LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

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ˆby
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On Some Classes of Splicing Languages 103
⊢ (a,1;1,b)
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Fig. 10. Circular splicing using the semi-simple rule (a,1;1,b)
Lemma 13. The following equalities of functions from V ◦
a
(a, 1; b, 1)mi =(mi × mi)(1,a;1,b),
(a, 1; 1,b)mi =(mi × mi)(1,a; b, 1).
× V ◦
b
to 2V ◦
Theorem 8. The classes SSH ◦ (1, 3) and SSH ◦ (2, 4) are incomparable.
✩
✪
hold:
Proof. Note first that simple circular splicing languages (in SH◦ (1, 3) =
SH◦ (2, 4)) are in the intersection SSH◦ (1, 3) ∩ SSH◦ (2, 4).
The semi-simple H system S1 =({a, b, c}, {ˆaac,ˆb}, {(c, 1; b, 1)}) generates
a language L1 = L(S1) ∈ SSH◦ (1, 3) \ SSH◦ (2, 4).
Note that the words of L1 have the following two properties:
(I) All words in L1, with the exception of the two axioms, have occurrences
of all three letters a, b, andc.
(II) All occurrences of a in the words of L1 are in subwords of the form aac.
Now, suppose L1 is (2, 4) generated. Then it would be respected by one or
several of the rules of the forms:
(i) (1,c;1,b), (ii) (1,b;1,c), (iii) (1,a;1,b), (iv) (1,b;1,a),
(v) (1,c;1,a), (vi) (1,a;1,c), (vii) (1,a;1,a), (viii) (1,b;1,b), (ix)
(1,c;1,c).
But we have:
(v) (ˆaac,ˆaac) ⊢ (1,c;1,a) ˆaacaca /∈ L1 (no b’s),
(vi) (ˆaac,ˆaac) ⊢ (1,a;1,c) ˆaacaca /∈ L1 (no b’s),
(vii) (ˆaac,ˆaac) ⊢ (1,a;1,a) ˆcaacaa /∈ L1 (no b’s),
(ix) (ˆaac,ˆaac) ⊢ (1,c;1,c) ˆcaacaa /∈ L1 (no b’s),
(viii) (ˆb,ˆb) ⊢ (1,b;1,b) ˆbb /∈ L1 (no a’s, no c’s),
(i) (ˆaac,ˆb) ⊢ (1,c;1,b) ˆcaab /∈ L1 (contradicts (II)),
(ii) (ˆb,ˆaac) ⊢ (1,b;1,c) ˆbcaa /∈ L1 (contradicts (II)),
(iii) (ˆaac,ˆb) ⊢ (1,a;1,b) ˆacab /∈ L1 (contradicts (II)),
(iv) (ˆb,ˆaac) ⊢ (1,b;1,a) ˆbaca /∈ L1 (contradicts (II)).
The semi-simple H system S2 =({a, b, c}, {ˆaac,ˆb}, {(1,c;1,b)}) generates
a language L2 = L(S2) ∈ SSH◦ (2, 4) \ SSH◦ (1, 3). The proof that L2 is not in
SSH◦ (1, 3) is similar to the above one. We note that the words of L2 have the
following properties: