LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

108 Erzsébet Csuhaj-Varjú and Arto Salomaa
the characteristic feature of DNA computing, the massive parallelism, is used
very strongly. As an illustration we use the formula from the celebrated recent
paper, [3]. The third result shows how in the very simple case of four-letter
DNA alphabets we can obtain weird (not even Z-rational) patterns of population
growth.
2 Definitions
By a D0L system we mean a triple H =(Σ,g,w0), where Σ is an alphabet, g is
an endomorphism defined on Σ ∗ and w0 ∈ Σ ∗ is the axiom. The word sequence
S(H) ofH is defined as the sequence of words w0,w1,w2,...where wi+1 = g(wi)
for i ≥ 0.
In the following we recall the basic notions concerning Watson-Crick D0L
systems, introduced in [10,16].
By a DNA-like alphabet Σ we mean an alphabet with 2n letters, n ≥ 1,
of the form Σ = {a1,...,an, ā1,...,ān}. Letters ai and āi, 1≤ i ≤ n, are
said to be complementary letters; we also call the non-barred symbols purines
and the barred symbols pyrimidines. The terminology originates from the basic
DNA alphabet {A, G, C, T }, where the letters A and G are for purines and their
complementary letters T and C for pyrimidines.
We denote by hw the letter-to-letter endomorphism of a DNA-like alphabet
Σ mapping each letter to its complementary letter. hw is also called the Watson-
Crick morphism.
A Watson-Crick D0L system (a WD0L system, for short) is a pair W =
(H, φ), where H =(Σ,g,w0) is a D0L system with a DNA-like alphabet Σ,
morphism g and axiom w0 ∈ Σ + ,andφ : Σ ∗ →{0, 1} is a recursive function
such that φ(w0) =φ(λ) = 0 and for every word u ∈ Σ ∗ with φ(u) =1itholds
that φ(hw(u)) = 0.
The word sequence S(W ) of a Watson-Crick D0L system W consists of words
w0,w1,w2,..., where for each i ≥ 0
wi+1 =
� g(wi) if φ(g(wi)) = 0
hw(g(wi)) if φ(g(wi)) = 1.
The condition φ(u) = 1 is said to be the trigger for complementarity transition.
In the following we shall also use this term: a word w ∈ Σ ∗ is called correct
according to φ if φ(w) = 0, and it is called incorrect otherwise. If it is clear from
the context, then we can omit the reference to φ.
An important notion concerning Watson-Crick D0L systems is the Watson-
Crick road. Let W = (H, φ) be a Watson-Crick D0L system, where H =
(Σ,g,w0). The Watson-Crick road of W is an infinite binary word α over {0, 1}
such that the ith bit of α is equal to 1 if and only if at the ith step of the computation
in W a transition to the complementary takes place, that is, φ(g(wi−1)) = 1,
i ≥ 1, where w0,w1,w2,... is the word sequence of W.
Obviously, various mappings φ can satisfy the conditions of defining a trigger
for complementarity transition. In the following we shall use a particular
variant and we call the corresponding Watson-Crick D0L system standard.