LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

302 Peter Leupold, Victor Mitrana, and José M.Sempere
4 Bounded Duplication Languages
Unlike the case of unbounded duplication languages, we are able to determine
the place of bounded duplication languages in the Chomsky hierarchy. This is
the main result of this section.
Theorem 4. For any word r and any integer n ≥ 1, then-bounded duplication
language defined by r is context-free.
Proof. For our alphabet V = alph(r) we define the extended alphabet Ve by
Ve := V ∪{〈a〉 |a ∈ V }. Further we define L≤l := {w ∈ L ||w| ≤l} for any
language L and integer l. We now define the pushdown automaton
�� µ
v
A r n =
�
Q, V, Γ, δ,
� ε
ε
r
�
, ⊥,
�� ε
ε
ε
���
,
�
where Q = | µ ∈ (V
w
∗
e · V ∗ ∪ V ∗ · V ∗
e ) ≤n ,v ∈ (V ∗ ) ≤n ,w ∈ (V ∗ ) ≤|r|
�
,
� � �
µ
and Γ = {⊥} ∪ µ | v ∈ Q, v ∈ (V
w
∗ ) ≤n ,w ∈ (V ∗ ) ≤|r|
�
.
Here we call the three strings occurring in a state from bottom to top pattern,
memory, andguess, respectively. Now we proceed to define an intermediate
deterministic transition function δ ′ . In this definition the following variables
are always quantified universally over the following domains: u, v ∈ (V ∗ ) ≤n ,
w ∈ (V ∗ ) ≤|r| , µ ∈ (V ∗
e )≤n , η ∈ (V ∗
e · V ∗ ∪ V ∗ · V ∗
e )≤n , γ ∈ Γ , x ∈ V and Y ∈ Ve.
(i) δ ′
(ii) δ ′
(iii) δ ′
(iv) δ ′
�� ε
ε
xw
��
µxu
ε
w ��
u〈x〉Yµ
ε
�� u〈x〉
ε
w
� � �� � �
ε
,x,⊥ = ε , ⊥ and δ
w
′
�� � � �� � �
ε
ε
xu ,ε,⊥ = u , ⊥
xw
w
� � �� � �
µ〈x〉u
,x,γ = ε ,γ and δ
w
′
�� � � ��
µxu
µ〈x〉u
xv ,ε,γ = v
� � �� � �
uxY µ
,x,γ = ε ,γ and
w
w
w
w
δ ′
�� � � �� � �
u〈x〉Yµ
uxY µ
xv ,ε,γ = v ,γ
w
� � �� � �
η
,x,η = ux ,ε ,
w
δ
w
′
�� � � �� � �
u〈x〉
η
xv ,ε,η = uxv ,ε .
w
w
� �
,γ
For all triples (q, x, γ) ∈ Q×(V ∪{ε})×Γ not listed above, we put δ ′ (q, x, γ) =∅.
To ensure that our finite state set suffices, we take a closer look at the memory
of the states – since after every reduction the reduced word is put there (see
transition set (iv)), there is a danger of this being unbounded. However, during
any reduction of a duplication, which in the end puts k ≤ n letters into the
memory, either 2k letters from the memory or all letters of the memory (provided
the memory is shorter than 2k) have been read, since reading from memory has
priority (note that the tape is read in states with empty memory only). This
gives us a bound on the length of words in the memory which is n. Itisworth
noting that the transitions of δ ′ actually match either the original word r or the