LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

148 Franziska Freund, Rudolf Freund, and Marion Oswald
{C}W ∗ {X}
or
{X}W ∗ {D}
{ Ã}W ∗ {X}
or
{X}W ∗ { ˜ B}
{A}W ∗ {X}
or
{X}W ∗ {B}
✲ ✲
˜r
✲ ✲
r
✲ ✲
�ax
′
ax
i ((i, r, j),X,1) ((i, r, j),X,2) j
Fig. 6. Simulation of communication rule (i, r, j) in HTTS.
As we can derive from the results proved in [7], the results proved in this
section show that test tube systems communicating by splicing have universal
computational power; in contrast to splicing test tube systems, where according
to the results proved in [7] already systems with two test tubes are universal,
we cannot bound the number of membranes in the case of test tube systems
communicating by splicing.
4 Test Tube Systems Using Splicing Rules and Membrane
Systems with Splicing Rules Assigned to Membranes
Are Equivalent
In this section we show that when equipped with splicing rules assigned to membranes,
(sequential) P systems are equivalent to test tube systems using splicing
rules.
Theorem 3. For every HTTS we can construct an equivalent PSSRAM.
Proof. Let
σ =(M ′ W ∗ M ′ ,M ′ T W ∗ T M ′ T ,A′ 1 , ..., A′ n ,I′ 1 , ..., I′ n ,R′ 1 , ..., R′ n ,D)
be an HTTS.
Then we construct an equivalent PSSRAM (see Figure 7)
where
Π =(MW ∗ M,MT W ∗ T MT ,µ,A0, ..., An+l,I0, ..., In+l,R1, ..., Rn+l)
1. M = {Xi | X ∈ M ′ , 1 ≤ i ≤ n}∪{Z} ; MT = {Xi | X ∈ M ′ T , 1 ≤ i ≤ n} ;
2. for every test tube i we use the membrane i in Π; moreover, for every filter
f (i, j, m) ∈ D, 1 ≤ m ≤ ni,j, ni,j ≥ 0, of the form {A} W ∗ {B} between the
tubes i and j we take a membrane (i, j, m)inΠ; these additional membranes
altogether are l membranes within the skin membrane;
3. A0 = ∅, Ai = {CiwDi | CwD ∈ Ai} , 1 ≤ i ≤ n;
A (i,j,m) = {AjZ, ZBj | f (i, j, m) ={A} W ∗ {B} , (i, f (i, j, m) ,j) ∈ D} ;