LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Communicating Distributed H Systems with Alternating Filters 369
set of instructions of machine M. Each instruction is represented in the following
way: qiakDalqj. The meaning is the following: if the head of machine M being
in state qi is scanning a cell which contains ak then the contents of the scanned
cell is replaced by al, the head moves to the left (if D = L) ortotheright(if
D = R) and its state changes to qj.
By a configuration of a Turing machine we shall understand the string w1qw2,
where w1,w2 ∈ T ∗ (w2 �∈ ε) andq ∈ Q. A configuration represents the contents
of non-empty cells of the working tape of the machine (i.e., all other cells to the
left and to the right are blank), its state and the position of the head on the
tape.Themachineheadisassumedtoreadtheleftmostletterofw2. Initially
all cells on the tape are blank except finitely many cells.
2.2 The Splicing Operation
An (abstract) molecule is simply a word over some alphabet. A splicing rule (over
alphabet V ), is a quadruple (u1,u2,u ′ 1,u ′ 2)ofwordsu1,u2,u ′ 1,u ′ 2 ∈ V ∗ ,whichis
often written in a two dimensional way as follows: u1 u2
u ′ 1 u′ 2
. We shall denote the
empty word by ε.
A splicing rule r =(u1,u2,u ′ 1,u ′ 2) is said to be applicable to two molecules
m1,m2 if there are words w1,w2,w ′ 1,w ′ 2 ∈ V ∗ with m1 = w1u1u2w2 and
m2 = w ′ 1u′ 1u′ 2w′ 2 , and produces two new molecules m′ 1 = w1u1u ′ 2w′ 2 and
m ′ 2 = w ′ 1u ′ 1u2w2. In this case, we also write (m1,m2) ⊢r (m ′ 1,m ′ 2).
Apairh =(V,R), where V is an alphabet and R is a finite set of splicing
rules, is called a splicing scheme or an H scheme.
For an H scheme h =(V,R) and a language L ⊆ V ∗ we define:
σh(L) def
= {w, w ′ ∈ V ∗ | (w1,w2) ⊢r (w, w ′ )forsomew1,w2 ∈ L, r ∈ R},
σ 0 h(L) =L,
σ i+1
h (L) =σi h (L) ∪ σh(σ i h (L)), i ≥ 0,
σ ∗ h(L) =∪i≥0σ i h(L).
A Head splicing system [5,6], or H system, is a construct H = (h, A) =
((V,R),A) of an alphabet V ,asetA ⊆ V ∗ of initial molecules over V ,the
axioms, andasetR ⊆ V ∗ × V ∗ × V ∗ × V ∗ of splicing rules. H is called finite if
A and R are finite sets.
The language generated by H system H is:
L(H) def
= σ ∗ h (A).
Thus, the language generated by H system H is the set of all molecules that
can be generated in H starting with A as initial molecules by iteratively applying
splicing rules to copies of the molecules already generated.