LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

134 Claudio Ferretti and Giancarlo Mauri
Moreover, if a language satisfies property 1 above, then it is called solid/1
relative to L.
One last definition from [2] considers another relativised property:
Definition 3. Astringw in A ∗ is a join relative to a language L iff w is a
factor of L and for every u, v in A ∗ for which uwv in L, bothu and v are also
in L. Awordw in a language L is a join in L iff w is a join relative to L ∗ .
J(L) will denote the set of all joins in the language L.
In [2] it is proved that:
– if a language on alphabet A is solid, then it is solid relative to every language
in A ∗ ,
– if a language is solid relative to a language L, then it is solid relative to every
subset of L,
– if C is a code, then J(C) is solid relative to C ∗ (Proposition 4 in [2]).
The definitions from [3] deal with involutions. An involution θ : A → A is a
mapping such that θ 2 equals to the identity mapping I: I(x) =θ(θ(x)) = x for
all x ∈ A. The following definitions, presented here with notations made uniform
with those of [2], describe languages which avoid having a word mapped to a
substring of another word, or to a substring of the concatenation of two words.
Definition 4. If θ : A ∗ → A ∗ is an involution, then a language L ⊆ A ∗ is said
to be θ-compliant iff u and xθ(u)y in L can hold only if xy is null.
Moreover, if L ∩ θ(L) =∅, thenL is called strictly θ-compliant.
In [5] it is proved that a language is strictly θ-compliant iff u and xθ(u)y in L
never holds.
Definition 5. If θ : A ∗ → A ∗ is an involution, then a language L ⊆ A ∗ is said
to be θ-free iff u in L with xθ(u)y in L 2 can hold only if x or y are null.
Moreover, if L ∩ θ(L) =∅, thenL is called strictly θ-free.
In [3] it is proved that a language is strictly θ-free iff u in L with xθ(u)y in L 2
never holds.
In [3] it is also proved that if a language L is θ-free, then both L and θ(L)
are θ-compliant.
3 Relationships Between the Two Formalisms
If we restrict ourselves to the case of θ = I, withI being the identity mapping,
then we can discuss the similarities between the properties of a DNA code as
defined in [2] compared to those defined in [3].
First of all we observe that I-compliance and I-freedom cannot be strict,
since L ∩ I(L) =L (unlessweareinthecaseofL = ∅).