LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

300 Peter Leupold, Victor Mitrana, and José M.Sempere
of all strings (words) over an alphabet V is denoted by V ∗ and V + = V ∗ \{ε},
where ε denotes the empty string. The length of a string x is denoted by |x|,
hence |ε| = 0, while the number of all occurrences of a letter a in x is denoted
by |x|a . For an alphabet V = {a1,a2,...,ak} (we consider an ordering on V ),
the Parikh mapping associated with V is a homomorphism ΨV from V ∗ into
the monoid of vector addition on IN k , defined by ΨV (s) =(|s|a1, |s|a2,...,|s|ak );
moreover, given a language L over V , we define its image through the Parikh
mapping as the set ΨV (L) ={ΨV (x) | x ∈ L}. A subset X of IN k is said to be
linear if there are the vectors c0,c1,c2,...,cn ∈ IN k ,forsomen≥0such that
X = {c0 + �n i=1 xici | xi ∈ IN, 1 ≤ i ≤ n}. A finite union of linear sets is called
semilinear. For any positive integer n we write [n] fortheset{1, 2,...,n}.
Let V be an alphabet and X ∈{IN}∪{[k] | k ≥ 1}. Forastringw∈V + ,we
set
DX(w) ={uxxv | w = uxv, u, v ∈ V ∗ ,x∈ V + , |x| ∈X}.
We now define recursively the languages:
D 0 X(w) ={w}, D i X(w) = �
DX(x), i≥ 1,
D ∗ X (w) =� D
i≥0
i X (w).
x∈D i−1
X (w)
The languages D∗ IN (w) andD∗ [k] (w), k ≥ 1, are called the unbounded duplication
language and the k-bounded duplication language, respectively, defined by w. In
other words, for any X ∈{IN}∪{[k] | k ≥ 1}, D∗ X (w) is the smallest language
L ′ ⊆ V ∗ such that w ∈ L ′ and whenever uxv ∈ L ′ , uxxv ∈ L ′ holds for all
u, v ∈ V ∗ , x ∈ V + ,and|x| ∈X.
A natural question concerns the place of unbounded duplication languages
in the Chomsky hierarchy. In [7] it is shown that the unbounded duplication
language defined by any word over a two-letter alphabet is regular, while [13]
shows that these are the only cases when the unbounded language defined by a
word is regular. By combining these results we have:
Theorem 1. [7,13] The unbounded duplication language defined by a word w is
regular if and only if w contains at most two different letters.
3 Unbounded Duplication Languages
We do not know whether or not all unbounded duplication languages are contextfree.
A straightforward observation leads to the fact that all these languages are
linear sets, that is, the image of each unbounded duplication language through
the Parikh mapping is linear. Indeed, if w ∈ V + , V = {a1,a2,...,an}, then one
can easily infer that
ΨV (D ∗ IN(w)) = {ΨV (w)+
n�
i=1
xie (n)
i
| xi ∈ IN, 1 ≤ i ≤ n},