LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

The Power of Networks of Watson-Crick D0L Systems 113
After 46 computation steps the word
w = f1t2f3f4f5f6t7t8f9t10t11t12f13f14t15t16t17f18f19f20
appears in all nodes. This word indicates the only truth-value assignment satisfying
the propositional formula β.
The assignment and the word w can be found by the following direct argument.
(The argument uses certain special properties of β. The above proof
or the construction in [3] are completely independent of such properties.) The
conjunction of the clauses involving ∼ x9 is logically equivalent to ∼ x9. This
implies that x9 must assume the value F . Using this fact and the conjunction of
the clauses involving ∼ x5, weinferthatx5 must assume the value F .(x9 and
x5 are detected simply by counting the number of occurrences of each variable
in β.) From the 9th clause we now infer that x17 must have the value T .After
this the value of each remaining variable can be uniquely determined from a
particular clause. The values of the variables can be obtained in the following
order (we indicate only the index of the variable):
9, 5, 17, 15, 7, 8, 10, 16, 18, 3, 20, 2, 13, 19, 6, 4, 11, 1, 14, 12.
4 Hamiltonian Path Problem
In this section we show how another well-known NP-complete problem, namely
the Hamiltonian Path Problem (HPP) can be solved in linear time by standard
NWD0L systems. In this problem one asks whether or not a given directed graph
γ =(V,E), where V is the set of vertices or nodes of γ, and E denotes the set of
its edges, contains a Hamiltonian path, that is, a path which starting from a node
Vin and ending at a node Vout visits each node of the graph exactly once. Nodes
Vin and Vout can be chosen arbitrarily. This problem has a distinguished role in
DNA computing, since the famous experiment of Adleman in 1994 demonstrated
the solution of an instance of a Hamiltonian path problem in linear time.
Theorem 2 The Hamiltonian Path Problem can be solved in linear time by
standard NWD0L systems.
Proof. Let γ =(V,E) be a directed graph with n nodes V1,...,Vn, where
n ≥ 1, and let us suppose that Vin = V1 and Vout = Vn. We construct a standard
NWD0L2n+1 system Γ such that any word in the language of Γ identifies a
Hamiltonian path in γ in a unique manner. If the language is the empty set,
then there is no Hamiltonian path in γ. Moreover, the computation process of
any word in the language L(Γ )ofΓ ends in 2n +1steps.
Let us denote the nodes of Γ by M1,...,M2n+1, and let the alphabet Σ
of Γ consist of letters ai, $i, for 1 ≤ i ≤ n, Zi, with 1 ≤ i ≤ n, Xk, with
n +1≤ k ≤ 2n, Z, F, $, as well as of their barred versions.
We follow the communication protocol (b), that is, the copies of the correct
words are communicated.