13th International Conference on Membrane Computing - MTA Sztaki

Multigraphical membrane systems revisited
• for i = 2 the diagram Dg(Γ ) is the following square:
Γ 2 Γ 3
Γ 0
Γ 1,
• for i = 3 the diagram Dg(Γ ) is the following cube:
Γ 4❈ Γ 5
❈❈❈❈❈❈
3 3333333
Γ 6 Γ 7
Γ 2
Γ 3❉ ❉❉❉❉❉❉❉
4 4444444
Γ 0
The above sketch-like multigraphical membrane systems drawn by using
Venn diagrams (with discs d Γ corresponding to vertices Γ of T i n such that d Γ j
is an immediate subset of d Γ ) coincide with the drawings shown in [25].
The following interpretation of Sn i by an i · n-dimensional hypercube [[S n]]
i
(n > 0 and i ∈ {1, 2, 3}) completes the proposed formal approach to the idea of
drawing hypercubes in [25].
We introduce the following notion to define hypercubes [[S n i ]]. For a natural
number n ≥ 0 and a finite directed graph G whose vertices are natural numbers
and the set E(G) of edges of G is such that E(G) ⊆ V (G) × V (G) we define a
new graph G ↑ n, called the translation of G to n, by
V (G ↑ n) = {i + n | i ∈ V (G)},
Γ 1.
E(G ↑ n) = {(i + n, j + n) | (i, j) ∈ E(G)}.
The hypercubes [[S n i ]] (n > 0, i ∈ {1, 2, 3}) are defined by induction on n in
the following way:
– for every i ∈ {1, 2, 3} the hypercube [[S 1 i]] is the diagram Dg(Λ) of Si 1 , where
Λ is the empty string and the digits in V (Dg(Λ)) are identified with corresponding
natural numbers,
– for all n > 0 and i ∈ {1, 2, 3} the hypercube [[S n+1 i ]] is such that
V ([[S n+1 i ]]) =
⋃
V ( [[S n i ]] ↑ (j · 2i·n ) ) ,
0≤j