LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Communicating Distributed H Systems with Alternating Filters 371
This is very similar to the previous system. The only difference is that instead
of one stationary filter Fi there is a tuple of filters F (1)
i ,...,F (m)
i and the filter
to be used depends on the number of communications steps being done.
The language generated by Γ is:
L(Γ )={w ∈ T ∗ | w ∈ L (k)
1 for some k ≥ 1}.
We denote by CDHn (or TTn) the family of communicating distributed H
systems having at most n tubes and by CDHFn,m (or TTFn,m) the family of
communicating distributed H systems with alternating filters having at most n
tubes and m filters.
3 TTF2,2 Are Computationally Universal
In this section we present a CDHF system with two components and two filters
which simulates a type-0 grammar. Moreover, both filters in the first component
are identical.
Theorem 1. For any type-0 grammar G =(N,T,P,S) there is a communicating
distributed H system with alternating filters having two components and two
filters Γ =(V,T,(A1,R1,F (1)
1 ,F (2)
1 ), (A2,R2,F (1)
2 ,F (2)
2 )) which simulates G and
L(Γ )=L(G). Additionally F (1) (2)
1 = F 1 .
We construct Γ as follows. Let
N ∪ T ∪{B} = {a1,a2,...,an}
(where we assume B = an).
In what follows we will assume the following:
1 ≤ i ≤ n, a ∈ N ∪ T ∪{B}, γ∈{α, β},
b ∈ N ∪ T ∪{B} ∪ {⋄},
V = N ∪ T ∪{B}∪{α, β} ∪ {⋄}
∪{X, Y, Xα,Xβ,X ′ α ,Yα,Y ′ α ,Yβ,Z,Z ′ }.
The terminal alphabet T is the same as for the grammar G.
Test tube I:
Rules of R1:
1.1 :
1.4 :
ε uY
Z vY
X a
Xαβ Z
; 1.2 : a aiY
Z βα i−1 Y ′ α
; 1.5 : X a
Xβββ Z
1.7 : X′ α α
Xα Z ; 1.8 : X′ α α
Xββ Z
; 1.3 :
a BY
Z ⋄Yβ
; 1.6 : b βYa
Z Yβ
; 1.9 : γ αYα
Z Y ′ α
;
;
;