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13th International Conference on Membrane Computing - MTA Sztaki

13th International Conference on Membrane Computing - MTA Sztaki

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Multigraphical membrane systems revisited<br />

• for i = 2 the diagram Dg(Γ ) is the following square:<br />

Γ 2 Γ 3<br />

Γ 0<br />

Γ 1,<br />

• for i = 3 the diagram Dg(Γ ) is the following cube:<br />

Γ 4❈ Γ 5<br />

❈❈❈❈❈❈<br />

3 3333333<br />

Γ 6 Γ 7<br />

Γ 2<br />

Γ 3❉ ❉❉❉❉❉❉❉<br />

4 4444444<br />

Γ 0<br />

The above sketch-like multigraphical membrane systems drawn by using<br />

Venn diagrams (with discs d Γ corresp<strong>on</strong>ding to vertices Γ of T i n such that d Γ j<br />

is an immediate subset of d Γ ) coincide with the drawings shown in [25].<br />

The following interpretati<strong>on</strong> of Sn i by an i · n-dimensi<strong>on</strong>al hypercube [[S n]]<br />

i<br />

(n > 0 and i ∈ {1, 2, 3}) completes the proposed formal approach to the idea of<br />

drawing hypercubes in [25].<br />

We introduce the following noti<strong>on</strong> to define hypercubes [[S n i ]]. For a natural<br />

number n ≥ 0 and a finite directed graph G whose vertices are natural numbers<br />

and the set E(G) of edges of G is such that E(G) ⊆ V (G) × V (G) we define a<br />

new graph G ↑ n, called the translati<strong>on</strong> of G to n, by<br />

V (G ↑ n) = {i + n | i ∈ V (G)},<br />

Γ 1.<br />

E(G ↑ n) = {(i + n, j + n) | (i, j) ∈ E(G)}.<br />

The hypercubes [[S n i ]] (n > 0, i ∈ {1, 2, 3}) are defined by inducti<strong>on</strong> <strong>on</strong> n in<br />

the following way:<br />

– for every i ∈ {1, 2, 3} the hypercube [[S 1 i]] is the diagram Dg(Λ) of Si 1 , where<br />

Λ is the empty string and the digits in V (Dg(Λ)) are identified with corresp<strong>on</strong>ding<br />

natural numbers,<br />

– for all n > 0 and i ∈ {1, 2, 3} the hypercube [[S n+1 i ]] is such that<br />

V ([[S n+1 i ]]) =<br />

⋃<br />

V ( [[S n i ]] ↑ (j · 2i·n ) ) ,<br />

0≤j

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