LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

On Some Classes of Splicing Languages 87
|w|a. Also the mirror image function mi : V ∗ −→ V ∗ can be readily extended to
mi : V ◦ −→ V ◦ ;ifˆw is read in a clockwise manner, mi(ˆw) =ˆmi(w) consists of
the letters of w read in a counter-clockwise manner. We will denote composition
of functions algebraically.
To a language L ⊆ V ∗ we can associate the circular language Cir(L) ={ˆw |
w ∈ L} which will be called the circularization of L. The notation L ◦ is also
used in the literature.
To a circular language C ⊆ V ◦ we can associate several linearizations, i.e.,
several languages L ⊆ V ∗ such that Cir(L) =C. Thefull linearization of the
circular language C will be Lin(C) ={w | ˆw ∈ C}.
Having a family FLof languages we can associate to it its circular counterpart
FL ◦ = {Cir(L) | L ∈ FL}. WecanthusspeakofFIN ◦ , REG ◦ ,etc.Acircular
language is in REG ◦ if some linearization of it is in REG.
Next we recall some notions on splicing. A splicing rule is a quadruple r =
(u1,u2; u3,u4) withui ∈ V ∗ for i =1, 2, 3, 4. The action of the splicing rule r on
linear words is given by
(x1u1u2x2,y1u3u4y2) ⊢r x1u1u4y2.
In other words, the string x = x1u1u2x2 is cut between u1 and u2, the string
y = y1u3u4y2 between u3 and u4, and two of the resulting substrings, namely
x1u1 and u4y2, are pasted together producing z = x1u1u4y2.
For a language L ⊆ V ∗ and a splicing rule r, wedenote
r(L) ={z | (x, y) ⊢r z for some x, y ∈ L}.
We say that a splicing rule r respects a language L if r(L) ⊆ L. Inother
words, L is closed w.r.t. the rule r. AsetofrulesR respects the language L if
R(L) ⊆ L.
For an alphabet V , a (finite) A ⊂ V ∗ and a set R of splicing rules, a triple
S =(V,A,R) is called a splicing system, orH system. The pair σ =(V,R) is
called a splicing (or H) scheme.
For an arbitrary language L ⊆ V ∗ ,wedenoteσ 0 (L) =L, σ(L) =L ∪{z |
for some r ∈ R, x, y ∈ L, (x, y) ⊢r z}, andforanyn ≥ 1 we define σ n+1 (L) =
σ n (L) ∪ σ(σ n (L)). The language generated by the H scheme σ starting from
L ⊆ V ∗ is defined as
σ ∗ (L) = �
σ n (L).
n≥0
The language generated by the H system S =(V,A,R), denoted L(S), is
L(S) =σ ∗ (A), i.e., what we obtain by iterated splicing starting from the strings
in the axiom set A. We recall that L(S) can be also characterized as being the
smallest language L ⊆ V ∗ such that:
(i) A ⊆ L ;
(ii) r(L) ⊆ L for any r ∈ R.
In a similar manner we define splicing on circular languages. The only difference
is that we start from circular strings as axioms, A ⊂ V ◦ , and the application