LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

On Some Classes of Splicing Languages 97
✓✏ ✓✏
S
✲ T
✒✑ ✒✑
❏ ❏❪
❏❏❏❫ ❏
✡ ❏
✓✏ ✡ ❏✓✏
b
a
✒✑ ✒✑
✡✡✣ aba
b
1,ab a
✻ 1
✡ ✠
Fig. 4. The (2, 4) prime arrow graph of L4 = b + a ∪ ab + a
✓✏
✓✏
S ❍ T
✒✑❍❍❍❍❍❍❍❍❍❥
✒✑
✟
✓✏ ✟ ✓✏
b
c
✒✑ ✒✑
✟✟✟✟✟✟✟✟✯
❏
❏❏❏❫ ✡ ✡✡✡✣
a a
a a
a
❥
❨
a
Fig. 5. The (1, 3) and also the (1, 4) prime arrow graph of L = (abac) + a ∪
(abac) ∗ aba ∪ (acab) + a ∪ (acab) ∗ aca
Note that the axiom set of S1 is closed to mirror image, and equals the axiom
set of S2. The(1, 3) arrow graph of L coincides with its (2, 4) arrow graph, and
is depicted in Figure 5.
Due to the construction of the (2, 4)-arrow graph, other results of [8] can
be extended from languages in SSH(1, 3) to languages in the class SSH(2, 4).
We give the following, without proofs, since the proofs in the (1, 3) case are
all based on the prime (1, 3) arrow graph construction, and can be replaced by
similar proofs, based on the prime (2, 4) arrow graph.
First, some relations with the notion of constant introduced by
Schützenberger [16]. A constant of a language L ⊆ V ∗ is a string c ∈ V ∗ such
that for all x, y, x ′ ,y ′ ∈ V ∗ ,ifxcy and x ′ cy ′ are in L, thenxcy ′ is in L. Therelation
with the splicing operation is obvious, since xcy ′ =(c, 1; c, 1)(xcy, x ′ cy ′ )=
(1,c;1,c)(xcy, x ′ cy ′ ). Thus c is a constant of L iff r(L) ⊆ L for r =(c, 1; c, 1), or
iff r ′ (L) ⊆ L for r ′ =(1,c;1,c).
Theorem 4. A language L ⊆ V ∗ is a simple splicing language (in the class
SH=SH(1, 3) = SH(2, 4)) iff there exists an integer K such that every factor
of L of length at least K contains a symbol which is a constant of L.
There are many constants in the (1, 3) and the (2, 4) semi-simple cases.
Theorem 5. If L ∈ SSH(1, 3) ∪ SSH(2, 4) then there exists a positive integer
K such that every string in V ∗ of length at least K is a constant of L.