LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

92 Rodica Ceterchi, Carlos Martín-Vide, and K.G. Subramanian
Let S =(V,A,R) bea(1, 3) semi-simple splicing system. Again, making
an abuse of notation, we let ϕ(S) denotethe(2, 4) semi-simple splicing system
ϕ(S) =(V,mi(A),ϕ(R)). For S ′ =(V,A ′ ,R ′ )a(2, 4) semi-simple splicing
system, let ψ(S ′ ) denote the (1, 3) semi-simple splicing system ψ(S ′ ) =
(V,mi(A ′ ),ψ(R ′ )).
Now we define ϕ and ψ on the respective classes of languages by: ϕ(L(S)) =
L(ϕ(S)), for L(S) ∈ SSH(1, 3), and ψ(L(S ′ )) = L(ψ(S ′ )), for L(S ′ ) ∈
SSH(2, 4). Note that L(ϕ(S)) = mi(L(S)), and also L(ψ(S ′ )) = mi(L(S ′ )),
thus ϕ and ψ are given by the mirror image function, and are obviously inverse
to each other. ✷
The above results emphasize the symmetry between the (1, 3) and the (2, 4)
cases, symmetry which makes possible the construction which follows next.
Let V be an alphabet, L ⊂ V ∗ a language, and R(L) the set of splicing rules
(of a certain predefined type) which respects L. For a subset of rules R ⊆ R(L)
we denote by σ =(V,R) a splicing scheme. We are looking for necessary and
sufficient conditions for the existence of a finite set A ⊂ V ∗ such that L = σ ∗ (A).
For R(L) thesetof(1, 3)-semi-simple splicing rules which respect L, Goode and
Pixton have given such a characterization, in terms of a certain directed graph,
the (1, 3)-arrow graph canonically associated to a pair (σ, L). Their construction
can be modified to accomodate the (2, 4)-semi-simple splicing rules.
We present the main ingredients of this second construction, which we will
call the (2, 4)-arrow graph associated to a pair (σ, L), where σ =(V,R) isa
(2, 4)-semi-simple H scheme (all rules in R are of the (2, 4)-semi-simple type).
Enrich the alphabet with two new symbols, ¯ V = V ∪{S, T }, enrich the set of
rules ¯ R = R ∪{(1,S;1,S)}∪{(1,T;1,T)}, take¯σ =( ¯ V, ¯ R), and ¯ L = SLT.The
(2, 4)-arrow graph of (σ, L) will be a directed graph G, with the set of vertices
V (G) = ¯ V ,andasetof(2, 4)-edges E(G) ⊂ ¯ V × V ∗ × ¯ V defined in the following
way: a triple e =(b ′ ,w,b)isanedge(fromb ′ to b) ifthereexistsa(2, 4) rule
r =(1,a;1,b) ∈ ¯ R such that b ′ wa is a factor of ¯ L.
In order to stress the “duality” of the two constructions, let us recall from
[8] the definition of a (1, 3)-edge: a triple e =(a, w, a ′ )isanedge(froma to a ′ )
if there exists a (1, 3) rule r =(a, 1; b, 1) ∈ ¯ R such that bwa ′ is a factor of ¯ L.
We list below some results and notions, which are analogous to the results
obtained for the (1, 3) case in [8].
Lemma 1. We have ¯σ( ¯ L) ⊂ ¯ L.IfA ⊂ V ∗ then ¯σ ∗ (SAT)=Sσ ∗ (A)T .
Proof. For x = Sw1T and y = Sw2T two words in ¯ L we have
(x, y) ⊢ (1,S;1,S) y ∈ ¯ L,
(x, y) ⊢ (1,T ;1,T ) x ∈ ¯ L,
(x, y) ⊢r Sr(w1,w2)T ∈ ¯ L, for any other r ∈ R.
Lemma 2. (S, w, T ) is a (2, 4) edge iff SwT ∈ ¯ L.
✷