LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

378 Sergey Verlan
both components differ only in one letter. Also the proof is different: we shall
use a simulation of a Turing machine instead of a Chomsky grammar.
We define a coding function φ as follows. For any configuration w1qw2 of a
Turing machine M we define φ(w1qw2) =Xw1Sqw2Y ,whereX, Y and S are
three symbols which are not in T .
We also define the notion of an input for a TTF system. An input word
for a system Γ is simply a word w over the non-terminal alphabet of Γ .The
computation of Γ on input w is obtained by adding w to the axioms of the first
tube and after that by making Γ to evolve as usual.
Lemma 1. For any Turing machine TM =(Q, T, a0,s0,F,δ) and for any input
word w there is a communicating distributed H system with alternating
filters having two components and two filters, Γ =(V,TΓ , (A1,R1,F (1)
1 ,F (2)
1 ),
(A2,R2,F (1)
2 ,F (2)
2 )) which given the input φ(w) simulates M on input w, i.e.,
such that:
1. for any word w ∈ L(M) that reaches a halting configuration w1qw2, Γ will
produce a unique result φ(w1qw2);
2. for any word w �∈ L(M), Γ will produce the empty language.
Let T = {a0,...,am−1}, Q = {q0,...,qn−1}, a ∈ T ∪{X}, b, d, e ∈ T ,
c ∈ T ∪{Y }, q ∈ Q, a0 –blanksymbol.
We construct Γ as follows:
V = T ∪ Q ∪{X, Y, S, S ′ ,RY ,R,L,R ′ ,L ′ ,R R ,Z R 1 ,ZX,R L 1 ,ZL 1 ,R′ 1 ,Z′ 1 }.
TΓ = T ∪{X, Y, S}∪{q | q ∈ F }.
Test tube I:
Rules of R1:
For any rule qiakRalqj ∈ δ we have the following group of 4 rules:
1.1.1.1. aSqiak Y
RY a0Y
a Sqiakb
, 1.1.1.2. ,
R L
1.1.1.3. RSqiak b
R R 1 alS ′ qj Z R , 1.1.1.4.
1
a L
R R 1 bS ′ qd ,
For any rule qiakLalqj ∈ δ we have the following group of 5 rules:
1.1.2.1.
1.1.2.1 ′ .
X Sqiak
, 1.1.2.2.
Xa0 ZX
X bSqiak
Xa0 ZX
1.1.2.4.
b dSqiak
, 1.1.2.3.
R L
RbSqiak c
R L 1 S ′ qjbal Z L ,
1
b L
R L 1 S ′ qdec ,
We have also the following group of 3 rules:
1.2.1. ab S′ qd
R ′ L ′ , 1.2.2. R′ S ′ qb
R ′ 1S Z ′ , 1.2.3.
1
b L′
R ′ 1 Sqd