LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

The Power of Networks of Watson-Crick D0L Systems 109
In this case a word w satisfies the trigger for turning to the complementary
word (it is incorrect) if it has more occurrences of pyrimidines (barred
letters) than purines (non-barred letters). Formally, consider a DNA-like alphabet
Σ = {a1,...,an, ā1,...,ān} , n ≥ 1. Let ΣPUR = {a1,...,an} and
ΣPYR = {ā1,...,ān}. Then, we define φ : Σ ∗ →{0, 1} as follows: for w ∈ Σ ∗
φ(w) =
� 0 if |w|ΣPUR ≥|w|ΣPYR and
1 if |w|ΣPUR < |w|ΣPYR.
Following [7], we now define the basic notion in this paper, a network of
Watson-Crick D0L systems (an NWD0L system). It is a finite collection of WD0L
systems over the same DNA-like alphabet and with the same trigger, where the
component WD0L systems act in a synchronized manner by rewriting their own
sets of strings in the WD0L manner and after each derivation step communicate
some of the obtained words to each other. The condition for communication is
determined by the trigger for complementarity transition.
Definition 1 By an NrWD0L system (a network of Watson-Crick D0L systems)
with r components or nodes, where r ≥ 1, we mean an r +2-tuple
where
Γ =(Σ,φ,(g1, {A1}),...,(gr, {Ar})),
– Σ = {a1,...,an, ā1,...,ān}, n ≥ 1, is a DNA-like alphabet, the alphabet of
the system,
– φ : Σ ∗ →{0, 1} is a mapping defining a trigger for complementarity transition,
and
– (gi, {Ai}), 1 ≤ i ≤ r, called the ith component or the ith node of Γ, is a pair
where gi is a D0L morphism over Σ and Ai is a correct nonempty word over
Σ according to φ, called the axiom of the ith component.
If the number of the components in the network is irrelevant, then we speak
of an NWD0L system. An NWD0L system is called standard if φ is defined in
the same way as in the case of standard WD0L systems.
Definition 2 For an NrWD0L system Γ =(Σ,φ,(g1, {A1}),...,(gr, {Ar})),
r ≥ 1, the r-tuple (L1,...,Lr), where Li, 1 ≤ i ≤ r, is a finite set of correct
strings over Σ according to φ, is called a state of Γ. Li, 1 ≤ i ≤ r, is called the
state or contents of the ith component. ({A1},...,{Ar}) is said to be the initial
state of Γ.
NWD0L systems change their states by direct derivation steps. A direct
change of a state to another one means a rewriting step followed by communication
according to the given protocol of the system. In the following we define
two variants of communication protocols, representing different communication
philosophies.