LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

The Power of Networks of Watson-Crick D0L Systems 107
A Watson-Crick D0L system (a WD0L system, for short) is a D0L system over
a so-called DNA-like alphabet Σ and a mapping φ called the mapping defining
the trigger for complementarity transition. In a DNA-like alphabet each letter
has a complementary letter and this relation is symmetric. φ is a mapping from
the set of strings (words) over the DNA-like alphabet Σ to {0, 1} with the following
property: the φ-value of the axiom is 0 and whenever the φ-value of a
string is 1, then the φ-value of its complementary string must be 0. (The complementary
string of a string is obtained by replacing each letter in the string
with its complementary letter.) The derivation in a Watson-Crick D0L system
proceeds as follows: when the new string has been computed by applying the
homomorphism of the D0L system, then it is checked according to the trigger.
If the φ-value of the obtained string is 0 (the string is a correct word), then
the derivation continues in the usual manner. If the obtained string is an incorrect
one, that is, its φ-value is equal to 1, then the string is changed for its
complementary string and the derivation continues from this string.
The idea behind the concept is the following: in the course of the computation
or development things can go wrong to such extent that it is of worth to continue
with the complementary string, which is always available. This argument is
general and does not necessarily refer to biology. Watson-Crick complementarity
is viewed as an operation: together with or instead of a word w we consider its
complementary word.
A step further was made in [7]: networks of Watson-Crick D0L systems were
introduced. The notion was a particular variant of a general paradigm, called
networks of language processors, introduced in [6] and discussed in details in [5].
A network of Watson-Crick D0L systems is a finite collection of Watson-Crick
D0L systems over the same DNA-like alphabet and with the same trigger. These
WD0L systems act on their own strings in a synchronized 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. Two variants of communication protocols were discussed in [7]. In
the case of protocol (a), after performing a derivation step, the node keeps every
obtained correct word and the complementary word of each obtained incorrect
word (each corrected word) and sends a copy of each corrected word to every
other node. In the case of protocol (b), as in the previous case, the node keeps
all the correct words and the corrected ones (the complementary strings of the
incorrect strings) but communicates a copy of each correct string to each other
node. The two protocols realize different strategies: in the first case, if some error
is detected, it is corrected but a note is sent about this fact to the others. In the
second case, the nodes inform the other nodes about their correct strings and
keep for themselves all information which refers to the correction of some error.
The purpose of this paper is to establish three results about the power of such
networks, where the trigger φ is defined in a very natural manner. The results are
rather surprising because the underlying D0L systems are very simple. Two of
them show how it is possible to solve in linear time two well-known NP-complete
problems, the Hamiltonian Path Problem and the Satisfiability Problem. Here