LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

I0
Ai
i
Splicing Test Tube Systems 149
A(i,j,m)
Fig. 7. PSSRAM simulating HTTS.
(i, j, m)
4. I0 = {AiwBi | AwB ∈ I ′ i , 1 ≤ i ≤ n} ; Ik = ∅, 1 ≤ k ≤ l;
5. Ri = �� in, out; Ciu1#v1Di$Aiu2#v2; out � | (Cu1#u2D$Au2#v2) ∈ R ′ �
�� i ∪
out, in; u1#v1Bi$Ciu2#v2Di; out � | (u1#v1B$Cu2#v2D) ∈ R ′ �
i ,
1 ≤ i ≤ n;
moreover, for i �= j (which can be assumed without loss of generality) let
f (i, j, m) ={A}W ∗ ��
{B} be a filter between
� �
tubes i and j; thenwetake
��
R (i,j,m) = in, out; Aj#Z$Ai#; in , in, in;#Bi$Z#Bj; out .
The construction elaborated above proves that for every splicing test tube
system we can effectively construct an equivalent membrane system with splicing
rules assigned to membranes.
Theorem 4. For every PSSRAM we can construct an equivalent HTTCS.
Proof. Let
Π =(MW ∗ M,MT W ∗ T MT ,µ,A0, ..., An,I0, ..., In,R1, ..., Rn)
be a PSSRAM. Without loss of generality, we assume that every axiom needed
in the splicing rules of Ri, 1 ≤ i ≤ n, is available in the corresponding given sets
of axioms Aj, 0 ≤ j ≤ n. Then we can easily construct an equivalent HTTCS
where A =
� n�
Ai
i=0
σ =(MW ∗ M,MT W ∗ T MT ,A,I0, ..., In,C)
�
and the communication rules in C are defined in the
following way:
Let k be a membrane, 1 ≤ k ≤ n (remember that there are no rules assigned
to the skin membrane, which is labbelled by 0), and let the surrounding membrane
be labelled by l, and let (or1,or2; r; tar) be a rule assigned to membrane
k for some splicing rule r. Then (or1,or2; r; tar) canbesimulatedbyacommunication
rule (i, r, j) , where j = k for tar = in and j = l for tar = out as well
as
0