LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

100 Rodica Ceterchi, Carlos Martín-Vide, and K.G. Subramanian
systems and languages can be defined as usual. For instance, a k-fat semi-simple
(2, 4) H system, orbrieflyanSSkH(2, 4) system, is an H system with rules of
the form (1,x;1,y), with x, y ∈ V ∗ ,and|x|, |y| ≤k. For every k ≥−1 we will
have the corresponding class of languages, SSkH(2, 4). One can readily prove
analogues of Proposition 1 and Corollary 1 for arbitrary k ≥−1.
Proposition 2. Let u, v ∈ V ∗ be two strings. The following equality of functions
∗
V holds:
from V ∗ × V ∗ to 2
(u, 1; v, 1)mi = inv(mi × mi)(1,mi(v); 1,mi(u)).
Corollary 4. For every k ≥−1, there exists a bijection between the classes of
languages SSHk(1, 3) and SSHk(2, 4).
The bijection between H systems will be given by standard passing from
a k-fat (1, 3) rule (u, 1; v, 1), to the k-fat (2, 4) rule (1,mi(v); 1,mi(u), and by
mirroring the axiom set. On languages it will again be the mirror image.
Using this fact, an analogue of Lemma 11 can be readily proved for the
sequence {SSkH(2, 4) | k ≥−1}.
6 Simple Circular H Systems
Splicing for circular strings was consideredbyTomHeadin[9].In[11]thecircular
splicing operation which uses a rule of the general type r =(u1,u2; u3,u4) is
defined by:
(ˆxu1u2,ˆyu3u4) ⊢r ˆxu1u4yu3u2
andisdepictedinFigure6.
ˆxu1u2 ˆyu3u4
✬ ✩✬
✩
x
u1
u2
✫
u4
u3
✪✫
✪
y
⊢r
Fig. 6. Circular splicing
ˆxu1u4yu3u2
✬
�� ❅
x
u1
u2
u4
u3
✫❅
�� y
✩
✪
We mention the following easy to prove properties of circular splicing, which
are not shared with the linear splicing.
Lemma 12. For every splicing rule (u1,u2; u3,u4), and every ˆx,ˆy,ˆz ∈ V ◦
such that (ˆx,ˆy) ⊢ (u1,u2;u3,u4) ˆz we have:
the length preserving property: |ˆx| + |ˆy| = |ˆz|,
thesymbolpreservingproperty:|ˆx|b + |ˆy|b = |ˆz|b for every b ∈ V.