LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

244 Nataˇsa Jonoska and Kalpana Mahalingam
We follow the definitions initiated in [13] and used in [15,16].
An involution θ : Σ → Σ of a set Σ is a mapping such that θ 2 equals the
identity mapping, θ(θ(x)) = x, ∀x ∈ Σ.
The mapping ν : ∆ → ∆ defined by ν(A) = T , ν(T ) = A, ν(C) = G,
ν(G) =C is an involution on ∆ and can be extended to a morphic involution of
∆ ∗ . Since the Watson-Crick complementarity appears in a reverse orientation,
we consider another involution ρ : ∆ ∗ → ∆ ∗ defined inductively, ρ(s) =s for
s ∈ ∆ and ρ(us) =ρ(s)ρ(u) =sρ(u) for all s ∈ ∆ and u ∈ ∆ ∗ . This involution is
antimorphism such that ρ(uv) =ρ(v)ρ(u). The Watson-Crick complementarity
then is the antimorphic involution obtained with the composition νρ = ρν. Hence
for a DNA strand u we have that ρν(u) =νρ(u) = ← u. The involution ρ reverses
the order of the letters in a word and as such is used in the rest of the paper.
For the general case, we concentrate on morphic and antimorphic involutions
of Σ ∗ that we denote with θ. The notions of θ-free and θ-compliant in 2, 3 of
Definition 21 below were initially introduced in [13]. Various other intermolecular
possibilities for cross hybridizations were considered in [16] (see Fig. 3). All of
these properties are included with θ-k-code introduced in [14] (4 of Definition
21).
Definition 21 Let θ : Σ∗ → Σ∗ be a morphic or antimorphic involution. Let
X ⊆ Σ∗ be a finite set.
1. The set X is called θ(k, m1,m2)-subword compliant if for all u ∈ Σ∗ such
that for all u ∈ Σk we have Σ∗uΣmθ(u)Σ ∗ ∩ X = ∅ for m1 ≤ m ≤ m2.
2. We say that X is called θ-compliant if Σ∗θ(X)Σ + ∩X = ∅ and Σ + θ(X)Σ ∗∩ X = ∅.
3. The set X is called θ-free if X2 ∩ Σ + θ(X)Σ + = ∅.
4. The set X is called θ-k-code for some k>0 if Subk(X) ∩ Subk(θ(X)) = ∅.
5. The set X is called strictly θ if X ′ ∩ θ(X ′ )=∅ where X ′ = X \{1}.
The notions of prefix, suffix (subword) compliance can be defined naturally
from the notions described above, but since this paper does not investigate these
properties separately, we don’t list the formal definitions here.
We have the following observations:
Observation 22 In the following we assume that k ≤ min{|x| : x ∈ X}.
1. X is strictly θ-compliant iff Σ∗θ(X)Σ ∗ ∩ X = ∅.
2. If X is strictly θ-free then X and θ(X) are strictly θ-compliant and θ(X) is
θ-free.
3. If X is θ-k-code, then X is θ-k ′ -code for all k ′ >k.
4. X is a θ-k-code iff θ(X) isaθ-k-code.
5. If X is strictly θ such that X2 is θ(k, 1,m)-subword compliant, then X is
strictly θ-k-code.
6. If X is a θ-k-code then both X and θ(X) areθ(k, 1,m)-subword compliant,
θ(k, 1,m) prefix and suffix compliant for any m ≥ 1, θ-compliant. If k ≤ |x|
2
for all x ∈ X then X is θ-free and hence avoids the cross hybridizations as
showninFig.1and2.