF -MAT ) is closed
under intersection with regular languages.
– In order to filter out the desired terminal results of L (Π C ) from L (G ′ M ), we
need a morphism h which maps any symbol [f, b] ′ to the terminal symbol
b for b ∈ T and all other symbols to λ. As L (CF -MAT ) is closed under
morphisms, we can construct a matrix grammar G ′′ M with
L (G ′′ M ) = h (L (G ′ M )) = h (L (G M ) ∩ L r ) = L (Π C ) .
These observations conclude the proof.
□
It is somehow surprising that the proof technique elaborated in the proof of
Theorem 1 also works for cooperative multiset rules, which type is abbreviated
by coo.
Corollary 1. For all α ∈ {λ, w}, β ∈ {λ, ac, ut}, and n ≥ 1,
)
L
(coo-αC (REG) β
OP n ⊆ P sL (CF -MAT ) .
Proof. We proceed exactly as in the proof of Theorem 1, except that
for any cooperative rule a 1 · · · a k → u ∈ R, we now take the matrix
(q → p, a 1 → λ, · · · , a k−1 → λ, a k → u) into M if and only if a 1 · · · a k → u is
in the set of rules R q labeled by q and (q, R q , p) ∈ δ. Moreover, the regular set
L r has to check for the (non-)appearance of a bounded number of symbols for
each rule, yet the main parts of the proof remain valid as elaborated before. □
As P sL (CF -MAT ) = L (mCF -MAT ), from Theorem 1 and Corollary 1
we finally obtain a characterization of L (mARB-MAT ) via specific families of
languages generated by P systems with regular control:
Theorem 2. For all α ∈ {λ, w}, β ∈ {λ, ac, ut}, and k, n, p ≥ 1,
L (mARB-MAT ) = L (mCF -MAT )
= P sL (CF -MAT )
= L (mARB)
= L (coo-T V (p) OP n ) )
= L
(coo-αC (REG) β
OP n
(
= L ncoo-αC ( REG 1∗ (∗, p + 1) ) OP β n
)
= L
(ncoo-αC (REG) β
OP n .
Proof. We first show that
L (mCF -MAT ) ⊆ L ( ncoo-C ( REG 1∗ (∗, 2) ) OP 1
)
.
)
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