LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

On Some Classes of Splicing Languages 93
Proof. By definition, if (S, w, T )isa(2, 4) edge, then there exists a (2, 4) rule
r =(1,a;1,T) ∈ ¯ R such that Swa is a factor of ¯ L. The only rule in ¯ R involving
T is (1,T;1,T), thus a = T ,andSwT is a factor of ¯ L.But ¯ L ⊂ SV ∗ T ,thus
SwT ∈ ¯ L.Conversely,ifSwT ∈ ¯ L,thenSwT is a factor of ¯ L, and since rule
(1,T;1,T) ∈ ¯ R,wehavethat(S, w, T )isa(2, 4) edge. ✷
The product of two adjacent edges in G, e1 =(b0,w1,b1)ande2 =(b1,w2,b2),
is defined as the triple e1e2 =(b0,w1b1w2,b2).
Lemma 3. (The closure property) Whenever e1 and e2 are adjacent edges in
G, e1e2 is an edge of G.
Proof. If e1 =(b0,w1,b1) ande2 =(b1,w2,b2) are adjacent (2, 4) edges in G,
then there exist the (2, 4) rules r1 =(1,a1;1,b1) andr2 =(1,a2;1,b2) in ¯ R,and
there exist the strings x0,y1,x1,y2 ∈ ¯ V , such that the words x0b0w1a1y1 and
x1b1w2a2y2 are in ¯ L.Usingr1 to splice these words we obtain
(x0b0w1|a1y1,x1|b1w2a2y2) ⊢r1 x0b0w1b1w2a2y2 ∈ ¯ L,
so b0w1b1w2a2 is a factor of ¯ L. This fact, together with having rule r2 in ¯ R,
makes e1e2 =(b0,w1b1w2,b2) anedgeofG. ✷
A path in G is a sequence π =< e1, ···,en > of edges ek =(bk−1,wk,bk),
1 ≤ k ≤ n, every two consecutive ones being adjacent. The label of a path as
above is λ(π) =b0w1b1 ···wnbn. Asingleedgee is also a path, ,thusits
label λ(e) is also defined.
Lemma 4. For π =< e1, ···,en > apathinG, the product e = e1 ···en exists
and is an edge of G, whose label equals the label of the path, i.e., λ(e) =λ(π).
Proof. Using Lemma 3 and the definitions, one can prove by straightforward
calculations that the product of adjacent edges is associative, hence the product
of n edges of a path can be unambiguously defined, and is an edge. ✷
The language of the (2, 4) graph G, denoted L(G), is the set of all labels of
paths in G, fromS to T .
Lemma 5. L(G) = ¯ L.
Proof. From Lemma 2 we have ¯ L ⊆ L(G), since SwT ∈ ¯ L implies (S, w, T )is
an edge, and λ(S, w, T )=SwT ∈ L(G) by definition. For the other implication,
if π is a path in G from S to T , then, according to Lemma 4 there exists the
product edge e =(S, w, T )ofalledgesinπ and λ(π) =λ(e) =SwT ∈ ¯ L. ✷
The following is the analogue of the prefix edge property of (1, 3) arrow
graphs from [8].
Lemma 6. (Suffix edge) If (b ′′ ,ub ′ v, b) is an edge of G, with b ′ ∈ V , then
(b ′ ,v,b) is an edge of G.